# **The Essence of Mathematics**

**Through Elementary Problems**

**ALEXANDRE BOROVIK AND TONY GARDINER**

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# THE ESSENCE OF MATHEMATICS THROUGH ELEMENTARY PROBLEMS

# The Essence of Mathematics Through Elementary Problems

Alexandre Borovik and Tony Gardiner

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c 2019 Alexandre Borovik and Tony Gardiner

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Alexandre Borovik and Tony Gardiner, The Essence of Mathematics through Elementary Problems. Cambridge, UK: Open Book Publishers, 2019. http://dx.doi.org/10.11647/OBP.0168

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This is the third volume of the OBP Series in Mathematics:

ISSN 2397-1126 (Print) ISSN 2397-1134 (Online) ISBN Paperback 9781783746996 ISBN Hardback: 9781783747009 ISBN Digital (PDF): 9781783747016 DOI: 10.11647/OBP.0168

Cover photo: Abstract Spiral Pattern (2015) by Samuel Zeller, https://unsplash.com/photos/j0g8taxHZa0

Cover design by Anna Gatti.

# Contents








Simon Phillips Norton 1952–2019

In memoriam

# Preface

Understanding mathematics cannot be transmitted by painless entertainment . . . actual contact with the content of living mathematics is necessary. The present book . . . is not a concession to the dangerous tendency toward dodging all exertion. Richard Courant (1888–1972) and Herbert Robbins (1915–2001) Preface to the first edition of What is mathematics?

Interested students of mathematics, who seek insight into the "essence of the discipline", and who read more widely with a view to discovering what the subject is really about, may emerge with the justifiable impression of serious mathematics as an austere, but distant mountain range – accessible only to those who devote their lives to its exploration. And they may conclude that the beginner can only appreciate its rough outline through a haze of unbridgeable distance. The best popularisers sometimes manage to convey more than this – including hints of the human story behind recent developments, and the way different branches and results interact in unexpected ways; but the essence of mathematics still tends to remain elusive, and the picture they paint is inevitably a broad brush substitute for the detail of living mathematics.

This collection takes a different approach. We start out by observing that mathematics is not a fixed entity – as one might unconsciously infer from the metaphor of an "austere mountain range". Mathematics is a mental universe, a work-in-progress in our collective imagination, which grows dramatically over time, and whose eventual extent would seem to be unconstrained – without any obvious limits. This boundlessness also works in reverse, when applied to small details: features which we thought we had understood are repeatedly filled in, or reinterpreted, in new ways to reveal finer and finer micro-structures.

Hence whatever the essence of the discipline may be, it is clearly not something which can only be accessed through the complete exploration of some fixed corpus of knowledge. Rather the essential character of mathematics seems to be related to


There are a number of books giving excellent general advice to prospective students about how university mathematics differs from school mathematics. In contrast, this collection – which we hope will be enjoyed by interested high school students and their teachers, by undergraduates and postgraduates, and by many others is more like a messy workshop than a polished exposition. Here the reader is asked to tackle a sequence of problems, to reflect on what they discover, and mostly to draw their own conclusions (though some key messages are explicitly discussed in the text, or in the solutions at the end of each chapter). This attempt to engage the reader as an active participant along the way is inevitably untidy – and may sometimes prove frustrating. In particular, whereas a polished exposition would break up the text with eye-catching diagrams, an untidy workshop will usually leave the reader to draw their own figures as an essential part of the struggle. This temporary untidiness and frustration is an integral part of "the essence" that we seek to capture – provided it leads to occasional glimpses of the power, and the elegance of mathematics.

Young children and students of all ages regularly experience the power, the economy, the beauty, and the elegance of mathematics and of mathematical thinking on a small scale, through struggling with certain elementary results and problems (or groups of problems). For example, one of the problems we have included in Chapter 3 was mentioned explicitly in an interview<sup>1</sup> with the leading Russian mathematician Vladimir Arnold (1937–2010):

Interviewer: Please tell us a little bit about your early education. Were you already interested in mathematics as a child?

Arnold: [. . . ] The first real mathematical experience I had was when our schoolteacher I.V. Morotzkin gave us the following problem [VA then formulated Problem 89 in Chapter 3].

I spent a whole day thinking on this oldie, and the solution (based on what are now called scaling arguments, dimensional analysis, or toric variety theory, depending on your taste) came as a revelation.

<sup>1</sup> Notices of the AMS, vol 44, no. 4.

The feeling of discovery that I had then (1949 ) was exactly the same as in all the subsequent much more serious problems – be it the discovery of the relation between algebraic geometry of real plane curves and four-dimensional topology (1970 ), or between singularities of caustics and of wave fronts and simple Lie algebras and Coxeter groups (1972 ). It is the greed to experience such a wonderful feeling more and more times that was, and still is, my main motivation in mathematics.

This suggests that school mathematics need not be seen solely as an extended apprenticeship, which is somehow different from the craft of mathematics itself. Maybe some aspects of elementary mathematics can be experienced as if they were a part of mathematics proper, in which case suitably chosen elementary material, addressed in the appropriate spirit, might serve as a microcosm, or mini-universe, in which many features of the larger mathematical cosmos can be directly, and faithfully experienced by a relative novice (at least to some extent).

This collection of problems (and solutions) is an attempt to embody this idea in a form that might offer students, teachers, and interested readers a glimpse of "the essence of mathematics" – where this insight is experienced, not vicariously through the authors' elegant prose, or broad-brush descriptions, but through the reader's own engagement with carefully chosen, accessible problems from elementary mathematics.

Our understanding of the human body and how it works owes much to those (such as the ancient Greeks from 500 BC to Galen in the 2nd century AD, and much later Vesalius in the 16th century AD), who went beyond merely writing about such things in high-sounding prose, and who got their hands dirty by procuring cadavers, and cutting them up in order to see things from the inside – while asking themselves all the time how the different parts of the body were connected, and what function they served. In a similar way, the European discovery of the New World in the 15th century, and the confirmation that the Earth can be circumnavigated, depended on those who dared to set sail into uncharted waters and to keep a careful record of what they found.

The process of trying to understand things from the inside is not a deterministic procedure: it depends on a mixture of experience and inspiration, intelligence and inference, error and self-criticism. At any given time, the prevailing view may be incomplete, or misguided. But the underlying approach (of checking current ideas against the reality they purport to describe) is the only way we human beings know that allows us to gradually overcome errors and to gain fresh insight.

Our goal in this book is universal (namely to illustrate the idea that a suitably selected elementary microcosm can capture something of the essence of mathematics): hence the problems have all been chosen because we believe they convey something universal in a relatively elementary setting. But the particular set of problems chosen to illustrate the central goal is personal. So we encourage the reader to engage with these problems and results in the same way that old anatomists engaged with cadavers, or old explorers set out on voyages of discovery – getting their hands dirty while asking questions, such as:

How do the things we see relate to what we know? What does this tell us about the subject of mathematics that we want to understand better?

In recent years schools and teachers in many countries have been under increasing political pressure to concentrate on measurable, short term "improvements". Such pressures have often been linked to central testing, with negative consequences for low scores. This has encouraged teachers to play safe, and to focus on backward-looking methods that allow students to produce answers to predictable one-step problems. The effect has been to downgrade the more important challenges which every student should face: namely


Focusing on short-term goals is incompatible with good mathematics teaching. Learning mathematics is a long game; and teachers and students need the freedom to digress, to look ahead, and to build slowly over time. Teachers at each stage must be free to recognise that their primary responsibility is not just to improve their students' performance on the next test, but to establish a firm platform on which subsequent stages can build.

The pressures referred to above will be recognised in many countries, where well-intentioned, but ill-considered, centrally imposed accountability mechanisms have given rise to short-sighted "reforms". A didactical and pedagogical framework that is consistent with the essence, and the educational value of elementary mathematics cannot be rooted in false alternatives to mathematics (such as numeracy, or mathematical literacy). Nor can it be based on tests measuring cheap success on questions that require only one-step routines. We need a framework that encourages a rich combination of childlike curiosity, persistence, fruitful frustration, and the solid satisfaction of structural sense-making.

A problem sequence such as ours should ideally be distilled and refined over decades. However, the best is sometimes the enemy of the good:

Striving to better, Oft we mar what's well. (William Shakespeare, King Lear )

Hence, as a mild contribution to this process of rediscovering the essence of elementary mathematics, we risk this collection in its present form. And we encourage interested readers to take up pencil and paper, and to join us on this voyage of discovery through elementary mathematics.

Those who enjoy watching professional football (i.e. soccer) must sometimes marvel at the way experienced players seem to be instinctively aware of the movements of other players, and manage to feed the ball into gaps and spaces that we mere spectators never even noticed were there. What we overlook is that the best players practise the art of constantly looking around them, and updating their mental record – "viewing the field of play, with their heads up" – so that when the ball arrives and their eyes have to focus on the ball, their ever-changing mental record keeps updating itself to tell them (sometimes apparently miraculously) where the best tactical options lie. Implementing those tactical options depends in part on endless practice of skills; but practice is only one part of the story. What we encourage readers to develop here is the mathematical equivalent of this habit of "viewing the field of play, with one's head up", so that what is noticed can continue to guide the choice of tactical options when one is subsequently immersed in the thick of calculation.

Ours is a unique discipline, which is so much richer than the predictable routines that dominate many contemporary classrooms and assessments. We hope that all readers will find that the experience of struggling with, and savouring, this little collection reveals the occasional fresh and memorable insight into "the essence of mathematics".

> We should not worry if students don't know everything, but only if they know everything badly. Peter Kapitsa, (1894–1984) Nobel Prize for Physics 1978

To ask larger questions is to risk getting things wrong. George Steiner (1929– )

### Acknowledgements

Our thanks for suggestions, corrections comments and other contributions go to: Jean Bacon, Ay¸se Berkman, Anna Borovik, Raul Cordovil, Serkan Dogan, Gwyneth Gardiner, Dick Hudson, Martin Hyland, Hovhannes Khuderverdyan, Ali Nesin, Martin Richards, Simon Singh, Gunnar Traustason, Ozge Uklem, Yusuf Ulum, and numerous students from the 2014 UKMT Summer School in Apperley Bridge.

# About this text

And as this is done, so all similar problems are done. Paolo dell'Abbaco (1282–1374) Trattato d'aritmetica

It is better to solve one problem in five different ways than to solve five problems in one way. George P´olya (1887–1985)

> If you go on hammering away at a problem, it seems to get tired, lies down, and lets you catch it. Sir William Lawrence Bragg (1890–1971) Nobel Prize for Physics 1915

Young man, in mathematics you don't understand things. You just get used to them. John von Neumann (1903–1957)

This is not a random collection of nice problems. Each item or problem, and each group of problems, is included for two reasons:

• they constitute good mathematics – mathematics which repays the effort of engaging with it for the first time, or revisiting it (should it already be familiar);

and

• they embody in a distilled form the quintessential spirit of elementary (initially pre-university) mathematics in a style which can be actively enjoyed by committed students and teachers in schools and colleges, and by the interested general reader.

Some items exemplify core general methods, which can be used over and over again (as hinted by the dell'Abbaco quotation). Some items require us to take different views of ostensibly the same material (as illustrated by the contrasting P´olya quote). Many items will at first seem elusive; but persistence may sometimes lead to an unexpected reward (in the spirit of the Bragg quote). In other instances, a correct answer may be obtained – yet leave the solver less than fully satisfied (at least in the short term, as illustrated by the von Neumann quote). And some items are of little importance in themselves – except that they force the solver to engage in a kind of thinking which is mathematically important.

Almost all of the included items are likely to involve – in some degree – that frustration which characterises all fruitful problem solving (as represented by the Bragg quote, and the William Golding quotation below), where, if we are lucky, a bewildering initial fog of incomprehension is sometimes magically dissipated by the process of struggling intelligently to make sense of things. And since one cannot always expect to succeed, there are bound to be occasions when the fog fails to lift. One may then have no choice but to consult the solutions (either because some essential idea or technique is not yet part of one's stock-in-trade, or because one has overlooked some simple connection). The only advice we can give here is: the longer you can delay looking at the solutions the better. But these solutions have been included both to help you improve your own efforts, and to show the way when you get truly stuck.

The "essence of mathematics", which we have tried to capture in these problems is mostly implicit, and so is often left for the reader to extract. Occasionally it has seemed appropriate to underline some aspect of a particular problem or its solution. Some comments of this kind have been included in the text that is interspersed between the problems. But in many instances, the comment or observation that needs to be made can only be appreciated after readers have struggled to solve a problem for themselves. In such cases, positioning the observation in the main text might risk spilling the beans prematurely. Hence, many important observations are buried away in the solutions, or in the Notes which follow many of the solutions. More often still, we have chosen to make no explicit remark, but have simply tried to shape and to group the problems in such a way that the intended message is conveyed silently by the problems themselves.

Roughly speaking, one can distinguish three types of problems: these may be labelled as Core, as Gems, or as focusing on more general Cognition.

1. Core problems or ideas encapsulate important mathematical concepts and mathematical knowledge in a relatively mundane way, yet in a manner that is in some way canonical. These have sometimes been included here to emphasise some important aspect, which contemporary treatments may have forgotten.


The items are grouped into chapters – each with a recognisable theme. Later chapters tend to have a higher level of technical demand than earlier chapters; and the sequence is broadly consistent with a rising level of sophistication. However, this is not a didactically organised text. Each problem is listed where it fits most naturally, even if it involves an idea which is not formally introduced until somewhat later. Detailed solutions, together with any comments which would be out of place in the main text, are grouped together at the end of each chapter.

The first few chapters tend to focus on more elementary material – partly to emphasise the hierarchical structure of mathematics, partly as a reminder that the essence of mathematics can be experienced at all levels, and partly to offer a gentle introduction to readers who may appreciate something slightly more structured before they tackle selected parts of later chapters. Hence these early chapters include more discursive commentary than later chapters. Readers who choose to skip these nursery slopes on a first reading may wish to return to them later, and to consider what this relatively elementary material tells us about the essence of mathematics.

The collection is offered as a supplement to the standard school curriculum. Some items could (and perhaps should) be incorporated into any official curriculum. But the collection as a whole is mainly designed for those who have good reason, and the time and inclination, to go beyond the usual institutional constraints, and to begin to explore the broader landscape of elementary mathematics in order to experience real, "free range" mathematics – as opposed to artificially reconstituted, or processed products.

It has come to me in a flash! One's intelligence may march about and about a problem, but the solution does not come gradually into view. One moment it is not. The next and it is there.

William Golding (1911–1993), Rites of Passage

Even a superficial glance at history shows . . . great innovators . . . did vast amounts of computation and gained much of their insight in this way. I deplore the fact that contemporary mathematical education tends to give students the idea that computation is demeaning drudgery to be avoided at all costs. Harold M. Edwards (1936– ) Fermat's Last Theorem

We start our journey in a way that should be accessible to everyone – with a quick romp through important ideas from secondary school mathematics. The content is at times very elementary; but the problems often hint at something more challenging. The items included in this first chapter also highlight selected facts, techniques and ideas. Some of this early material is included to introduce certain ideas and techniques that later chapters will assume to be "known". A few problems appeal to more advanced ideas (such as complex numbers), and are included here to indicate that "mental skills" are not restricted to elementary material.

Pencil and paper will be needed, but the items tend to focus on things which a student of mathematics should know by heart, or should learn to see at a glance, or should be able to calculate inside the head. In later problems (e.g. from Problem 18 onwards) the emphasis on mental skills should be interpreted as "ways of thinking", rather than being taken to mean that everything should be done in your head. This is especially true where extended calculations or proofs are required.

Some of the items in this chapter (such as Problems 1 and 2) should be thoroughly familiar, and are included to underline this fact, rather than because we anticipate that they will need much active attention. Most of the early items in this first chapter are either core or auxiliary. However, there are also some real gems, which may even warrant a place in the the standard core.

The chapter is largely devoted to underlining the need for mastery of a repertoire of instantly available techniques, that can be used mentally, quickly, and flexibly to analyse less familiar problems at sight. But it also seeks to emphasise connections. Hence readers should be prepared to challenge their previous experience, in case it may have led to methods and results being perceived too narrowly.

We repeat the comment made in the section About this book. The "essence of mathematics", which is referred to in the title, is largely implicit in the problems, and is there for the reader to extract. There is some discussion of this essence in the text interspersed between the problems. But, to avoid spilling the beans prematurely, and hence spoiling the problems, many important observations are buried away in the solutions, or in the Notes which follow many of the solutions.

### 1.1. Mental arithmetic and algebra

### 1.1.1 Times tables.

Problem 1 Using only mental arithmetic:


(i) 0.004 ˆ 0.02 (ii) 0.0008 ˆ 0.07 (iii) 0.007 ˆ 0.12 (iv) 1.08 ˜ 1.2 (v) p0.08q <sup>2</sup> 4

Multiplication tables are important for many reasons. They allow us to appreciate directly, at first hand, the efficiency of our miraculous place value system – in which representing any number, and implementing any operation, are reduced to a combined mastery of


Fluency in mental and written arithmetic then leaves the mind free to notice, and to appreciate, the deeper patterns and structures which may be lurking just beneath the surface.

### 1.1.2 Squares, cubes, and powers of 2.

Algebra begins in earnest when we start to calculate with expressions involving powers. As one sees in the language we use for squares and cubes (i.e. 2nd and 3rd powers), these powers were interpreted geometrically for hundreds and thousands of years – so that higher powers, beyond the third power, were seen as being somehow unreal (like the 4th dimension). Our uniform algebraic notation covering all powers emerged in the 17th century (with Descartes (1596–1650)). But before one begins to work with algebraic powers, one should first aim to achieve complete fluency in working with numerical powers.

### Problem 2

	- (i) the squares of positive integers: first up to 12<sup>2</sup> ; then to 31<sup>2</sup>
	- (ii) the cubes of positive integers up to 11<sup>3</sup>
	- (iii) the powers of 2 up to 2<sup>10</sup> .

Evaluating powers, and the associated index laws, constitute an example of a direct operation. For each direct operation, we need to think carefully about the corresponding inverse operation – here "extracting roots". In particular, we need to be clear about the distinction between the fact that the equation x <sup>2</sup> " 4 has two different solutions, while ? 4 has just one value (namely 2).

### Problem 3

	- Every positive number arises as an output ("is the square of something").
	- Since x <sup>2</sup> " p´xq 2 , each output (other than 0) arises from at least two different inputs.
	- If a <sup>2</sup> " b 2 , then 0 " a <sup>2</sup> ´ b <sup>2</sup> " pa ´ bqpa ` bq, so either a " b, or a " ´b. Hence no two positive inputs have the same square, so each output (other than 0) arises from exactly two inputs (one positive and one negative).
	- Hence each positive output y corresponds to just one positive input, called ? y.

Find:


Moreover, they have the same square, since

$$(\sqrt{ab})^2 = ab = (\sqrt{a})^2 \cdot (\sqrt{b})^2 = (\sqrt{a} \times \sqrt{b})^2.$$

6 ? a ˆ b " ? a ˆ ? b.

Use this fact to simplify the following:


(c) [This part requires some written calculation.] Exact expressions involving square roots occur in many parts of elementary mathematics. We focus here on just one example – namely the regular pentagon.

Suppose that a regular pentagon ABCDE has sides of length 1.


(viii) Find the exact values of sin 36˝ , sin 72˝

. 4

Every calculation with square roots depends on the fact that "? is a function". That is: given y ą 0,

? y denotes a single value – the positive number whose square is y.

The equation x <sup>2</sup> " <sup>y</sup> has two roots, namely <sup>x</sup> " ˘? <sup>y</sup>; however, ? y has just one value (which is positive).

The mathematics of the regular pentagon is important – and generally neglected. It is included here to underline the way exact expressions involving square roots arise naturally.

In Problem 3(c), parts (iii) and (vi) require one to identify similar triangles using angles. The fact that "corresponding sides are then proportional" leads to a quadratic equation – and hence to square roots.

Parts (vii) and (viii) illustrate the fact that basic tools, such as


should be part of one's stock-in-trade. Notice that the exact values for

cos 36˝ , cos 72˝ , sin 36˝ , and sin 72˝

also determine the exact values of

sin 54˝ " cos 36˝ , sin 18˝ " cos 72˝ , cos 54˝ " sin 36˝ , and cos 18˝ " sin 72˝ .

### 1.1.3 Primes

### Problem 4

	- (ii) Find another such prime. 4

There are 4 prime numbers less than 10; 25 prime numbers less than 100; and 168 prime numbers less than 1000.

Problem 4(c) is included to emphasise a frequently neglected message:

Words and images are part of the way we communicate. But most of us cannot calculate with words and images.

To make use of mathematics, we must routinely translate words into symbols. For example, unknown numbers need to be represented by symbols, and points in a geometric diagram need to be properly labelled, before we can begin to calculate, and to reason, effectively.

### 1.1.4 Common factors and common multiples

To add two fractions we need to find a common multiple, or the LCM, of the two given denominators. To cancel fractions, or to simplify ratios, we need to be able to spot common factors and to find HCFs. Two positive integers a, b which have no (positive) common factors other than 1 (that is, with HCFpa, bq " 1) are said to be relatively prime, or coprime.

Problem 5 [This problem requires a mixture of serious thought and written proof.]

	- (i) Prove that some pair of integers among my chosen six must be relatively prime.
	- (ii) Is it also true that some pair must have a common factor?
	- (i) Prove that some pair among my chosen integers must be relatively prime.
	- (ii) Is it also true that some pair must have a common factor?
	- (i) Prove that some pair among the chosen integers must be relatively prime.
	- (ii) Is it also true that some pair must have a common factor? 4

### 1.1.5 The Euclidean algorithm

School mathematics gives the impression that to find the HCF of two integers m and n, one must first obtain the prime power factorisations of m and of n, and can then extract the HCF from these two expressions. This is fine for beginners. But arithmetic involves unexpected subtleties. It turns out that, as the numbers get larger, factorising integers quickly becomes extremely difficult – a difficulty that is exploited in modern encryption systems. (The limitations of any method that depends on first finding the prime power factorisation of an integer should have become clear in Problem 4(b), where it is all too easy to imagine that 91 is prime, and in Problem 4(c)(ii), where students regularly think that 143, or that 323 are prime.)

Hence we would like to have a simple way of finding the HCF of two integers without having to factorise each of them first. That is what the Euclidean algorithm provides. We will look at this in more detail later. Meanwhile here is a first taste.

### Problem 6

	- (ii) Prove that

$$HCF(m,n) = HCF(m-n,n).$$

	- (ii) Find HCFpm, 2m ` 1q.
	- (iii) Find HCFpm<sup>2</sup> ` 1, m ´ 1q. 4

### 1.1.6 Fractions and ratio

Problem 7 Which is bigger: 17% of nineteen million, or 19% of seventeen million? 4

### Problem 8

(a) Evaluate

$$
\left(1 + \frac{1}{2}\right)\left(1 + \frac{1}{3}\right)\left(1 + \frac{1}{4}\right)\left(1 + \frac{1}{5}\right)\dots
$$

(b) Evaluate

$$
\sqrt{1+\frac{1}{2}} \times \sqrt{1+\frac{1}{3}} \times \sqrt{1+\frac{1}{4}} \times \sqrt{1+\frac{1}{5}} \times \sqrt{1+\frac{1}{6}} \times \sqrt{1+\frac{1}{7}}.
$$

(c) We write the product "4 ˆ 3 ˆ 2 ˆ 1" as "4!" (and we read this as "4 factorial"). Using only pencil and paper, how quickly can you work out the number of weeks in 10! seconds? 4

Problem 9 The "DIN A" series of paper sizes is determined by two conditions. The basic requirement is that all the DIN A rectangles are similar ; the second condition is that when we fold a given size exactly in half, we get the next smaller size. Hence


(a) Find the constant ratio

r " "(longer side length) : (shorter side length)"

for all DIN A paper sizes.

	- (ii) To "enlarge" A4 size to A5 size (e.g. on a photocopier), each length is "enlarged" by a factor of <sup>1</sup> r . What is the enlargement factor to get from A5 size back to A4 size? 4

### Problem 10


### Problem 11

	- (i) Determine which is bigger:

$$
\frac{1}{2} + \frac{1}{5} \quad \text{or} \quad \frac{1}{3} + \frac{1}{4}?
$$

	- (i) For positive real numbers x, compare

$$
\frac{1}{x+2} + \frac{1}{x+5} \quad \text{and} \quad \frac{1}{x+3} + \frac{1}{x+4}.
$$

(ii) What happens in part (i) if x is negative? 4

2q 3

### 1.1.7 Surds

### Problem 12

(a) Expand and simplify in your head: (i) p ? 2 ` 1q 2 (ii) p ? 2 ´ 1q 2 (iii) p1 ` ?

(b) Simplify:

$$\begin{array}{ll} \text{(i)} \ \sqrt{10+4\sqrt{6}} & \text{(ii)} \ \sqrt{5+2\sqrt{6}}\\ \text{(iii)} \ \sqrt{\frac{3+\sqrt{5}}{2}} & \text{(iv)} \ \sqrt{10-2\sqrt{5}} \end{array}$$

The expressions which occur in exercises to develop fluency in working with surds often appear arbitrary. But they may not be. The arithmetic of surds arises naturally: for example, some of the expressions in the previous problem have already featured in Problem 3(c). In particular, surds will feature whenever Pythagoras' Theorem is used to calculate lengths in geometry, or when a proportion arising from similar triangles requires us to solve a quadratic equation. So surd arithmetic is important. For example:


$$
\underline{AB} = \sqrt{10 - 2\sqrt{5}}.
$$

### 1.2. Direct and inverse procedures

We all learn to calculate – with numbers, with symbols, with functions, etc. But we may not notice that most calculating procedures come in pairs.


To master inverse procedures requires a surprising amount of time and effort. And because they are harder to master, they can easily get neglected. Even where they receive a lot of time, there are aspects of inverse procedures which tend to go unnoticed.

Problem 13 In how many different ways can the missing digits in this short multiplication be completed?

l 6 ˆ l l 2 8 4

One would like students not only to master the direct operation of multiplying digits effectively, but also to notice that the inverse procedure of

"identifying the multiples of a given integer that give rise to a specified output"

depends on

the HCF of the multiplier and the base (10) of the numeral system.


• Multiplying by 0 induces a ten-to-one mapping onto the multiples of 0 (namely 0); so an inverse problem such as "0 ˆ l ends in 0" has ten digit-solutions and an inverse problem such as "0 ˆ l ends in 3 (or any digit other than 0)" has no digit-solutions at all.

The next problem shows – in a very simple setting – how elusive inverse problems can be. Here, instead of being asked to perform a direct calculation, the rules and the answer are given, and we are simply asked to invent a calculation that gives the specified output.

### Problem 14

	- (i) using the four numbers 3, 3, 6, 6
	- (ii) using the four numbers 3, 3, 7, 7
	- (iii) using the four numbers 3, 3, 8, 8.
	- (i) Which of the numbers 0–10 cannot be made?
	- (ii) What if one is allowed to use squaring and square roots as well as the four basic operations? What is the first inaccessible integer? 4

Calculating by turning the handle deterministically (as with addition, or multiplication, or multiplying out brackets, or differentiating) is a valuable skill. But such direct procedures are usually only the beginning. Using mathematics and solving problems generally depend on the corresponding inverse procedures – where a certain amount of juggling and insight is needed in order to work backwards (as with subtraction, or division, or factorisation, or integration). For example, in applications of calculus, the main challenge is to solve differential equations (an inverse problem) rather than to differentiate known functions.

Problem 14 captures the spirit of this idea in the simplest possible context of arithmetic: the required answer is given, and we have to find how (or whether) that answer can be generated. We will meet more interesting examples of this kind throughout the rest of the collection.

### 1.2.1 Factorisation

### Problem 15

	- (ii) Without doing any more work, write out the expanded forms of pa´bq 2 and pa ´ bq 3 .
	- (ii) Use (c)(i) and (a)(i) to write down (with no extra work) the expanded form of

$$(a-b-c)(a+b+c)$$

and of

$$(a-b+c)(a+b-c).$$

(d) Factorise 3x <sup>2</sup> ` 2x ´ 1. 4

### 1.3. Structural arithmetic

Whenever the answer to a question turns out to be unexpectedly nice, one should ask oneself whether this is an accident, or whether there is some explanation which should perhaps have led one to expect such a result. For example:


11, 101, 1001, 10 001, 100 001, . . .?

It may at first be tempting to think so – until, that is, you remember what you found in Problem 6(a)(iii).

Problem 16 Write out the first 12 or so powers of 4:

4, 16, 64, 256, 1024, 4096, 16 384, 65 536, . . .

Now create two sequences:

the sequence of final digits: 4, 6, 4, 6, 4, 6, . . . the sequence of leading digits: 4, 1, 6, 2, 1, 4, 1, 6, . . .

Both sequences seem to consist of a single "block", which repeats over and over for ever.


Problem 17 The 4 by 4 "multiplication table" below is completely familiar.


What is the total of all the numbers in the 4 by 4 square? How should one write this answer in a way that makes the total obvious? 4

### 1.4. Pythagoras' Theorem

From here on the idea of "mental skills" tends to refer to ways of thinking rather than to doing everything in your head.

Pythagoras' Theorem is one of the first truly surprising results in school mathematics: it is hard to see why anyone would think of "adding the squares of the two shorter sides". Despite the apparent attribution to a named person (Pythagoras), the origin of the theorem, and its proof, are unclear. There certainly was someone called Pythagoras (around 500 BC). But the main ancient references to him were written many hundreds of years after he died, and are not very reliable. The truth is that we know very little about him, or his theorem. The proof in Problem 18 below appeared in Book I of Euclid's thirteen books of Elements (written around 300 BC – two hundred years after Pythagoras). Much that is said (wrongly) to stem from Pythagoras is attributed in some sources to the Pythagoreans – a loose term which refers to any philosopher in what is seen as a tradition going back to Pythagoras. (This is a bit like interpreting anything called Christian in the last 2000 years as stemming directly from Christ himself.)

Clay tablets from around 1700 BC suggest that some Babylonians must have known "Pythagoras' Theorem"; and it is hard to see how one could know the result without having some kind of justification. But we have no evidence of either a clear statement, or a proof, at that time. There are also Chinese texts that refer to Pythagoras' Theorem (or as they call it, "Gougu"), which are thought to have originated BC – though the earliest surviving edition is from the 13th century AD. There is even an interesting little book by Frank Swetz, with the tongue-in-cheek title Was Pythagoras Chinese?.

The history may be confused, but the result – and its Euclidean proof – embodies something of the surprise and elegance of the very best mathematics. The Euclidean proof is included here partly because it is one that can, and should, be remembered (or rather, reconstructed – once one realises that there is really only one possible way to split the "square on the hypotenuse" in the required way). But, as we shall see, the result also links to exact mental calculation with surds, to trigonometry, to the familiar mnemonic "CAST", to the idea of a "converse", to sums of two squares, and to Pythagorean triples.

### 1.4.1 Pythagoras' Theorem, trig for special angles, and CAST

Problem 18 (Pythagoras' Theorem) Let 4ABC be a right angled triangle, with a right angle at C. Draw the squares ACQP, CBSR, and BAUT on the three sides, external to 4ABC. Use the resulting diagram to prove in your head that the square BAUT on BA is equal to the sum of the other two squares by:


The proof in Problem 18 is the proof to be found in Euclid's Elements Book 1, Proposition 47. Unlike many proofs,


### Problem 19

	- (ii) Extend the definitions of cos θ and sin θ to apply to angles beyond the first quadrant, so that for any point P on the unit circle, where OP makes an angle θ measured anticlockwise from the positive x-axis, the coordinates of P are pcos θ,sin θq. Check that the resulting functions sin and cos satisfy:
		- ∗ sin and cos are both positive in the first quadrant,
		- ∗ sin is positive and cos is negative in the second quadrant,
		- ∗ sin and cos are both negative in the third quadrant, and
		- ∗ sin is negative and cos is positive in the fourth quadrant.
	- (iii) Use (a), (b) to calculate the exact values of cos 315˝ , sin 225˝ , tan 210˝ , cos 120˝ , sin 960˝ , tanp´135˝ q.

(i) when n " 3 (ii) when n " 4 (iii) when n " 6 (iv) when n " 8 (v) when n " 12.

(e) Given a circle of radius 1, work out the area of a regular n-gon circumscribed around the circle:

(i) when n " 3 (ii) when n " 4 (iii) when n " 6 (iv) when n " 8 (v) when n " 12. 4

Knowing the exact values of sin, cos and tan for the special angles 0˝ , 30˝ , 45˝ , 60˝ , 90˝ is like knowing one's tables. In particular, it allows one to evaluate trigonometric functions mentally for related angles in all four quadrants (using the CAST mnemonic – C being in the SE of the unit circle, A in the NE quadrant, S in the NW quadrant, and T in the SW quadrant – to remind us which functions are positive in each quadrant). These special angles arise over and over again in connection with equilateral triangles, squares, regular hexagons, regular octagons, regular dodecagons, etc., where one can use what one knows to calculate exactly in geometry.

### Problem 20


Pythagoras' Theorem holds the key to calculating exact distances in the plane. To calculate distances on the Earth's surface one needs a version of Pythagoras for "right angled triangles" on the sphere. We address this in Chapter 5.

### 1.4.2 Converses and Pythagoras' Theorem

Each mathematical statement of the form

"if . . . (Hypothesis H), then . . . (Consequence C)"

has a converse statement – namely

"if C, then H".

If the first statement is true, there is no a priori reason to expect its converse to be true. For example, part (c) of Problem 25 below proves that

"if an integer has the form 4k ` 3, then it cannot be written as the sum of two squares".

However, the converse of this statement

"if an integer cannot be written as a sum of two squares, then it has the form 4k ` 3"

is false – since 6 cannot be written as the sum of two squares.

Despite this counterexample, whenever we prove a standard result, it makes sense to ask whether the converse is also true. For example,

"if P QRS is a parallelogram, then opposite angles are equal: =P " =R, and =Q " =S" (see Problem 157(ii)).

However you may not have considered the truth (or otherwise) of the converse statement:

If ABCD is a quadrilateral in which opposite angles are equal (=A " =C and =B " =D), is it true that ABCD has to be a parallelogram?

The next problem invites you to prove the converse of Pythagoras' Theorem. You should not use the Cosine Rule, since this is a generalisation of both Pythagoras' Theorem and its converse.

Problem 21 Let ABC be a triangle. We use the standard labelling convention, whereby the side BC opposite A has length a, the side CA opposite B has length b, and the side AB opposite C has length c. Prove that, if c <sup>2</sup> " a <sup>2</sup> ` b 2 , then =BCA is a right angle. 4

### 1.4.3 Pythagorean triples

The simplest example of a right angled triangle with integer length sides is given by the familiar triple 3, 4, 5:

$$3^2 + 4^2 = 5^2.$$

Any such integer triple is called a Pythagorean triple.

The classification of all Pythagorean triples is a delightful piece of elementary number theory, which is included in this chapter both because the result deserves to be memorised, and because (like Pythagoras' Theorem itself) the proof only requires one to juggle a few simple ideas that should be part of one's armoury.

Pythagorean triples arise in many contexts (e.g. see the text after Problem 180). The classification given here shows that Pythagorean triples form a family depending on three parameters p, q, s (in which s is simply a "scaling" parameter, so the most important parameters are p, q). As a warm-up we consider two "one-parameter subfamilies" related to the triple 3, 4, 5.

Problem 22 Suppose a <sup>2</sup> ` b <sup>2</sup> " c <sup>2</sup> and that b, c are consecutive integers.


Problem 22 reveals the triple p3, 4, 5q as the first instance (m " 1) of a one-parameter infinite family of triples, which continues

p5, 12, 13q pm " 2q, p7, 24, 25q pm " 3q, p9, 40, 41q pm " 4q, . . . ,

whose general term is

$$(2m+1, \ 2m(m+1), \ 2m(m+1)+1).$$

The triple p3, 4, 5q is also the first member of a quite different "one-parameter infinite family" of triples, which continues

$$(6, 8, 10), \ (9, 12, 15), \ \dots \ \dots$$

Here the triples are scaled-up versions of the first triple p3, 4, 5q.

In general, common factors simply get in the way:

If a <sup>2</sup> ` b <sup>2</sup> " c <sup>2</sup> and HCFpa, bq " s, then s <sup>2</sup> divides a <sup>2</sup> ` b 2 , and a <sup>2</sup> ` b <sup>2</sup> " c 2 ; so s divides c. And if a <sup>2</sup>`b <sup>2</sup> " c <sup>2</sup> and HCFpb, cq " s, then s <sup>2</sup> divides c <sup>2</sup>´b <sup>2</sup> " a 2 , so s divides a.

Hence a typical Pythagorean triple has the form psa, sb, scq for some scale factor s, where pa, b, cq is a triple of integers, no two of which have a common factor: any such triple is said to be primitive (that is, basic – like prime numbers). Every Pythagorean triple is an integer multiple of some primitive Pythagorean triple. The next problem invites you to find a simple formula for all primitive Pythagorean triples.

Problem 23 Let pa, b, cq be a primitive Pythagorean triple.


$$
\left(\frac{b}{2}\right)^2 = \left(\frac{c-a}{2}\right)\left(\frac{c+a}{2}\right),
$$

where

$$HCF\left(\frac{c-a}{2}, \frac{c+a}{2}\right) = 1$$

and <sup>c</sup>´<sup>a</sup> 2 , c`a 2 have opposite parity.

(c) Conclude that

$$\frac{c+a}{2} = p^2 \quad \text{and} \quad \frac{c-a}{2} = q^2,$$

where HCFpp, qq " 1 and p and q have opposite parity, so that c " p <sup>2</sup>`q 2 , a " p <sup>2</sup> ´ q 2 , b " 2pq.

(d) Check that any pair p, q having opposite parity and with HCFpp, qq " 1 gives rise to a primitive Pythagorean triple

$$c = p^2 + q^2, \quad a = p^2 - q^2, \quad b = 2pq$$
 
$$\text{satisfying } a^2 + b^2 = c^2. \tag{7}$$

Problem 24 The three integers a " 3, b " 4, c " 5 in the Pythagorean triple p3, 4, 5q form an arithmetic progression: that is, c ´ b " b ´ a. Find all Pythagorean triples pa, b, cq which form an arithmetic progression – that is, for which c ´ b " b ´ a. 4

### 1.4.4 Sums of two squares

The classification of Pythagorean triples tells us precisely which squares can be written as the sum of two squares. We now turn to the wider question: "Which integers are equal to the sum of two squares?"

### Problem 25


(e) For which integers N ă 100 is it possible to construct a square of area N, with vertices having integer coordinates? 4

In Problem 25 parts (a) and (d) you had to decide which integers ă 100 can be written as a sum of two squares as an exercise in mental arithmetic. In part (b) the fact that this set of integers is closed under multiplication turned out to be an application of the arithmetic of norms for complex numbers. Part (e) then interpreted sums of two squares geometrically by using Pythagoras' Theorem on the square lattice. These exercises are worth engaging in for their own sake. But it may also be of interest to know that writing an integer as a sum of two squares is a serious mathematical question – and in more than one sense.

Gauss (1777–1855), in his book Disquisitiones arithmeticae (1801) gave a complete analysis of when an integer can be represented by a 'quadratic form', such as x <sup>2</sup> ` y 2 (as in Problem 25) or x <sup>2</sup> ´ 2y 2 (as in Problem 54(c) in Chapter 2).

A completely separate question (often attributed to Edward Waring (1736–1798)) concerns which integers can be expressed as a k th power, or as a sum of n such powers. If we restrict to the case k " 2 (i.e. squares), then:

• When n " 2, Euler (1707–1783) proved that the integers that can be written as a sum of two squares are precisely those of the form

$$m^2 \times p\_0 \times p\_1 \times p\_2 \times \dots \times p\_s,$$

where p<sup>0</sup> " 1 or 2, and p<sup>1</sup> ă p<sup>2</sup> ă ¨ ¨ ¨ ă p<sup>s</sup> are odd primes of the form 4l ` 1.


### 1.5. Visualisation

Problem 26 (Pages of a newspaper) I found a (double) sheet from an old newspaper, with pages 14 and 27 next to each other. How many pages were there in the original newspaper? 4

Problem 27 (Overlapping squares) A square ABCD of side 2 sits on top of a square P QRS of side 1, with vertex A at the centre O of the small square, side AB cutting the side P Q at the point X, and =AXQ " θ.


Problem 28 (A folded triangle) The equilateral triangle 4ABC has sides of length 1 cm. D and E are points on the sides AB and AC respectively, such that folding 4ADE along DE folds the point A onto A<sup>1</sup> which lies outside 4ABC.

What is the total perimeter of the region formed by the three single layered parts of the folded triangle (i.e. excluding the quadrilateral with a folded layer on top)? 4

Problem 29 (A ` B " C) The 3 by 1 rectangle ADEH consists of three adjacent unit squares: ABGH, BCF G, CDEF left to right, with A in the top left corner. Prove that

$$
\angle DAE + \angle DBE = \angle DCE. \qquad \qquad \qquad \triangle
$$

### Problem 30 (Dissections)


Problem 31 (Yin and Yang) The shaded region in Figure 1, shaped like a large comma, is bounded by three semicircles – two of radius 1 and one of radius 2.

Cut each region (the shaded region and the unshaded one) into two 'halves', so that all four parts are congruent (i.e. of identical size and shape, but with possibly different orientations). 4

Figure 1: Yin and Yang

In Problem 31 your first thought may have been that this is impossible. However, since the wording indicated that you are expected to succeed, it was clear that you must be missing something – so you tried again. The problem then tests both flexibility of thinking, and powers of visualisation.

### 1.6. Trigonometry and radians

### 1.6.1 Sine Rule

School textbooks tend to state the Sine Rule for a triangle ABC without worrying why it is true. So they often fail to give the result in its full form:

Theorem If R is the radius of the circumcircle of the triangle ABC, then

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R.$$

This full form explains that the three ratios

$$\frac{a}{\sin A}, \quad \frac{b}{\sin B}, \quad \frac{c}{\sin C}$$

are all equal because they are all equal to the diameter 2R of the circumcircle of 4ABC – an additional observation which may well suggest how to prove the result (see Problem 32).

Problem 32 Given any triangle ABC, construct the perpendicular bisectors of the two sides AB and BC. Let these two perpendicular bisectors meet at O.

	- (i) The centre O lies on one of the sides of triangle ABC.
	- (ii) The centre O lies inside triangle ABC.
	- (iii) The centre O lies outside triangle ABC.

Case (i) leads directly to the Sine Rule for a right angled triangle ABC (remembering that sin 90˝ " 1). We address case (ii), and leave case (iii) to the reader.

(ii) Extend the line BO to meet the circle again at the point A<sup>1</sup> . Explain why =BA1C " =BAC " =A, and why =A1CB is a right angle. Conclude that

$$
\sin A = \frac{BC}{A'B} = \frac{a}{2R},
$$

and hence that

$$\frac{a}{\sin A} = 2R \quad \left(= \frac{b}{\sin B} = \frac{c}{\sin C}\right). \tag{7}$$

Problem 33 Let ∆ " areap4ABCq.

(a) Prove that

$$
\Delta = \frac{1}{2} \cdot ab \cdot \sin C.
$$

(b) Prove that 4R∆ " abc. 4

### 1.6.2 Radians and spherical triangles

There is no God-given unit for measuring distance; different choices of unit give rise to answers that are related by scaling. However the situation is different for angles. In primary and secondary school we measure turn in degrees – where a half turn is 180˝ , a right angle is 90˝ , and a complete turn is 360˝ . This angle unit dates from the ancient Babylonians (" 2000 BC). We are not sure why they chose 360 units in a full turn, but it seems to be related to the approximate number of days in a year (the time required for the heavens to make a complete rotation in the night sky), and to the fact that they wrote their numbers in "base 60". However the choice is no more objectively mathematical than measuring distance in inches or in centimetres.

After growing up with the idea that angles are measured in degrees, we discover towards the end of secondary school that:

there is another unit of measure for angles – namely radians.

It may not at first be clear that this is an entirely natural, God-given unit. The size of, or amount of turn in, an angle at the point A can be captured in an absolute way by drawing a circle of radius r centred at the point A, and measuring the arc length which the angle cuts off on this circle. The angle size (in radians) of the angle at A is then defined to be the ratio

$$\frac{\text{arc length}}{\text{radius}}.$$

That is,

size of angle at the point A " arc length cut off on a circle of radius 1, centred at the apex A.

Hence a right angle is of size <sup>π</sup> 2 radians; a half turn is equal to π (radians); a full turn is equal to 2π (radians); each angle in an equilateral triangle is equal to <sup>π</sup> 3 (radians); the three angles of a triangle have sum π; and the angles of a polygon with n sides have sum pn ´ 2qπ (see Problem 230 in Chapter 6).

For a while after the introduction of radians we continue to emphasise the word radians each time we give the measure of an angle in order to stress that we are no longer using degrees. But this is not really a switch to a new unit: this new way of measuring angles is in some sense objective – so we soon drop all mention of the word "radians" and simply refer to the size of an angle (in radians) as if it were a pure number.

This switch affects the meaning of the familiar trigonometric functions. And though we continue to use the same names (sin, cos, tan, etc.), they become slightly different as functions, since the inputs are now always assumed to be in radians.

The real payoff for making this change stems from the way it recognizes the connection between angles and circles. This certainly makes calculating circular arc lengths and areas of sectors easy (an arc with angle θ on a circle of radius r now has length θr; and a circular sector with angle 2θ now has area θr<sup>2</sup> ). But the main benefit – which one hopes all students appreciate eventually – is that this change of perspective highlights the fundamental link between sin x, cos x, and e x :


$$e^{i\theta} = \cos\theta + i\sin\theta.$$

The next problem draws attention to a beautiful result which reveals, in a pre-calculus, pre-complex number setting, a beautiful consequence of thinking about angles in terms of radians. The goal is to discover a formula for the area of a spherical triangle in terms of its angles and π, which links the formula for the circumference of a circle with that for the surface area of a sphere.

Suppose we wish to do geometry on the sphere. There is no problem deciding how to make sense of points. But it is less clear what we mean by (straight) lines, or line segments.

Before making due allowance for the winds and the tides, an airline pilot and a ship's Captain both need to know how to find the shortest path joining two given points A, B on a sphere. If the two points both lie on the equator, it is plausible (and correct) that the shortest route is to travel from A to B along the equator. If we think of the equator as being in a horizontal plane through the centre O of the sphere, then we may notice that we can change the equator into a circle of longitude by rotating the sphere so that the "horizontal" plane (through O) becomes a "vertical" plane (through O). So we may view two points A and B which both lie on the same circle of longitude as lying on a "vertical equator" passing through A, B and the North and South poles: the shortest distance from A to B must therefore lie along that circle of longitude.

If we now rotate the sphere through some other angle, we get a "tilted equator" passing through the images of the (suitably tilted) points A and B: these "tilted equators" are called great circles. Each great circle is the intersection of the sphere with a plane through the centre O of the sphere. So

to find the shortest path from A to B:


Once we have points and line segments (i.e. arcs of great circles) on the sphere, we can think about triangles, and about the angles in such a triangle. In a triangle ABC on the sphere, the sides AB and AC are arcs of great circles meeting at A. By rotating the sphere we can imagine A as being at the North pole; so the two sides AB and AC behave just like arcs of two circles of longitude emanating from the North pole. In particular, we can measure the angle between them (this is exactly how we measure longitude): the two arcs AB, AC of circles of longitude set off from the North pole A in different horizontal directions before curving southwards, and the angle between them is the angle between these two initial horizontal directions (that is, the angle between the plane determined by O, A, B and the plane determined by O, A, C).

Problem 34 Imagine a triangle ABC on the unit sphere (with radius r " 1), with angle α between AB and AC, angle β between BC and BA, and angle γ between CA and CB. You are now in a position to derive the remarkable formula for the area of such a spherical triangle.

Figure 2: Angles on a sphere

	- (i) What fraction of the surface area of the whole sphere is contained in this lune of angle α? Write an expression for the actual area of this lune.
	- (ii) If the sides AB and AC are extended backwards through A, these backward extensions define another lune with the same angle α, and the same surface area. Write down the total area of these two lunes with angle α.
	- (ii) Do the same for the two sides CA, CB meeting at the vertex C, to find the total area of the two lunes meeting at C and C <sup>1</sup> with angle γ.
	- (ii) Which parts of the sphere have been covered more than once? How many times have you covered the area of the original triangle ABC? And how many times have you covered the area of its sister triangle A<sup>1</sup>B<sup>1</sup>C 1 ?
	- (iii) Hence find a formula for the area of the triangle ABC in terms of its angles – α at A, β at B, and γ at C. 4

#### 1.6.3 Polar form and sin(A+B)

The next problem is less elementary than most of Chapter 1, but is included here to draw attention to the ease with which the addition formulae in trigonometry can be reconstructed once one knows about the polar form representation of a complex number. Those who are as yet unfamiliar with this material may skip the problem – but should perhaps remember the underlying message (namely that, once one is familiar with this material, there is no need ever again to get confused about the trig addition formulae).

### Problem 35


sin X ` sin Y, sin X ´ sin Y, cos X ` cos Y, cos X ´ cos Y.

	- (ii) Check your answer to (b) for cos X ´ cos Y by substituting X " 60˝ , and Y " 0 ˝ .

sin A sin B ` sin C sin D " sinpB ` CqsinpB ` Dq.

(ii) Given a cyclic quadrilateral W XY Z, with =XW Y " A, =W XZ " B, =Y XZ " C, =W Y X " D, deduce Ptolemy's Theorem:

$$WX \times YZ + WZ \times XY = WY \times XZ. \tag{\Delta}$$

### 1.7. Regular polygons and regular polyhedra

Regular polygons have already featured rather often (e.g. in Problems 3, 12, 19, 27, 28, 29). This is a general feature of elementary mathematics; so the neglect of the geometry of regular polygons, and their 3D companions, the regular polyhedra, is all the more unfortunate. We end this first chapter with a first brief look at polygons and polyhedra.

### 1.7.1 Regular polygons are cyclic

Problem 36 A polygon ABCDE ¨ ¨ ¨ consists of n vertices A, B, C, D, E, . . . , and n sides AB, BC, CD, DE . . . which are disjoint except that successive pairs meet at their common endpoint (as when AB, BC meet at B). A polygon is regular if any two sides are congruent (or equal), and any two angles are congruent (or equal). Can a regular n-gon ABCDE ¨ ¨ ¨ always be inscribed in a circle? In other words, does a regular polygon automatically have a "centre", which is equidistant from all n vertices? 4

### 1.7.2 Regular polyhedra

### Problem 37 (Wrapping)


Problem 38 (Cross-sections) Can a cross-section of a cube be:


Problem 39 (Shadows) Can one use the Sun's rays to produce a plane shadow of a cube:



The imparting of factual knowledge is for us a secondary consideration. Above all we aim to promote in the reader a correct attitude, a certain discipline of thought, which would appear to be of even more essential importance in mathematics than in other scientific disciplines. . . .

General rules which could prescribe in detail the most useful discipline of thought are not known to us. Even if such rules could be formulated, they would not be very useful. Rather than knowing the correct rules of thought theoretically, one must have assimilated them into one's flesh and blood ready for instant and instinctive use. Therefore for the schooling of one's powers of thought only the practice of thinking is really useful.

G. P´olya (1887–1985) and G. Szeg˝o (1895–1985)

### 1.8. Chapter 1: Comments and solutions

#### 1.

(a) Assuming that the 2ˆ, 3ˆ, 4ˆ, and 5ˆ tables are known, and that one has understood that the order of the factors does not matter, all that remains to be learned is 6 ˆ 6, 6 ˆ 7, 6 ˆ 8, 6 ˆ 9; 7 ˆ 7, 7 ˆ 8, 7 ˆ 9; 8 ˆ 8, 8 ˆ 9; and 9 ˆ 9.

$$\text{(b)} \text{ (i)} \text{ (4} \times 10^{-3}) \times \text{(2} \times 10^{-2}) = 8 \times 10^{-5} = 0.00008$$


#### 2.

	- (ii) 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331
	- (iii) 1 " 2 0 , 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
	- (ii) 99 (100<sup>2</sup> " 10<sup>4</sup> " 10 000)
	- (iii) 316 (310<sup>2</sup> " 96 100 ă 100 000 ă 320<sup>2</sup> ; so look more carefully between 310 and 320)
	- (ii) 21 (20<sup>3</sup> " 8000, 22<sup>3</sup> " 10 648)
	- (iii) 99 (100<sup>3</sup> " 10<sup>6</sup> " 1 000 000)
	- (ii) Those powers 2<sup>e</sup> of the form 2<sup>3</sup><sup>n</sup> for which the exponent e is a multiple of 3: i.e. e " 0 pmod 3q.

3.

	- (ii) AX is parallel to ED; similarly DX is parallel to EA. Hence AXDE is a parallelogram, with EA " ED.
	- (iii) The two triangles are both isosceles and =AXD " =CXB " 108˝ (vertically opposite angles).

.

Hence =XAD " =XCB " 36˝ , and =XDA " =XBC " 36˝


$$\text{(iii)}\text{ Use }\sin^2 36^\circ + \cos^2 36^\circ = 1; \text{ }\sin 36^\circ = \frac{\sqrt{10 - 2\sqrt{5}}}{4}; \sin 72^\circ = \frac{\sqrt{10 + 2\sqrt{5}}}{4}.$$

Note: The Golden Ratio crops up in many unexpected places (including the regular pentagon, and the Fibonacci numbers). Unfortunately much that is written about its ubiquity is pure invention. One of the better popular treatments, that highlights the number's significance, while taking a sober view of spurious claims, is the book The Golden Ratio by Mario Livio.

4.

(a) 12 345 " 5 ˆ 2469 " 3 ˆ 5 ˆ 823. But is 823 a prime number?

It is easy to check that 823 is not divisible by 2, or 3, or 5, or 7, or 11. The Square Root Test (displayed below) tells us that it is only necessary to check four more potential prime factors.

Square Root Test: If N " a ˆ b with a ď b, then a ˆ a ď a ˆ b " N, so the smaller factor a ď ? N.

? Hence, if N (" 823 say) is not prime, its smallest factor ą 1 is at most equal to N (" ? 823 ă 29). Checking a " 13, 17, 19, 23 shows that the required prime factorisation is 12 345 " 3 ˆ 5 ˆ 823.

(b) There are 25 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97).

#### Notes:

(i) For small primes, mental arithmetic should suffice. But one should also be aware of the general Sieve of Eratosthenes (a Greek polymath from the 3rd century BC). Start with the integers 1–100 arranged in ten columns, and proceed as follows:


Delete 1 (which is not a prime: see (ii) below).

Circle the first undeleted integer; remove all other multiples of 2.

Circle the first undeleted integer; remove all other multiples of 3.

Circle the first undeleted integer; remove all other multiples of 5.

Circle the first undeleted integer; remove all other multiples of 7.

All remaining undeleted integers ă 100 must be prime (by the Square Root Test (see part (a)).

(ii) The multiplicative structure of integers is surprisingly subtle. The first thing to notice is that 1 has a special role, in that it is the multiplicative identity: for each integer n, we have 1 ˆ n " n. Hence 1 is "multiplicatively neutral" – it has no effect.

The "multiplicative building blocks" for integers are the prime numbers: every integer ą 1 can be broken down, or factorised as a product of prime numbers, in exactly one way. The integer 1 has no proper factors, and has no role to play in breaking down larger integers by factorisation. So 1 is not a prime.

(If we made the mistake of counting 1 as a prime number, then we would have to make all sorts of silly exceptions – for example, to allow for the fact that 2 " 2 ˆ 1 " 2 ˆ 1 ˆ 1 " . . . , so 2 could then be factorised in infinitely many ways.)

(iii) Notice that 91 = 7 ˆ 13 is not a prime; so there is exactly one prime in the 90s – namely 97.

How many primes are there in the next run of 10 (from 100–109)? How many primes are there from 190–199? How many from 200–210?

	- (ii) Many students struggle with this, and may suggest 143, or 323, or even 63. The problem conceals a (very thinly) disguised message:

#### One cannot calculate with words.

To make use of mathematics, we must routinely translate words into symbols. As soon as "one less than a square" is translated into symbols, bells should begin to ring. For you know that x <sup>2</sup> ´ 1 " px ´ 1qpx ` 1q, so x <sup>2</sup> ´ 1 can only be prime if the smaller factor (x ´ 1) is equal to 1.

#### 5.

	- (ii) If we try to avoid such a pair, then we can choose at most one even integer. So we are then forced to choose all five available odd integers, and our list will be: "unknown even, 11, 13, 15, 17, 19". If the even integer is chosen to be 14, or 16, then every pair in my list has LCM = 1. So the answer is "No".
	- (ii) If we try to avoid such a pair, then we can choose at most one even integer. So we are then forced to choose all five available odd integers, and our list will be: "unknown even, 91, 93, 95, 97, 99", and so must include the pair 93, 99 – with common factor 3. So the answer is "Yes".

Note: The question says that "I choose", and asks whether "you" can be sure. So you have to find either a general argument that works for any n, or a counterexample. And the theme of this chapter indicates that it should not require any extended calculation.

The relevant "general idea" is the Pigeon Hole Principle which we may meet in the second part of this collection. So this problem may be viewed as a gentle introduction.

(i) Group the 2n consecutive integers

a ` 1, a ` 2, . . . , a ` 2n

into n pairs of consecutive integers

ta ` 1, a ` 2u, ta ` 3, a ` 4u, . . . , ta ` p2n ´ 1q, a ` 2nu.


However, as soon as n is at least 6, we show that the argument in part (a)(ii) breaks down. As before, if we try to choose a subset in which no pair has a common factor, then we can choose at most one even integer. So we are forced to choose all the odd integers. But any run of at least six consecutive odd integers includes two multiples of 3. So for n ě 6, the answer is "Yes".

#### 6.

	- (ii) Any factor of m and n is also a factor of their difference m ´ n; so the set of common factors of m and n is a subset of the set of common factors of m ´ n and n.

And any factor of m ´ n and n is also a factor of their sum m; so the set of common factors of m ´ n and n is a subset of the set of common factors of m and n.

Hence the two sets of common factors are identical. In particular, the two "highest common factors" are equal.

(iii) Subtract 91 from 1001 ten times to see that

$$HCF(1001, 91) = HCF(1001 - 910, 91) = 91.1$$

(b)(i) Subtract m from m ` 1 once to see that

$$HCF(m+1,m) = HCF(1,m) = 1.$$

(ii) Subtract m from 2m ` 1 twice to see that

$$HCF(2m+1,m) = HCF(m+1,m) = HCF(1,m) = 1.$$

(iii) Subtract m ´ 1 from m<sup>2</sup> ` 1 "m ` 1 times" to see that

$$HCF(m^2+1, m-1) = HCF((m^2+1)-(m^2-1), m-1) = HCF(2, m-1).$$

Hence, if m is odd, the HCF " 2; if m is even, the HCF " 1.

7. They are equal. (The first is

$$
\frac{17}{100} \times 19\,000\,000,
$$

the second is

$$\frac{19}{100} \times 17\,000\,000,1$$

which are equal since multiplication is commutative and associative.)

8.

(a)

(b)

3 2 ˆ 4 3 ˆ 5 4 ˆ 6 5 " 6 2 " 3

$$
\sqrt{\frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times \frac{6}{5} \times \frac{7}{6} \times \frac{8}{7}} = \sqrt{\frac{8}{2}} = 2
$$

(c)

10 ˆ 9 ˆ 8 ˆ 7 ˆ ˆ5 ˆ 4 ˆ 3 ˆ 2 ˆ 1 seconds

$$=\frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{60}\text{ minutes}$$

$$=\frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{60\times60}\text{ hours}$$

$$=\frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{60\times60\times24}\text{ days}$$

$$=\frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{60\times60\times24\times7}\text{ weeks}$$

$$=6\text{ weeks (after cancelling)}.$$

Note: These three questions underline what we mean by structural arithmetic. Fractions should never be handled by evaluating numerators and denominators. Instead one should always be on the lookout for structural features which simplify calculation – such as cancellation.

9.

(a) Suppose a rectangle in the "DIN A" series has dimensions a by b, with a ă b. Folding in half produces a rectangle of size <sup>b</sup> 2 by a. Hence b : a " a : b 2 , so b <sup>2</sup> " 2a 2 , and b : a " ? 2 : 1.

$$\text{(b) (i) } \frac{1}{r} \text{. (ii) } r .$$

10.

(a) "15% discount" means the price actually charged is "85% of the marked price". Hence each marked price needs to be multiplied by 0.85.

The distributive law says we may add the marked prices first and then multiply the total (exactly £80) by 0.85 to get

$$\pounds \left(\frac{85}{100} \times 80\right) = \pounds (17 \times 4) = \pounds 68.5$$

Note: The context (shopping, sales tax, and discount) is mathematically uninteresting. What matters here is the underlying multiplicative structure of the solution, which arises in many different contexts.

(b) "Add 20% VAT" means multiplying the discounted pre-VAT total (£68) by 1.2, or <sup>6</sup> 5 . Hence the final price, with VAT added, is £p1.2 ˆ 0.85 ˆ 80q. If the VAT were added first, the price before discount would be £p1.2ˆ80q, and the final price after allowing for the discount would be £p0.85 ˆ 1.2 ˆ 80q. Since multiplication is commutative, the two calculations have the same result, so the order does not matter (just as the final result in Problem 9 is the same whether one first enlarges A4 to A3 and then reduces A3 to A4, or first reduces A4 to A5 and then enlarges A5 to A4).

Note: Notice that we did not evaluate the two answers to see that they gave the same output £81.60. If we had, then the equality might have been a fluke due to the particular numbers chosen. Instead we left the answer unevaluated, in structured form, which showed that the equality would hold for any input.

(c) To cancel out multiplying by <sup>6</sup> <sup>5</sup> we need to multiply by <sup>5</sup> 6 – a discount of <sup>1</sup> 6 , or 16 <sup>2</sup> <sup>3</sup>%.

Note: This question has nothing to do with financial applications. It is included to underline the fact that although percentage change questions use the language of addition and subtraction ("increase", or "decrease"), the mathematics suggests they should be handled multiplicatively.

7

11.

(a)(i) 2 ˆ 5 ă 3 ˆ 4, so

Hence

$$2 \times 5 \quad 3 \times 4$$

$$\frac{1}{2} + \frac{1}{5} > \frac{1}{3} + \frac{1}{4} .$$

ą 7

.

(ii) At first sight, "10 ă 12" may not seem related to " <sup>1</sup> <sup>2</sup> ` 1 <sup>5</sup> ą 1 <sup>3</sup> ` 1 4 ". Yet the crucial fact we started from in part (i) was "2 ˆ 5 " 10 ă 12 " 3 ˆ 4".

(b) 10 ă 12, so

$$(x+2)(x+5) = x^2 + 7x + 10 < x^2 + 7x + 12 = (x+3)(x+4).$$

(i) If all four brackets are positive (i.e. if x ą ´2), then we also have 2x ` 7 ą 0, and it follows that

$$\begin{array}{rcl} \frac{1}{x+2} + \frac{1}{x+5} &=& \frac{2x+7}{(x+2)(x+5)}\\ &>& \frac{2x+7}{(x+3)(x+4)}\\ &=& \frac{1}{x+3} + \frac{1}{x+4} \end{array}$$

(ii) When calculating with the given algebraic expression, the values

$$x = -2, -3, -4, -5$$

are "forbidden values".

If x ą ´2, then (as in part (i)) we have

$$\begin{array}{rcl} \frac{1}{x+2} + \frac{1}{x+5} &=& \frac{2x+7}{(x+2)(x+5)}\\ &>& \frac{2x+7}{(x+3)(x+4)}\\ &=& \frac{1}{x+3} + \frac{1}{x+4} \end{array}$$

For permitted values of x ă ´2, one or more of the brackets px ` 2q, px ` 5q, px ` 3q, px ` 4q will be negative. However, one can still carry out the algebra to simplify

$$\frac{1}{x+2} + \frac{1}{x+5} = \frac{2x+7}{(x+2)(x+5)} \quad \text{and} \quad \frac{1}{x+3} + \frac{1}{x+4} = \frac{2x+7}{(x+3)(x+4)}$$

When x " ´ 7 2 both expressions are equal, and equal to 0. The simplified numerators are both positive if x ą ´<sup>7</sup> 2 , and both negative if x ă ´<sup>7</sup> 2 ; and the sign of the denominators changes as one moves through the four intervals ´3 ă x ă ´2, ´4 ă x ă ´3, ´5 ă x ă ´4, x ă ´5, with the inequality switching

from " ą " (for x ą ´2) to " ă " (for ´3 ă x ă ´2), to " ą " (for ´3.5 ă x ă ´3), to " ă " (for ´4 ă x ă ´3.5), to " ą " (for ´5 ă x ă ´4), to " ă " (for x ă ´5).

$$^{12.}$$

(a) (i) 3 ` 2 ? 2; (ii) 3 ´ 2 ? 2; (iii) 7 ` 5 ? 2. Note: Notice that you can write down the answer to (ii) as soon as you have finished (i), without doing any further calculation.

(b) (i) 2 ` ? 6; (ii) ? 2 ` ? 3; (iii) <sup>1</sup>` ? 5 2 (the Golden Ratio <sup>1</sup>` ? 5 <sup>2</sup> " τ is the larger root of the quadratic equation x <sup>2</sup> ´ x ´ 1 " 0. Hence <sup>3</sup>` ? 5 <sup>2</sup> " τ ` 1 " τ 2 ); (iv) <sup>a</sup> 10 ´ 2 ? 5: this does not simplify further.

Note: Some readers may think an apology is in order for part (iv). The lesson here is that, while one should always try to simplify, there is no way of knowing in advance whether a simplification is possible. And there is no way out of this dilemma. So one is reduced to thinking: any simplification would involve ? 5, and if one tries to solve pa ` b ? 5q <sup>2</sup> " 10 ´ 2 ? 5, then the solutions for a and b do not lead to anything "simpler". (This repeated surd should perhaps have rung bells, as it was equal to the exact expression for 4 sin 36˝ in Problem 3(c). It was included here partly because the question of its simplification should already have arisen when it featured in that context.)

13. In reconstructing the missing digits the number of possible solutions is determined by the highest common factor of the multiplier and 10. At the first step (in the units column):

because HCFp6, 10q " 2, l ˆ 6 " 8 pmod 10q has two solutions which differ by 5 – namely 3 and 8.

The first possibility then requires us to solve pl ˆ 3q ` 1 " 2 pmod 10q: because HCFp3, 10q " 1, this has just one solution – namely 7. This gives rise to the solution 76 ˆ 3 " 228.

The second possibility requires us to solve pl ˆ 8q ` 4 " 2 pmod 10q: because HCFp8, 10q " 2, this has two solutions which differ by 5 – namely 1 and 6. This gives rise to two further solutions: 16 ˆ 8 " 128, and 66 ˆ 8 " 528.

### 14.

	- (ii) With squaring and ? allowed we can manage 10 " <sup>4</sup> ` <sup>4</sup> ` <sup>4</sup> ´ ? 4. Indeed, one can make everything up to 40 except (perhaps) 39.

15. (a)(i) a <sup>2</sup> ` 2ab ` b 2 ; a <sup>3</sup> ` 3a 2 b ` 3ab<sup>2</sup> ` b 3 (ii) Replace b by p´bq: a <sup>2</sup> ´ 2ab ` b 2 ; a <sup>3</sup> ´ 3a 2 b ` 3ab<sup>2</sup> ´ b 3 (b) (i) px ` 1q 2 ; (ii) px <sup>2</sup> ´ 1q 2 ; (iii) px <sup>2</sup> ´ 1q 3 (c) (i) a <sup>2</sup> ´ b 2 (ii) Replace "b" by "b ` c": a <sup>2</sup> ´ pb ` cq <sup>2</sup> " a <sup>2</sup> ´ b <sup>2</sup> ´ c <sup>2</sup> ´ 2bc Replace "b" by "b ´ c": a <sup>2</sup> ´ pb ´ cq <sup>2</sup> " a <sup>2</sup> ´ b <sup>2</sup> ´ c <sup>2</sup> ` 2bc (d) One way is to rewrite this expression as a difference of two squares:

$$\begin{aligned} \left(2x\right)^2 - \left(x^2 - 2x + 1\right) &=& \left(2x\right)^2 - \left(x - 1\right)^2 \\ &=& \left(2x - \left(x - 1\right)\right)\left(2x + \left(x - 1\right)\right) \\ &=& (x + 1)\left(3x - 1\right) \end{aligned}$$

Note: As so often, the messages here are largely implicit. In part (a)(ii) we explicitly highlight the intention to use what you already know (by simply substituting "´b" in place of "b". In part (b), you are expected to recognise (i), and then to see (ii) and (iii) as mild variations on the expansions of pa ´ bq 2 and pa ´ bq 3 in part (a). Part (c) repeats (in silence) the message of (a)(ii): think – don't slog it out. And part (d) encourages you to keep an eye out for thinly disguised instances of "a difference of two squares".

#### 16.

(a) Final digits: 'block' 4, 6 of length 2; leading digits: "block" 4, 1, 6, 2, 1 of length 5.

(b) Claim The sequence of "units digits" really does recur. Proof Given a power of 4 that has units digit 4, the usual multiplication algorithm for multiplying by 4 produces a number with units digit 6.

Given this new power of 4 with units digit 6, the usual multiplication algorithm for multiplying by 4 produces a number with units digit 4.

At this stage the sequence of units digits begins a new cycle.

[Alternatively: The units digit is simply equal to the relevant power of 4 pmod 10q. Multiplying by 4 changes 4 to 6 pmod 10q; multiplying by 4 changes 6 to 4 pmod 10q; – and the cycle repeats.]

(c) The sequence of leading digits seems to recur every 5 terms, because 4<sup>5</sup> " 2 <sup>10</sup> " 1024 is almost exactly equal to 1000. Each time we move on 5 steps in the sequence, we multiply by 4<sup>5</sup> " 1024. As far as the leading digit is concerned, this has the same effect as multiplying the initial term (4) by slightly more than 1.024 (then adding any 'carries'), which is very like multiplying by 1 – and so does not change the leading digit (yet).

However, each time we move on 10 steps in the sequence, we multiply by 4<sup>10</sup> " 1024<sup>2</sup> " 1 048 576. As far as the leading digit is concerned, this has the same effect as multiplying by slightly more than 1.048576.

When we move on 25 steps, we multiply by 4<sup>25</sup> " 1 125 899 906 842 624. And as far as the leading digit is concerned, this has the same effect as multiplying by slightly more than 1.12599906842624. And so on.

Eventually, the multiplier becomes large enough to change one of the leading digits.

#### 17. The total is 100.

Having found this by direct calculation, we should think indirectly and notice that 100 " 10<sup>2</sup> .

And we should then ask: "Why 10? What has 10 got to do with the 4ˆ multiplication table?"

A quick check of the 1ˆ multiplication table (total " 1), the 2ˆ multiplication table (total " 9), etc. may suggest what we should have seen immediately.


19.

$$\text{(a) } \sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} = \cos 45^\circ; \tan 45^\circ = 1$$

	- (i) <sup>3</sup> 4 ; (ii) 2; (iii) <sup>3</sup> 2 2; (v) 3

Note: There is no hidden trig here: all you need is Pythagoras' Theorem. For example, in part (e)(iv) we can extend alternate sides of the regular octagon to form the circumscribed 2 by 2 square. The four corner triangles are isosceles right angled triangles with hypotenuse of length s (the side of the octagon). Hence each side of the square is equal to s ` 2 ¨ ?<sup>s</sup> 2 " 2, whence s " 2p ? 2 ´ 1q.

$$\text{20. (a) } \sqrt{8}; \text{ (b) } \sqrt{2}, \sqrt{2}; \text{ (c) } \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$

21. Construct the perpendicular from A to BC (possibly extended); let this meet the line BC at X. There are four possibilities:


We analyse case (ii) and leave cases (iii) and (iv) to the reader.

(ii) 4AXC and 4AXB are both right angled triangles; so by Pythagoras' Theorem we know that

$$\begin{aligned} AC^2 &=& A X^2 + X C^2,\text{ and} \\ AB^2 &=& A X^2 + X B^2 \\ &=& A X^2 + (X C + C B)^2 \\ &=& A X^2 + X C^2 + C B^2 + 2 X C \cdot C B \\ &=& A C^2 + C B^2 + 2 X C \cdot C B. \end{aligned}$$

Since we are told that AC<sup>2</sup> ` CB<sup>2</sup> " AB<sup>2</sup> , it follows that 2XC ¨ CB " 0, contrary to X ‰ C.

Note: Notice that the proof of the converse of Pythagoras' Theorem makes use of Pythagoras' Theorem itself.

#### 22.


Hence b " 2n is even and c " 2n ` 1 is odd. But then

$$(2m+1)^2 + (2n)^2 = a^2 + b^2 = c^2 = (2n+1)^2,$$

so 4pm<sup>2</sup> ` mq " 4n, and n " mpm ` 1q.

23.

(a) If a and b are both even, then HCFpa, bq ‰ 1, so the triple would not be primitive.

If a and b are both odd, we use the idea from Problem 22(b). Suppose a " 2m ` 1, b " 2n ` 1; then a <sup>2</sup> " 4pm<sup>2</sup> ` mq ` 1, and b <sup>2</sup> " 4pn <sup>2</sup> ` nq ` 1, so a <sup>2</sup> ` b <sup>2</sup> " 2 ˆ p2pm<sup>2</sup> ` m ` n <sup>2</sup> ` nq ` 1q. But this is "twice an odd number", so cannot be equal to c 2 (since c would have to be even, and any even square must be a multiple of 4).

Hence we may assume that a is odd and b is even: so c is is odd.

(b) Then a <sup>2</sup> ` b <sup>2</sup> " c 2 yields b <sup>2</sup> " c <sup>2</sup> ´ a <sup>2</sup> " pc ´ aqpc ` aq, so

$$
\left(\frac{b}{2}\right)^2 = \left(\frac{c-a}{2}\right)\left(\frac{c+a}{2}\right)\dots
$$

Any common factor of <sup>c</sup>`<sup>a</sup> 2 and <sup>c</sup>´<sup>a</sup> 2 divides their sum c and their difference a, so HCFp c´a 2 , c`a 2 q " 1. Since the difference of these two factors is a, which is odd, they have opposite parity.

(c) If two integers are relatively prime, and their product is a square, then each of the factors has to be a square (consider their prime factorisations). Hence c`a <sup>2</sup> " p 2 and <sup>c</sup>´<sup>a</sup> <sup>2</sup> " q 2 , where HCFpp, qq " 1 and p and q have opposite parity. Therefore

$$c = p^2 + q^2, \quad a = p^2 - q^2, \quad b = 2pq.$$

(d) It is easy to check that any triple of the given form is (i) primitive, and (ii) satisfies a <sup>2</sup> ` b <sup>2</sup> " c 2 .

#### 24. Claim The only such triples are those of the form p3s, 4s, 5sq.

Proof We show that the only primitive Pythagorean triple which forms an arithmetic progression is the familiar triple p3, 4, 5).

By Problem 23, one of the numbers in any primitive Pythagorean triple is even (namely 2pq) and two are odd (p and q are of opposite parity, so p <sup>2</sup>´q 2 and p <sup>2</sup>`q 2 are both odd).


$$.\;^\cdot . \;2pq = p^2 \,, \text{so } p = 2q.\text{.}\;^\cdot$$

Finally, since p and q are relatively prime, we must have q " 1, p " 2. QED

Note: Alternatively, let pa, b, cq be any Pythagorean triple (not necessarily primitive), which forms an arithmetic progression. Then

$$a^2 = c^2 - b^2 = (c - b)(c + b) = (b - a)(c + b).$$

So bpc ` bq " apa ` b ` cq. Hence a ¨ 3b " apa ` b ` cq " bpc ` bq. It then follows that 3b <sup>2</sup> " bpa ` b ` cq " 4ba, so 3b " 4a and a : b " 3 : 4.

25.

(a)  $2 = 1^2 + 1^2$ ,  $5 = 2^2 + 1^2$ ,  $13 = 3^2 + 2^2$ ,  $17 = 4^2 + 1^2$ ,  $29 = 5^2 + 2^2$ ,  $37 = 6^2 + 1^2$ ,  $41 = 5^2 + 4^2$ ,  $53 = 7^2 + 2^2$ ,  $61 = 6^2 + 5^2$ ,  $73 = 8^2 + 3^2$ ,  $89 = 8^2 + 5^2$ ,  $97 = 9^2 + 4^2$ .
(b)  $(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2$ .

Note: It is easy to check this identity once it is given, but most of us are not so fluent in algebra as to spot this handy identity without help! However, Chapter 1 is about "Mental skills", and one such technique (once you have mastered it) arises from the arithmetic of complex numbers. If you have met complex numbers, then this identity can be written down immediately. Let us explain briefly how.

Every complex number w " a ` bi (where i <sup>2</sup> " ´1) can be represented as a point in the complex plane with coordinates pa, bq. The "size", or modulus, of w is its length |w| (the distance of pa, bq from the origin p0, 0q); and the square of this length a <sup>2</sup> ` b 2 is referred to as the norm of the complex number w " a ` bi. The required identity is an immediate consequence of the two facts:


Once we know these facts:


the product of the two norms pa <sup>2</sup> ` b 2 qpc <sup>2</sup> ` d 2 q is then equal to the norm of the product w ¨ z " pac ´ bdq ` pad ` bcqi.

Note: If we choose z " c ´ di, then wz " pac ` bdq ` pbc ´ adqi, and we get a second identity: pa <sup>2</sup> ` b 2 qpc <sup>2</sup> ` d 2 q " pac ` bdq <sup>2</sup> ` pbc ´ adq 2 .

	- (i) both squares are even and their sum is a multiple of 4, or
	- (ii) one square is even and one is odd and their sum is of the form 4k ` 1, or
	- (iii) both squares are odd and their sum is of the form 4k ` 2.

Hence no number of the form 4k ` 3 can be written as a sum of two squares.

(d) We are told that 2 " 1 <sup>2</sup> ` 1 2 , and that Euler showed every prime of the form 4k ` 1 can be written as the sum of two squares. Part (b) then shows that any product of such primes can be written as the sum of two squares. And if we multiply a sum of two squares by a square, the result can again be written as the sum of two squares. This allows us to construct the list of forty six integers N ă 100 which can be so written. These are precisely the integers of the form

"(a square) ˆ (a product of distinct primes p, where p " 2 or p " 4k`1)":

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 60, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 83, 85, 87, 89, 90, 97, 98.

(e) The side of such a square is the hypotenuse of a right angled triangle whose legs run in the x- and y- directions, and have integer lengths (because their vertices are at points with integer coordinates). Hence the answer is exactly the same as for part (d) (provided one does not quibble about the idea of a square with side 0 and area 0).

26. Most sheets in a newspaper are double sheets with four pages. If we assume that all sheets are double sheets, then the 13 pages before page 14 match up with the 13 pages after page 27, so there are 27 ` 13 " 40 pages in all. (If the paper included inserted 'single sheets' – with just two pages, then there is no solution.)

### 27.


The key is to realise how the fraction "one quarter" arises for a regular 4-gon. There 2<sup>n</sup> " 4, so <sup>n</sup> " 2, and each vertex angle is equal to ´ 360˝ 2n ¯ pn ´ 1q " 90˝ , which is exactly <sup>n</sup>´<sup>1</sup> <sup>2</sup><sup>n</sup> " 1 4 of 360˝ . For a regular 2n-gon, the large polygon always covers a fraction equal to exactly <sup>n</sup>´<sup>1</sup> 2n of the small polygon.

28. 3 cm, the same as the perimeter of triangle ABC. 2 What if A were folded to some point A 2 on BC?

29. In extremis one may reach for trigonometry: if we denote the three angles by α (at A), β (at B), and γ (at C), then the arrangement of squares implies that tan α " 1 3 , tan β " 1 2 , and tan γ " 1, so we can use the standard identity

$$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$

to see that tanpα ` βq " 1 " tan γ.

<sup>2</sup> From: Y. Wu, The examination system in China: the case of zhongkao mathematics. 12th International Congress on Mathematical Education. 8 July – 15 July, 2012, COEX, Seoul, Korea

However, it is worth looking for a more elementary explanation than 'brute force calculation'. If we embed the horizontal 3 by 1 rectangle ADEH in the top right hand corner of a 4 by 4 square ZDXY , (with Z labelling the top left hand corner), then we can complete the square AEP Q, which has AE as one side, with P on side XY and Q on side Y Z.

Then =AEH " =DAE, and =AEQ " =DCE. So all we need to explain is why =HEQ " =DBE – and this follows from the fact that EQ passes through the centre of the 4 by 4 square ZDXY .

30.


31. To divide the shaded region in 2 congruent parts, rotate the lower small semicircle through the angle <sup>π</sup> 2 anticlockwise about the centre of the large circle.

Note: The same idea allows one to divide the shaded region into n congruent parts: rotate the lower small semicircle successively through the angle <sup>π</sup> n anticlockwise about the centre of the large circle.<sup>3</sup>

### 32.


$$
\sin A = \sin A' = \frac{a}{2R}.
$$

If we now switch attention from the angle at A to the angle at B, and then to the angle at C, we can show that sin B " b 2R , and that sin C " c 2R .

<sup>3</sup> From: Introductory Assignment, Gelfand Correspondence Program in Mathematics.

33.

(a) Drop a perpendicular from A to meet the line BC at X. Then AX " b ¨ sin C, so

$$
\Delta = \frac{1}{2} \cdot (a \times b \sin C).
$$

(b) Substitute "sin C " c 2R " (from the Sine Rule) into the formula in part (a).

34.

	- (ii) Triangle ABC and its sister triangle A 1B 1C 1 are congruent, and each is covered 3 times.

(iii)

4pα ` β ` γq " ptotal surface area of the unit sphereq `p4 ˆ parea of the spherical triangle ABCqq 6 areap4ABCq " pα ` β ` γq ´ π.

Note: In particular, the formula for the area of a spherical triangle implies:


35.

(a)

$$\begin{aligned} \left(\cos(A+B) + i\sin(A+B)\right) &=& e^{i(A+B)} \\ &=& e^{iA} \cdot e^{iB} \\ &=& (\cos A + i\sin A)(\cos B + i\sin B) .\end{aligned}$$

Hence

$$
\sin(A+B) = \sin A \cos B + \cos A \sin B,
$$

and

$$\cos(A+B) = \cos A \cos B - \sin A \sin B.$$

To reconstruct tanpA ` Bq, divide these two expressions, and then divide numerator and denominator by "cos A cos B" to get

$$
\tan(A+B) = \frac{\sin(A+B)}{\cos(A+B)} = \frac{\tan A + \tan B}{1 - \tan A \tan B}.
$$

(b)

$$\begin{aligned} \sin X &= \quad \sin(A+B) \\ &= \quad \sin A \cos B + \cos A \sin B \\ &= \quad \sin \left(\frac{X+Y}{2}\right) \cos \left(\frac{X-Y}{2}\right) + \cos \left(\frac{X+Y}{2}\right) \sin \left(\frac{X-Y}{2}\right), \end{aligned}$$

and

$$\begin{aligned} \sin Y &= \sin(A - B) \\ &= \sin A \cos B - \cos A \sin B \\ &= \sin \left(\frac{X + Y}{2}\right) \cos \left(\frac{X - Y}{2}\right) - \cos \left(\frac{X + Y}{2}\right) \sin \left(\frac{X - Y}{2}\right), \end{aligned}$$
 
$$\therefore \sin X + \sin Y = 2 \sin \left(\frac{X + Y}{2}\right) \cos \left(\frac{X - Y}{2}\right).$$

For sin X ´ sin Y , substitute "´Y " in place of Y to get:

$$
\sin X - \sin Y = 2\sin\left(\frac{X - Y}{2}\right)\cos\left(\frac{X + Y}{2}\right) \dots
$$

Similarly

$$\begin{aligned} \cos X + \cos Y &= \begin{aligned} \cos(A+B) + \cos(A-B) \\ &= \left(\cos A \cos B - \sin A \sin B\right) + \left(\cos A \cos B + \sin A \sin B\right) \\ &= 2 \cos A \cos B \\ &= 2 \cos \left(\frac{X+Y}{2}\right) \cos \left(\frac{X-Y}{2}\right) \end{aligned}$$

$$\begin{aligned} \cos X - \cos Y &= \quad \cos(A+B) - \cos(A-B) \\ &= \quad \left(\cos A \cos B - \sin A \sin B\right) - \left(\cos A \cos B + \sin A \sin B\right) \\ &= \quad -2\sin A \sin B \\ &= \quad -2\sin\left(\frac{X+Y}{2}\right)\sin\left(\frac{X-Y}{2}\right) .\end{aligned}$$

(c)(i) sinpA ` Bq " sin 90˝ " 1;

$$
\sin A \cos B + \cos A \sin B = \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1.
$$

.

 $\text{(ii) } \cos X - \cos Y = \frac{1}{2} - 1 = -\frac{1}{2};$ 
$$-2\sin\left(\frac{X+Y}{2}\right)\sin\left(\frac{X-Y}{2}\right) = -2\sin^2 30^\circ = -\frac{1}{2}$$

(d)(i) 2 sin A sin B " cospA ´ Bq ´ cospA ` Bq.

$$\begin{aligned} \text{If } 2\sin A \sin B + 2\sin C \sin D \\ &= \left[ \cos(A - B) - \cos(A + B) \right] \\ &\quad + \left[ \cos(C - D) - \cos(C + D) \right] \\ &= \cos(A - B) + \cos(C - D) \\ &\quad \text{(since } C + D = \pi - (A + B)) \\ &= 2\cos\left(\frac{A + C - (B + D)}{2}\right)\cos\left(\frac{A + D - (B + C)}{2}\right) \\ &= 2\cos\left(\frac{\pi}{2} - (B + D)\right)\cos\left(\frac{\pi}{2} - (B + C)\right) \\ &= 2\sin(B + D)\sin(B + C). \end{aligned}$$

Note: We can swap A and B without changing the expression "sin A sin B ` sin C sin D". Hence the same should be true of the RHS "sinpB`Cq sinpB`Dq". Fortunately, since A ` B ` C ` D " π, we know that sinpA ` Cq " sinpB ` Dq, and sinpA ` Dq " sinpB ` Cq.

(ii) In triangle W XY we see that A ` B ` C ` D " π. Hence

sin A sin B ` sin C sin D " sinpA ` Dq sinpB ` Dq.

Now let R be the radius of the circle. Use "equality of angles in the same segment" and the Sine Rule (in its full form: see Problem 32) to write:

2R sin A " XY , 2R sin B " W Z, 2R sin C " Y Z, 2R sin D " W X,

2R sinpA ` Dq " W Y , 2R sinpB ` Dq " XZ.

$$\vdots \,\, WX \times YZ + WZ \times XY = WY \times XZ.$$

36. Yes.

Let the perpendicular bisectors of AB and BC meet at the point O.

Then OA " OB and OB " OC, so the circle with centre O passing through A also goes through B and C.

We have to prove that this circle also passes through D, E, etc..

To do this we prove that 4OBC " 4OCD.

We know that 4OAB " 4OBC (by SSS-congruence: OA " OB and OB " OC by the construction of O; and AB " BC since both are sides of a regular n-gon). Moreover

$$
\begin{aligned}
\angle OAB &=& \angle OBA \text{ (base angles of the isosceles triangle } \triangle OAB) \\&=& \angle OCB \text{ (since } \triangle OAB = \triangle OBC) \\&=& \angle OBC \text{ (base angles of the isosceles triangle } \triangle OBC).
\end{aligned}
$$

And =ABC " =BCD (angles of the same regular n-gon).

6 =OCD " =BCD ´ =OCB " =ABC ´ =OBA " =OBC. 6 4OBC " 4OCD (by the SAS-congruence criterion).

Hence OC " OD.

Continuing in this way we can prove that OA = OD = OE, etc..

### 37.


"area of paper" : "surface area of cube" " 8 : 6.

The same 'wastage rate' can be achieved with a square 4? 2 by 4? 2 piece of paper. Position the cube centrally on the paper, but turned through an angle of 45˝ . Then fold the four corners of the paper up each of the four side faces (with folds to tuck in four 'wasted' isosceles right angled triangles – one in the middle of each edge of the paper, with total wasted area 8). Finally, the four isosceles right angled triangles at the four corners of the paper can be folded in to exactly cover the top face without further overlaps.

However, one can do significantly better if the paper can be folded back on itself. Take a 2 by 14 rectangle, and think of this as being marked into seven 2 by 2 squares. Place the cube to cover the central 2 by 2 square – leaving three 2 by 2 squares sticking out each side. Fold one 2 by 6 strip up to cover the top square, before folding back along a diagonal of the top square to reveal the inside of the paper and to cover half of the top square twice before folding down to cover one side square. Do the same with the other 2 by 6 strip, with the reverse fold along the diagonal of the top square resulting in the other half of the top square being covered twice, with the tail folding down to cover the other side square. Hence the ratio

"area of paper" : "surface area of cube" " 7 : 6.

(This lovely solution was provided by Julia Gog. We do not know whether one can do better.)

38.


### 39.

(i) No; (ii) Yes; (iii) No; (iv) Yes; (v) No.

The 12 edges of the cube come in three groups of four – namely the four parallel edges in each of three directions.

Consider the four edges in one of these parallel groups. If the Sun's rays are parallel to these four edges, then each of these edges projects to a single vertex of the outline of the shadow – which is a projection of a square.

In all other cases two of the four parallel edges in the group give rise to shadows that form part of the boundary of the shadow polygon, while the other two edges project to the inside of the shadow (and so do not feature in the boundary of the shadow). Hence each of the three groups provides two edges to the boundary of the shadow polygon, and we obtain a hexagon.

To obtain a regular hexagon, align the Sun's rays parallel to the line AG joining two opposite corners A and G of the cube, and position the shadow plane perpendicular to this direction. The three edges at these two corners A and G then project to the inside of the shadow, while the six remaining edges project to a regular hexagon. (Since there are four body diagonals like AG, there are four ways to make such a projection. In each case, the six edges of the cube that project to the regular hexagon form a non-planar hexagon on the surface of the cube, that zig-zags its way round the polyhedron like a 'wobbly equator', turning alternately left and right each time it reaches a vertex. Such a closed circuit on a regular polyhedron is called a Petrie polygon – named after John Flinders Petrie (1907–1972), son of the famous Egyptologist Flinders Petrie).

A child of the new generation Refused to learn multiplication He said, "Don't conclude That I'm stupid, or rude. I am simply without motivation." Joel Henry Hildebrand (1881–1983)

Many important aspects of serious mathematics have their roots in the world of arithmetic. This is a world everyone can enjoy and master. In this chapter we re-visit, or maybe meet for the first time, key aspects of arithmetic that are often overlooked – ending with an introduction to the basic result on the distribution of primes.

The place of arithmetic in elementary mathematics can only be understood if one realises that, from upper primary school onwards, mathematics should no longer focus on more and more complicated calculations. Rather it moves beyond a set of procedures for grinding out answers, and should become a structural laboratory, where we gain insight into simple phenomena, and where we begin to appreciate how calculation can be managed, or tamed. The focus on structure leads in the main to matters which can be best expressed algebraically. This chapter concentrates mainly on structural aspects of number that are strictly arithmetical (e.g. related to numerals and place value), or where the relevant structural approach is "pre-algebraic" – with occasional forays into the world of algebra.

We repeat the observation that the "essence of mathematics" in the title is mostly left implicit in the problems. And while there is some discussion of this "essence" in the text between the problems, most of the relevant observations that we make are to be found in the solutions, or in the Notes which follow many of the solutions.

### 2.1. Place value and decimals: basic structure

Problem 40 Without using a calculator:

(a) Work out


### (b) Divide


Problem 41 Work out in your head (i) 11<sup>2</sup> (ii) 11<sup>3</sup> (iii) 101<sup>2</sup> . 4

Problem 42 Try to answer the following questions using only mental arithmetic:

	- (ii) What if we multiply an m-digit integer by an n-digit integer?
	- (ii) Estimate ` 1 2 ˘<sup>20</sup> to 6 decimal places.

Problem 43 Imagine the sequence of positive integers from 1 to 60 written in a single row as the digits of a very large integer:

1234567891011121314151617181920212223 ¨ ¨ ¨ 5960.

You have to cross out 100 of these digits.


### 2.2. Order and factors

Problem 44 Find the remainder when we divide

$$1111\cdots1111\text{ (with 1111 digits 1)}$$

by 1111. 4

Problem 45 Which of the numbers


is bigger? 4

Problem 46 Show that the integer

#### 100 000 000 003 000 000 000 000 700 000 000 021

is not prime. 4

Problem 47 How many prime numbers are there in each of these sequences? (Can you identify infinitely many primes in either sequence? Can you identify infinitely many non-primes?)

(a) 1, 11, 111, 1111, 11 111, 111 111, 1 111 111, . . .

(b) 11, 1001, 100 001, 10 000 001, . . . 4

### 2.3. Standard written algorithms

Problem 48 Use standard column arithmetic (i.e. long multiplication) to evaluate 9009 ˆ 37. Why should you have foreseen the outcome? 4

Problem 49 In the long division shown here, all the digits are missing.

But the "shape" of the constituent numbers is clear. Can you work out all possibilities for the two-digit divisor? 4

Problem 50 (For those readers who can write simple computer code.) In these problems you may choose your favourite programming language, and a device of your choice.

	- (i) m ` n
	- (ii) m ´ n
	- (iii) m ˆ n
	- (iv) (if n is a divisor of m) m ˜ n
	- (v) (if n is not a divisor of m) the integer part q of the quotient m ˜ n and the remainder r.

### 2.4. Divisibility tests

An integer written in base 10:

is divisible by 10 precisely when the units digit is 0.

Because 10 " 2 ˆ 5, it follows that an integer (in base 10):

is divisible by 5 precisely when the units digits is 0 or 5 (i.e. a multiple of 5); and

is divisible by 2 precisely when the units digit is 0, 2, 4, 6, or 8 (i.e. a multiple of 2).

Because 100 " 4 ˆ 25, it follows that an integer:

is divisible by 4 precisely when the integer formed by its last two digits is a multiple of 4; and

is divisible by 25 precisely when its last two digits are 00, 25, 50, or 75 (that is, a multiple of 25).

Because 1000 " 8 ˆ 125, it follows that an integer:

is divisible by 8 precisely when the integer formed by its last three digits is a multiple of 8.

Hence simple tests for divisibility by 2, by 4, by 5, by 8, and by 10 all follow easily from the way we write numbers in base 10.

### Problem 51


### Problem 52


Problem 53 Prove than an integer written in base 11 is divisible by ten precisely when its digit-sum is divisible by ten. 4

### 2.5. Sequences

We have already met


We have also considered


### 2.5.1 Triangular numbers

### Problem 54

	- 1, 1 ` 2, 1 ` 2 ` 3, 1 ` 2 ` 3 ` 4, . . . , 1 ` 2 ` 3 ` ¨ ¨ ¨ ` 10 ` 11 ` 12.

$$T\_n = 1 + 2 + 3 + \dots + n.$$

(c) Which triangular numbers are also (i) powers of 2? (ii) prime? (iii) squares? (iv) cubes? 4

### 2.5.2 Fibonacci numbers

The Hindu-Arabic numeral system emerged in the Middle East in the 10th and 11th centuries. Fibonacci, also known as Leonardo of Pisa, is generally credited with introducing this system to Europe around 1200 – especially through his book Liber Abaci (1202). One of the problems in that book introduced the sequence that now bears his name.

The sequence of Fibonacci numbers begins with the terms F<sup>0</sup> " 0, F<sup>1</sup> " 1, and continues via the Fibonacci recurrence relation:

$$F\_{n+1} = F\_n + F\_{n-1} \cdot \frac{1}{2}$$

The sequence was introduced through a curious problem about breeding rabbits; but to this day it continues to feature in many unexpected corners of mathematics and its applications.

### Problem 55

(a)(i) Generate the first twelve terms of the Fibonacci sequence:

 $F\_0$ ,  $F\_1$ ,  $\dots$ ,  $F\_{11}$ .


$$2^0, \, 2^1, \, 2^2, \, \dots, \, 2^{11}.$$


The sequence of differences between successive terms in the sequence of triangular numbers is just the sequence of natural numbers (starting with 2):

2, 3, 4, 5, 6, . . . ;

and the sequence of "second differences" is then constant:

$$1, \ 1, \ 1, \ 1, \ 1, \ \dots$$

The sequences of powers of 2 and the Fibonacci numbers behave very differently from this, in that taking differences reproduces something very like the initial sequence. In particular, taking differences can never lead to a constant sequence.

Logically the next four problems should wait until Chapter 6, where we address the delicate matter of "proof by mathematical induction". However, that would deprive us of the chance to sample the kind of surprises that lie just beneath the surface of the Fibonacci sequence, and to experience the process of fumbling our way towards a structural understanding of the apparent patterns that emerge. Of course, each time we think we have managed to guess what seems to be true, we face the challenge of proof. Those who have not yet mastered "proof by induction" are encouraged to get what they can from the solutions, and to view this as an informal introduction to ideas that will be squarely addressed in Chapter 6.

### Problem 56

(a)(i) Generate the sequence of partial sums of the sequence of powers of 2:

2 0 , 2 <sup>0</sup> ` 2 1 , 2 <sup>0</sup> ` 2 <sup>1</sup> ` 2 2 , 2 <sup>0</sup> ` 2 <sup>1</sup> ` 2 <sup>2</sup> ` 2 3 , . . .


$$F\_0, \ F\_0 + F\_1, \ F\_0 + F\_1 + F\_2, \ F\_0 + F\_1 + F\_2 + F\_3, \ \dots$$

(ii) Prove that each partial sum is 1 less than the next but one Fibonacci number. 4

Problem 56(b) starts out with the observation that

$$F\_0 + F\_1 = F\_3 - 1$$

which is a consequence of the first two instances of the fundamental recurrence relation

$$F\_{n-1} + F\_n = F\_{n+1}$$

and derives a surprising value for the n th partial sum:

$$F\_0 + F\_1 + F\_2 + \cdots + F\_{n-1} \cdot \frac{}{}$$

Fibonacci numbers make their mathematical presence felt in a quiet way – partly through the almost spooky range of unexpected internal relations which they satisfy, as illustrated in Problem 56(b) and in the next few problems.

#### Problem 57

(a) Note that

$$F\_n^2 = F\_{n-0} F\_{n+0} = F\_n^2 + (-1)^{n-1} F\_0.$$

(i) Evaluate the succession of terms:

$$F\_{1-1}F\_{1+1}, \ F\_{2-1}F\_{2+1}, \ F\_{3-1}F\_{3+1}, \ F\_{4-1}F\_{4+1}, \ \dots, \ \dots$$

	- (i) Show that the parallelogram OABC spanned by the origin O, and the points A " pa, bq, C " pc, dq and their sum B " pa ` c, b ` dq has area |ad ´ bc|.
	- (ii) Find the area of the first parallelogram in the sequence of "Fibonacci parallelograms", spanned by the origin O, and the points A " pF0, F1q " p0, 1q, C " pF1, F2q " p1, 1q.
	- (iii) Show that the n th parallelogram OACB in this sequence, spanned by the origin O, and the points A " pFn´1, Fnq and B " pFn, Fn`1q, and the pn ` 1q th parallelogram OBDC spanned by the origin O, and the points B " pFn, Fn`1q and C " pFn`1, Fn`2q overlap in the triangle OBC, which is exactly half of each parallelogram.

Conclude that every such parallelogram has area 1. Relate this to the conclusion of (a)(ii). 4

The basic recurrence relation for Fibonacci numbers specifies the next term as the sum of two successive terms. We now consider what this implies about the sum of the squares of two successive terms.

### Problem 58

(a) Evaluate the first few terms of the sequence

$$\left| F\_0^2 + F\_1^2, \,\, F\_1^2 + F\_2^2, \,\, F\_2^2 + F\_3^2, \,\, \dots \right.$$

(b) Guess a simpler expression for the sum F 2 <sup>n</sup>´<sup>1</sup> ` F 2 n . Prove your guess is correct. 4

### Problem 59

(a) Note that

$$F\_0 F\_4 = 0 = F\_2^2 - 1, \quad F\_1 F\_5 = \dots = F\_3^2 + 1.$$

(i) Evaluate the succession of terms:

$$F\_{2-2}F\_{2+2}, \ F\_{3-2}F\_{3+2}, \ F\_{4-2}F\_{4+2}, \ F\_{5-2}F\_{5+2}, \ F\_{6-2}F\_{6+2}, \ \dots \ \ .$$

(ii) Guess a simpler expression for the product F<sup>n</sup>´<sup>2</sup>F<sup>n</sup>`<sup>2</sup>. Prove your guess is correct.

(b)(i) Evaluate the succession of terms:

$$\begin{aligned} &F\_{3-3}F\_{3+3}, \; F\_{4-3}F\_{4+3}, \; F\_{5-3}F\_{5+3}, \; F\_{6-3}F\_{6+3}, \dots \end{aligned}$$

(ii) Guess a simpler expression for the product Fn´3Fn`3. Prove your guess is correct. 4

### 2.6. Commutative, associative and distributive laws

In this short section we re-emphasise the shift away from blind calculation, and towards consideration of the structure of arithmetic, which was already implicit in Problems 7–10, and Problems 16–17 in Chapter 1.

Problem 60 Each of two positive numbers a and b is increased by 10%.


Problem 61 The numbers a, b, c, d, e, f are positive. How will the value of the expression

$$a \div (b \div (c \div (d \div (e \div f))))$$

change if the value of f is doubled? 4

Problem 62 In Problem 17 we saw that it is no accident that the sum of entries in the 4 by 4 'multiplication table' is equal to 100.

	- (i) Work out the subtotal in each of the four reverse L-shapes in the 4 by 4 multiplication table. What do you notice about these four subtotals?

(ii) Use the formulae for the k th and pk ´ 1q th triangular numbers T<sup>k</sup> and Tk´<sup>1</sup> to prove that, in the n by n multiplication table, the k th reverse L-shape always gives rise to a subtotal k 3 . Conclude that

$$T\_n^2 = 1^3 + 2^3 + 3^3 + \dots + n^3.$$

Hence find a simple formula for the sum C<sup>n</sup> of the first n cubes. 4

Now that we have a compact formula


we would naturally like to find a similar formula

• for the sum S<sup>n</sup> of the first n squares:

$$S\_n = 1^2 + 2^2 + 3^2 + \dots + n^2$$

(that is, the sum of the entries on the leading diagonal of the n by n multiplication square).

This can be surprisingly elusive. But one way of obtaining it is to look instead for the sum of the entries in the sloping diagonal 2, 6, 12, 20, . . . just above the main diagonal in the n by n multiplication square.

Problem 63 Consider the n by n multiplication square.

	- (i) Guess a formula for the successive sums of these terms (6, 6 ` 18, 6 ` 18 ` 36, . . . ), and prove that your formula is correct.
	- (ii) Hence derive a formula for the sum S<sup>n</sup> of the first n squares. 4

### 2.7. Infinite decimal expansions

The standard written algorithms for calculating with integers extend naturally to terminating decimals. But how is one supposed to calculate exactly with decimals that go on for ever?

Problem 64 The decimals listed here all continue forever, recurring in the expected way. Calculate:


### Problem 65


### Problem 66

	- (i) 0.037037037 ¨ ¨ ¨
	- (ii) 0.370370370 ¨ ¨ ¨
	- (iii) 0.703703703 ¨ ¨ ¨

(b) Let a, b, c be digits (0 ď a, b, c ď 9).


Problem 67 Find the lengths of the recurring blocks for:

$$\begin{array}{l} \text{(a)} \ \frac{1}{6}, \frac{5}{6} \\\\ \text{(b)} \ \frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7} \\\\ \text{(c)} \ \frac{1}{11}, \frac{2}{11}, \frac{3}{11}, \frac{4}{11}, \frac{5}{11}, \frac{6}{11}, \frac{7}{11}, \frac{8}{11}, \frac{9}{11}, \frac{10}{11} \\\\ \text{(d)} \ \frac{1}{13}, \frac{2}{13}, \frac{3}{13}, \frac{4}{13}, \frac{5}{13}, \frac{6}{13}, \frac{7}{13}, \frac{8}{13}, \frac{9}{13}, \frac{10}{13}, \frac{11}{13}, \frac{12}{13} \end{array}$$

Problem 68 Decide whether each of these numbers has a decimal that recurs. Prove each claim.


Problem 69 For which real numbers x is the decimal representation of x unique? 4

Problem 68 raises the question as to whether one person, who has total control, can specify the digits of a decimal so as to be sure that it neither terminates nor recurs: that is, so that it represents an irrational number. The next problem asks whether one person can achieve the same outcome with less control over the choice of digits.

Problem 70 Players A and B specify a real number between 0 and 1. The first player A tries to make sure that the resulting number is rational; the second player B tries to make sure that the resulting number is irrational. In each of the following scenarios, decide whether either player has a strategy that guarantees success.

(a) Can either player guarantee a "win" if the two players take turns to specify successive digits: first A chooses the entry in the first decimal place, then B chooses the entry in the second decimal place, then A chooses the entry in the third decimal place, and so on?

4


### 2.8. The binary numeral system

There are all sorts of reasons why one should give thought to numeral systems using bases different from the familiar base 10. This is especially true of base 2, which is the simplest system of all, and is also (in some sense) the most widely used. What follows is only intended to offer a restricted glimpse into this alternative universe.

Problem 71 The numbers in this item are all written in base 2.

(a) Carry out the addition

1 1 1 0 0 ` 1 1 1 0

without changing the numbers into their base 10 equivalents – simply by applying the rules for base 2 column addition and "carrying".

	- (i) 1 0 1 1 0 ˆ 1 0 (ii) 1 1 1 0 ˆ 1 1 (iii) 1 1 0 ˆ 1 1 1

$$\frac{110}{1111} + \frac{1}{10} + \frac{1001}{1110} \tag{7}$$

The next problem invites you to devise divisibility tests for integers written in base 2 like those for base 10 (that is, tests which implement some check involving the base 2 digits in place of carrying out the actual division).

Problem 72 Let N be a positive integer written in base 2. Describe and justify a simple test, based on the digits of Nbase2:


Problem 73 A mathematical merchant has a pair of scales and an infinite set of calibrated integral weights with values w0, w1, w2, . . . (where w<sup>0</sup> ă w<sup>1</sup> ă w<sup>2</sup> ă . . .), but with only one weight of each value.

	- (i) If for each weight w there is a unique choice of weights w<sup>i</sup> that balance w, prove that the collection of weights must consist of all the powers of 2.
	- (ii) If every object of unknown integral weight w can be balanced by some collection of the weights w<sup>i</sup> , but some weights w can be balanced, or "represented", in more than one way, is it true that the merchant's collection of weights has to include all the powers of 2?

Problem 74 Explain how to express any fraction

$$\frac{m}{2^n}$$

where 0 ă m ă 2 <sup>n</sup> as a sum of distinct unit fractions with denominator a power of 2. 4

You may have heard of an algorithm (a bit like long division) which allows one to compute by hand the square root of any number N given in base 10. The algorithm starts by grouping the digits of N in pairs, starting from the decimal point. It then extracts the square root, digit by digit, with the square root having one digit for each successive pair of digits of N, starting with the left-most pair (which may be a single digit).

We all know how to start the process. For example, if the left-most pair of digits in N is "12", then we know that the square root starts with a "3". Successive digits are then identified using the algebraic identity

$$N = (x+y)^2 = x^2 + 2xy + y^2,$$

where x is the sequence of leading digits in the "partial square root" extracted so far (followed by an appropriate string of 0s), and y stands for the residual part of the required square root.

The key is to concentrate each time on the leading digit Y of the residue "N ´ x <sup>2</sup>", and at each stage to choose the leading digit Y of y so that 2xy ` y <sup>2</sup> does not exceed N ´ x 2 . This sequence of steps is traditionally (and helpfully) laid out in much the same way as long division, where at each stage we subtract the square of the current approximate square root x, from the original number N, and "bring down" the next pair of digits, and then choose the next digit Y in the square root (the leading digit of y) so that "2xy ` y <sup>2</sup>" does not exceed the residue N ´ x 2 .

In base 10 each stage requires one to juggle possibilities to decide on the next digit in the partial square root. However, in base 2 the process should be simpler, since at each stage we only have to decide whether the next digit is a 1 or a 0.

Problem 75 Work out how to calculate the square root of any square given in base 2. 4

### 2.9. The Prime Number Theorem

We have already observed that there are 4 primes less than 10, 25 primes less than 100, and 168 primes less than 1000. And there are 78 498 primes less than 10<sup>6</sup> . So

40% of integers ă 10 are prime; 25% of integers ă 100 are prime; 16.8% of integers ă 1000 are prime; and 7.8498% of integers ă 10<sup>6</sup> are prime.

In other words, the fraction of integers which are prime numbers diminishes as we go up.

The first question to ask is whether prime numbers themselves "run out" at some stage, or whether they go on for ever. The answer is very like that for the counting numbers, or positive integers 1, 2, 3, 4, 5, . . . :

the counting process certainly gets started (with 1); and no matter how far we go, we can always "add 1" to get a larger counting number.

Hence we conclude that the counting numbers "go on for ever".

### Problem 76

	- (ii) Then define n<sup>1</sup> " p<sup>1</sup> ` 1.
	- (ii) Let p<sup>2</sup> be the smallest prime factor of n1.
	- (iii) Define n<sup>2</sup> " p<sup>1</sup> ˆ p<sup>2</sup> ` 1
	- (ii) Let p<sup>3</sup> be the smallest prime factor of n2.
	- (iii) Define n<sup>3</sup> " p<sup>1</sup> ˆ p<sup>2</sup> ˆ p<sup>3</sup> ` 1

Once we know that the prime numbers go on for ever, we would like to have a clearer idea as to the frequency with which prime numbers occur among the positive integers. We have already noted that


In other words, the distribution of prime numbers seems to be fairly chaotic. Our understanding of the full picture remains fragmentary, but we are about to see that the apparent chaos in the distribution of prime numbers conceals a remarkable pattern just below the surface.

The next item is only an experiment; but it is a very suggestive experiment. It is artificial, in that what you are invited to count has been carefully chosen to point you in the right direction. The resulting observation is generally referred to as the Prime Number Theorem. The result was conjectured by Legendre (1752–1833) and by Gauss (1777–1855) in the late 1790s – and was proved 100 years later (independently and almost simultaneously) in 1896 by the French mathematician Hadamard (1865–1963) and by the Belgian mathematician de la Vall´ee Poussin (1866–1962). You will need to access a list of prime numbers up to 5000 say.

Problem 77 Let πpxq denote the number of prime numbers ď x: so πp1q " 0, πp2q " 1, πp3q " πp4q " 2, πp100q " 25. You are invited to count the number of primes up to certain carefully chosen numbers, and then to study the results. The pattern you should notice works just as well for other numbers – but is considerably harder to spot.

The special values we choose for "x" are

the next integer above successive powers of the special number e,

where e is an important constant in mathematics – an irrational number whose decimal begins 2.7182818 ¨ ¨ ¨ , and which has its own button on most calculators (see Problem 248).


(a) Complete the following table.


(b) Find an expression that seems to specify πpNq as a function of n. Hence conjecture an expression for πpxq in terms of x. 4

Durch planm¨assiges Tattonieren. [Through systematic fumbling.] Carl Friedrich Gauss (1777–1855), when asked how he came to make so many profound discoveries in mathematics.

### 2.10. Chapter 2: Comments and solutions

40.

	- (ii) 9009 (1001 " 7 ˆ 11 ˆ 13 is a factorisation that is worth remembering for all sorts of reasons: for example, it incorporates 91 " 7 ˆ 13; and it lies behind certain tests for divisibility by 7).

41.


$$\text{(iii)} \ (100+1)^2 = 100^2 + 2 \times 100 + 1^2 = 10000 + 200 + 1 = 10 \,201$$

42.

	- (ii) Largest m ` n, smallest m ` n ´ 1. (The smallest m-digit number is 10<sup>m</sup>´<sup>1</sup> and the smallest n-digit number is 10<sup>n</sup>´<sup>1</sup> , so the smallest possible product is 10<sup>m</sup>`n´<sup>2</sup> – and so has m`n´1 digits. The largest m-digit number is just less than 10<sup>m</sup> and the largest n-digit number is just less than 10<sup>n</sup> , so the largest possible product is just less than 10<sup>m</sup> ˆ 10<sup>n</sup> " 10<sup>m</sup>`<sup>n</sup> – and so has m ` n digits.)
	- (ii) 2 <sup>20</sup> is very slightly larger than 10<sup>6</sup> . In fact

$$\left(10^3 + 24\right)^2 = 10^6 + 2 \times 10^3 + 24^2 = 10^6 + 2 \times 10^3 + 576 = 1002576.5$$

` 1 2 ˘<sup>20</sup> is its reciprocal, so is slightly smaller than 10´<sup>6</sup> " <sup>0</sup>.000001, so it starts with six 0s after the decimal point and rounds up to 0.000001 (to 6 d.p.).

(c) No. (It can be equal to the product of its digits if it has just 1 digit. If a number N has k digits, with leading digit " m, then N ě m ˆ 10<sup>k</sup>´<sup>1</sup> , but the product of its digits is at most m ˆ 9 k´1 .)

	- (ii) 4, 4. (Most of us will need some rough work to supplement mental arithmetic here.

$$\begin{aligned} 20! &=& 20 \times 19 \times 18 \times \cdots \times 2 \times 1 \\ &=& 2^{18} \times 3^8 \times 5^4 \times 7^2 \times 11 \times 13 \times 17 \times 19 \\ &=& 10^4 \times 2^{14} \times 3^8 \times 7^2 \times 11 \times 13 \times 17 \times 19. \end{aligned}$$

So 20! ends in 4 zeros, and its last non-zero digit is equal to the units digit of 2 <sup>14</sup> ˆ 3 <sup>8</sup> ˆ 7 <sup>2</sup> ˆ 11 ˆ 13 ˆ 17 ˆ 19. If we work "mod 10" this is equal to the units digit of 4 ˆ 1 ˆ 9 ˆ 1 ˆ 3 ˆ 7 ˆ 9.)

Note: The reader may notice that we have used "congruences", or "modular arithmetic" (mod 10) here and at several points in Chapter 1 (e.g. in the solutions to Problem 2(d), Problem 13, Problem 16(b)).

In all these contexts one only needs to know that, if we fix the divisor n, then the remainders on division by n can be added and multiplied like ordinary numbers, since

$$(an+r)+(bn+s)=(a+b)n+(r+s),$$

and

$$(an+r)(bn+s) = (abn+as+br)n+rs.$$

Division is more delicate. We leave the reader to look up the details in any book on elementary number theory.

#### 43. (a) 00 000 123 450 (b) 99 999 785 960

The initial number p12 ¨ ¨ ¨ 9 10 11 ¨ ¨ ¨ 59 60q has 9`50ˆ2`2 " 111 digits. Hence we are left with a number having exactly 11 digits.

For the smallest integer, we delete digits to leave the smallest initial digits (preferably 0s).

For the largest integer, we delete digits to leave as many 9s at the front as possible (and then sort out the tail).

44.

$$11\ 111\ 111 = 11\ 110\ 000 + 1111 = 1111 \times 10\ 001\ .$$

In much the same way

$$1111\cdots1111000$$

(with 1108 1s and three 0s) is exactly divisible by 1111. So the remainder is 111.

45. Compare p10<sup>5</sup> ` 1qp10<sup>7</sup> ` 2q and p10<sup>5</sup> ` 2qp10<sup>7</sup> ` 1q.

The second is 10<sup>7</sup> ´ 10<sup>5</sup> bigger than the first, so the second fraction is bigger than the first.

46. The fact that 3 ˆ 7 " 21, and the position of the zeros, suggests that we express the integer as:

$$10^{35} + 3 \times 10^{24} + 7 \times 10^{11} + 3 \times 7 = (10^{11} + 3)(10^{24} + 7).$$

Note: If you feel you should have been "given a hint", then pause for a moment. There is nothing misleading here. We have no standard techniques for analysing such large numbers. The very size of the number forces you to think whether there is anything familiar about it that you might use. And the number is so simple that the only thing that can possibly stand out is the 3, 7, and 21. The rest is up to you.

47.

(a) 11 is prime. And 111 is a multiple of 3: 111 " 3 ˆ 37. You should also be able to see that 1111 is a multiple of 11: 1111 " 11 ˆ 101.

It is unclear whether 11 111 is prime or not. The Square Root Test says that to decide, we only need to check possible prime factors up to ? 11 111 ă 107. We can eliminate 2, 3, 5, 7, 11 mentally, with very little effort. And with a calculator, it is easy to check 13, 17, 19, 23, 29, 31, 37, 41, . . . and to discover that 11 111 " 41 ˆ 271.

Clearly 111 111 " 11 ˆ 10 101 " 111 ˆ 1001.

So the sequence does not look too promising. All the even-numbered terms are divisible by 11; every third term is divisible by 111 (and of course, by 3); every fourth term is divisible by 1111 (and hence by 101); and so on. So the only possible candidates for primes are the second, third, fifth, seventh, eleventh, . . . terms: that is the terms in prime positions.

Each of these terms is equal to the second bracket in the factorisation:

$$10^p - 1 = (10 - 1)(10^{p-1} + 10^{p-2} + \dots + 10 + 1),$$

where p is a prime number.

We have seen that 111 " 3 ˆ 37, and that 11 111 " 41 ˆ 271, which is not very encouraging. The 7th, 11th, 13th, and 17th terms are also not prime. But the 19th term and the 23rd terms are prime.

So primes seem scarce, but 11 is not the only prime in the sequence.

Note: Again, if you feel the problem was misleading, then pause for a moment. Part of "the essence of mathematics" is learning that some problems have a tidy solution, while others open up a rather different agenda. The only obvious way to begin to recognise this distinction is occasionally to be left to struggle to solve something that is presented as if it were a closed problem (with a tidy solution), only to discover that it is messier than one expected.

(b) We have already seen that 1001 " 7 ˆ 11 ˆ 13.

Another reason for remembering this is that it is a simple instance of the standard factorisation:

$$10^3 + 1 = (10+1)(10^2 - 10 + 1)$$

Because the signs in the second bracket are alternately "`" and "´', this factorisation extends to all odd powers of 10: for example,

$$100\,001 = 10^5 + 1 = (10+1)(10^4 - 10^3 + 10^2 - 10 + 1)$$

So this time, 11 is the only prime in the list.

Note: The missing "odd" terms

$$101, 10\,001, 1\,000\,001, 100\,000\,001, \dots, \dots$$

are slightly different – each being of the form x <sup>2</sup> ` 1. The fact that there is an algebraic factorisation of

$$x^3 + 1 = (x+1)(x^2 - x + 1)$$

implies that 1001 " 10<sup>3</sup> ` 1 has to factorise. But the lack of an algebraic factorisation of x <sup>2</sup> ` 1 does not prevent any particular number of the form x <sup>2</sup> ` 1 from factorising: for example, 3<sup>2</sup> ` 1 " 2 ˆ 5, and 5<sup>2</sup> ` 1 " 2 ˆ 13 both factorise; but 4<sup>2</sup> ` 1, 6<sup>2</sup> ` 1, and 10<sup>2</sup> ` 1 do not.

One may be forgiven for not knowing that 10<sup>4</sup> ` 1 " 10 001 " 73 ˆ 137. But one should realize that

$$10^6 + 1 = 100^3 + 1 = (100+1)(100^2 - 100 + 1).$$

48. The prime factorisation 111 " 3 ˆ 37 is worth remembering. If this is second nature, then one can do better in this problem than merely grind out the answer using long multiplication, by seeing how the output to the calculation 1001 ˆ 333 simply positions "333 thousands" and "333 units" next to each other:

3 ˆ 37 " 111, so 9 ˆ 37 " 333. Hence 9009 ˆ 37 " 1001 ˆ 333 " 333 333.

Note: The prime factorisation of 1001 is not needed here. But it is important elsewhere.

49. The very first step shows that the leading digit of the dividend must be 1; and since "three-digit minus two-digit leaves one-digit (d say)" the divisor has a multiple in the 90s.

The very next stage again gives "three-digit minus two-digit leaves one-digit", and the remainder from the first division is now the hundreds digit, so d " 1. Hence the two-digit divisor has 99 as a multiple (at the first step of the long division) – so the divisor must be 11, 33, or 99.

The next division shows that the divisor has a two-digit multiple, which when subtracted from a two-digit number leaves a two-digit remainder, so the divisor cannot be 99.

The final stage shows that the divisor has a three-digit multiple, so it cannot be 11.

Hence the divisor must be 33.

50. Your solution will depend on the programming language used. We use this problem to attract the reader's attention to some not so frequently discussed issues:


structures" involved and starts by writing one base 10 integer under another keeping digits in the same decimal position aligned in a column (a computer scientist would call it "parsing the input").


51.

(a) This exploits the fact that

$$(10^k - 1) = (10 - 1)(10^{k-1} + 10^{k-2} + \dots + 10 + 1),$$

and so is divisible by p10´1q (a fact which is obvious when we write 10 ´1 " 9, 10<sup>2</sup> ´ 1 " 99, 10<sup>3</sup> ´ 1 " 999, etc.). For example:

$$\begin{aligned} \textbf{12.345} &= \begin{array}{c} \textbf{12.345} \\ = \end{array} + \textbf{2.345} \times \begin{aligned} &\textbf{12.345} \\ = \textbf{12.345} \end{aligned} + \textbf{3.345} \times \begin{aligned} &\textbf{12.345} \\ = \textbf{12.345} \end{aligned} + \textbf{2.345} \times \begin{aligned} &\textbf{12.345} \\ = \textbf{12.345} \end{aligned} + \textbf{2.345} \end{aligned}$$

" ra sum of terms, each of which is a multiple of 9s

` rthe sum of the digits of "12 345"s

If the LHS is divided by 9, the remainder from the first bracket on the RHS is zero, so the overall remainder is the same as the remainder from dividing the digit sum by 9.

Since 9 is a multiple of 3, the first bracket is exactly divisible by 3. Hence if the LHS is divided by 3, the remainder from the first bracket on the RHS is zero, and the overall remainder is the same as the remainder from dividing the digit sum by 3.

Note: If we were only interested in "divisibility by 9", then we could have managed without appealing to the algebraic factorisation

$$(10^k - 1) = (10 - 1)(10^{k-1} + 10^{k-2} + \dots + 10 + 1),$$

since

10 ´ 1 " 9, 10<sup>2</sup> ´ 1 " 99, 10<sup>3</sup> ´ 1 " 999, . . .

are all visibly "multiples of 9". However, the structure of the above solution extends naturally to prove that, when an integer is written in base b, the remainder on division by b ´ 1 is the same as the remainder on dividing the base b "digit sum" by b ´ 1.

(b) If an integer N is divisible by 6, then we can write N " 6m for some integer m.

Hence N " p2 ˆ 3qm " 2 ˆ p3mq, so N is a multiple of 2; and N " 3 ˆ p2mq, so N is a multiple of 3.

If an integer N is divisible by 2, then we can write N " 2k for some integer k. If N is also divisible by 3, then 3 divides exactly into 2k. But HCFp2, 3q " 1, so the 3 must go exactly into the second factor k, so k " 3m for some integer m, and N " 6m is divisible by 6.

Note: It is crucial that HCFp2, 3q " 1. (E.g. 12 is divisible by 6 and by 4; but 12 is not divisible by 6 ˆ 4.)

### 52.


53. (a) This exploits the fact that

$$(11^k - 1) = (11 - 1)(11^{k-1} + 11^{k-2} + \dots + 11 + 1),$$

and so is divisible by p11 ´ 1q – a fact which is obvious if we introduce a new digit X in base 11 to stand for "ten", and then notice that

$$11 - 1 = X\_{\text{base }11}, \ 11^2 - 1 = XX\_{\text{base }11}, \ 11^3 - 1 = XXX\_{\text{base }11}, \ \text{etc.}$$

For example:

$$\begin{aligned} \textbf{12.345}\_{\textbf{base}\ 11} &= \begin{array}{c} \textbf{1}\times 11^4 + \textbf{2}\times 11^3 + \textbf{3}\times 11^2 + \textbf{4}\times 11 + \textbf{5} \\ &= \begin{bmatrix} \textbf{1}\times (11^4 - 1) + \textbf{2}\times (11^3 - 1) + \textbf{3}\times (11^2 - 1) + \textbf{4}\times (11 - 1) \end{bmatrix} \\ &+ \begin{bmatrix} \textbf{1} + \textbf{2} + \textbf{3} + \textbf{4} + \textbf{5} \end{bmatrix} \\ &= \begin{bmatrix} \textbf{a}\text{ sum of terms, each of which is a multiple of ten} \end{bmatrix} \\ &+ \begin{bmatrix} \textbf{the sum of the digits of } \textbf{"12 345"} \end{bmatrix} \end{aligned}$$

If the LHS is divided by ten, the remainder from the first bracket on the RHS is zero, so the overall remainder is the same as the remainder from dividing the digit sum by ten.

54.


$$\begin{array}{ccccccccc}1 & + & 2 & + & 3 & + & \cdots & + & n\\n & + & n-1 & + & n-2 & + & \cdots & + & 1\end{array}$$

and observe that each of the n vertically aligned columns adds to n+1. Hence

$$T\_n = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}.$$

If we do the same geometrically, then we can combine two "staircases"


of dots (one of which is inverted) into an n by n ` 1 array of dots (either with n columns and n ` 1 dots in each column, or with n ` 1 columns and n dots in each column).

Note: The n th triangular number is defined by the "formula"

$$T\_n = 1 + 2 + 3 + \dots + n.$$

But this "formula" has serious limitations: in particular, there is no way to calculate T<sup>100</sup> without first calculating T1, then T2, then T3, . . . all the way up to T99. Hence it is just a "recurrence relation", which tells us how to find T<sup>n</sup> once we know T<sup>n</sup>´<sup>1</sup> (just "add n").

The formula

$$T\_n = \frac{n(n+1)}{2}$$

derived in part (b) is much more useful, in that it allows us to work out the value of T<sup>n</sup> as soon as we know the value of n. This is what we call a "closed formula". (The language may seem strange, but it refers to the fact that the calculation is direct, and that the formula involves a small, fixed number of operations – whereas using the recurrence requires more and more work as n gets larger.)

(c) Note: There are two reasons why these questions are worth asking. The first is that whenever we focus attention on certain special classes of objects, it is always good practice to consider whether the notions we have defined are completely separate, and to try to identify any overlaps. The second reason is less obvious, but can be surprisingly fruitful: sometimes two ideas may be interesting, yet have nothing to do with each other; but at other times, the two ideas may not only be interesting in their own right, but may "combine" in a way that gives rise to surprising subtleties. Here two of the combinations are routine and uninteresting; but two combinations generate more interesting mathematics than we have a right to expect.

(i) We know that one of the two factors n and n ` 1 in the numerator is odd, and the other is even. If the triangular number

$$T\_n = \frac{n(n+1)}{2}$$

is to be a power of 2, then any odd factor of T<sup>n</sup> must be equal to 1, so n ă 3: n " 2 does not give a power of 2. Hence n " 1, and T<sup>n</sup> " 1 is the only triangular number which is also a power of 2.

	- ∗ n is odd and one of n, n`1 2 is equal to 1 (so n " 1 and T<sup>1</sup> " 1 is not prime), or
	- ∗ n is even and one of <sup>n</sup> 2 , n ` 1 is equal to 1, so n " 2, and T<sup>2</sup> " 3 is the only triangular number which is also prime.

When n is even, we notice that a " n 2 and n ` 1 " 2a ` 1 are integers with no common factors. We want their product to be a square. Because HCFpa, 2a ` 1q " 1, this occurs precisely when both a (" b 2 ) and 2a ` 1 (" c 2 ) are both squares. So we see that solutions correspond to pairs of integers b, c which satisfy the Pell equation c <sup>2</sup> " 2b <sup>2</sup> ` 1. Notice that b " 2, c " 3 is one solution, and that they satisfy the equation c <sup>2</sup> ´ 2b <sup>2</sup> " 1. We have already met

$$a^2 + b^2 = (a + bi)(a - bi)$$

as the norm (or square of the length) of the complex number a ` bi (Problem 25). In a similar way, we can "factorise"

$$c^2 - 2b^2 = (c + b\sqrt{2})(c - b\sqrt{2}).$$

So once we have one solution of the equation c <sup>2</sup> ´2b <sup>2</sup> " 1, we can take powers to get more solutions:

$$[(c + b\sqrt{2})^2][(c - b\sqrt{2})^2] = 1^2 = 1, \quad \text{etc...}$$

Hence, for example,

.

$$(3+2\sqrt{2})^2 = 17+12\sqrt{2}$$

gives rise to the solution b " 12, c " 17 – corresponding to a " 144, n " 288. Similarly

$$(3+2\sqrt{2})^3 = \dots + \dots \sqrt{2}$$

gives rise to the solution b " . . . , c " . . . , corresponding to (a " . . .), n " . . .

Note: If you are not yet familiar with complex numbers, or with the idea of a norm, don't worry. Make a note of it as something that seems to be powerful and is worth learning. It will reappear later.

(iv) The only obvious cube triangular number is the first one – namely T<sup>1</sup> " 1. Basic algebra leads quickly to an equation as in part (i):

$$\frac{n(n+1)}{2} = m^3,$$

which is equivalent to

$$\left(2n+1\right)^{2}-1=\left(2m\right)^{3}.$$

So p2mq 3 and p2mq <sup>3</sup> ` 1 are consecutive integers that are both proper powers. Catalan (1814–1894) conjectured in 1844 that 8 " 2 3 and 9 " 3 2 are the only consecutive powers (other than 0 and 1). This simple-sounding conjecture was finally proved only in 2004. It follows that T<sup>1</sup> " 1 is the only triangular number that is also a cube.

55.

	- (ii) 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34; 1, 1, 0, 1, 1, 2, 3, 5, 8, 13
	- (iii) mth term of k th sequence of differences " F<sup>m</sup>´<sup>k</sup>
	- (ii) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024; 1, 2, 4, 8, 16, 32, 64, 128, 256, 512
	- (iii) mth term of k th sequence of differences " 2 m

56.

	- (ii) x <sup>n</sup>`<sup>1</sup> ´ 1 " px ´ 1qpx <sup>n</sup> ` x <sup>n</sup>´<sup>1</sup> ` ¨ ¨ ¨ ` x ` 1q. When x " 2, the first factor on the RHS px ´ 1q " 1, so

$$2^0 + 2^1 + 2^2 + \dots + 2^n = 2^{n+1} - 1.$$

[Alternatively:

$$\begin{aligned} \left(2^0 + \left(2^0 + 2^1 + 2^2 + \dots + 2^n\right)\right) &= \left\{2^0 + 2^0 \ \left[= 2^1\right]\right\} + \left\{2^1 + 2^2 + \dots + 2^n\right\} \\ &= \left\{2^1 + 2^1 \ \left[= 2^2\right]\right\} + \left\{2^2 + 2^3 + \dots + 2^n\right\} \\ &= \left\{2^2 + 2^2 \ \left[= 2^3\right]\right\} + \left\{2^3 + 2^4 + \dots + 2^n\right\} \\ &= \dots \\ &= \left\{2^n + 2^n\right\} = 2^{n+1}. \end{aligned}$$

(b)(i) 0, 1, 2, 4, 7, 12, 20, 33, 54, 88, . . .

(ii) F<sup>0</sup> ` F<sup>1</sup> " F<sup>2</sup> " F<sup>3</sup> ´ F<sup>1</sup> F<sup>0</sup> ` F<sup>1</sup> ` F<sup>2</sup> " pF<sup>3</sup> ´ F1q ` F<sup>2</sup> " pF<sup>3</sup> ` F2q ´ F<sup>1</sup> " F<sup>4</sup> ´ F<sup>1</sup> F<sup>0</sup> ` F<sup>1</sup> ` F<sup>2</sup> ` F<sup>3</sup> " pF<sup>4</sup> ´ F1q ` F<sup>3</sup> " pF<sup>4</sup> ` F3q ´ F<sup>1</sup> " F<sup>5</sup> ´ F1.

### Claim:

$$F\_0 + F\_1 + F\_2 + \dots + F\_{n-1} = F\_{n+1} - F\_1$$

holds for all n ě 1.

Proof: When n " 1, the LHS " F<sup>0</sup> " 0 " 1 ´ 1 " F<sup>2</sup> ´ F<sup>1</sup> " RHS. We proved the next few case n " 2, n " 3, n " 4 above.

Suppose we have already proved the required relation holds all the way up to the pk ´ 1q th equation:

$$F\_0 + F\_1 + F\_2 + \dots + F\_{k-1} = F\_{k+1} - F\_1 \dots$$

Then the k th equation follows like this:

$$\begin{aligned} \left( \left( F\_0 + F\_1 + F\_2 + \dots + F\_{k-1} \right) + F\_k \right) &= \left( F\_{k+1} - F\_1 \right) + F\_k \\ &= \left( F\_{k+1} + F\_k \right) - F\_1 \\ &= F\_{k+2} - F\_1. \end{aligned}$$

So we have shown


Hence the identity holds for all n ě 1. QED

Alternatively:

$$\begin{aligned} \left(F\_1 + \left(F\_0 + F\_1 + \dots + F\_k\right)\right) &= \left(F\_1 + F\_0 \ \left[= F\_2\right]\right) + \left(F\_1 + F\_2 + \dots + F\_k\right) \\ &= \left(F\_2 + F\_1 \ \left[= F\_3\right]\right) + \left(F\_2 + F\_3 + \dots + F\_k\right) \\ &= \left(F\_3 + F\_2 \ \left[= F\_4\right]\right) + \left(F\_3 + F\_4 + \dots + F\_k\right) \\ &= \dots \\ &= F\_{k+1} + F\_k = F\_{k+2} .\end{aligned}$$

Note: In 56(a)(ii) we appealed directly to the factorisation of x <sup>n</sup>`<sup>1</sup> ´ 1 as though this were a "known fact" which is easy to prove. And in the "alternative" proof, we repeatedly combined "2<sup>k</sup> + 2<sup>k</sup> " to make 2<sup>k</sup>`<sup>1</sup> , inserting dots ". . . " to indicate that this replacement operation is repeated n ` 1 times. Both of these involved thinly veiled applications of the principle of Mathematical Induction, which is addressed in detail in Chapter 6. In 56(b)(ii) we had no way of concealing the use of "proof by Mathematical Induction", which is likely to be lurking whenever we have

and

a proposition, or statement, Ppnq involving the parameter n

we wish to prove the infinite collection of assertions:

"Ppnq is true for every n " 1, 2, 3, . . . ".

The standard way of achieving this apparent miracle of proving infinitely many things at once is:

to check that Pp1q holds (that is, to check that Ppnq is true when n " 1);

then

to suppose that we have checked all of the instances Pp1q, Pp2q, . . . , up to Ppkq for some k ě 1,

and

to show that the next instance Ppk ` 1q must then also be true.

We then conclude that Ppnq is true for all ně 1.

57.

(a)(i) 0, 2, 3, 10, 24, 65, 168, . . .

(ii) Guess:

$$F\_{n-1}F\_{n+1} = F\_n^2 + \left(-1\right)^n F\_1, \text{ for all } n \gg 1.$$

Proof: By part (i), this identity holds for n " 1, 2, 3, 4, 5, 6, 7. Suppose we have checked it as far as the k th instance:

$$F\_{k-1}F\_{k+1} = F\_k^2 + \left(-1\right)^k F\_1 \dots$$

Then the next instance follows, since

$$\begin{aligned} F\_{\{k+1\}-1}F\_{\{k+1\}+1} &=& F\_k F\_{k+2} \\ &=& (F\_{k+1} - F\_{k-1})(F\_k + F\_{k+1}) \\ &=& F\_{k+1}^2 + \{F\_{k+1}F\_k - F\_{k-1}F\_k\} - F\_{k-1}F\_{k+1} \\ &=& F\_{k+1}^2 + (F\_k^2 - F\_{k-1}F\_{k+1}) \\ &=& F\_{k+1}^2 + \{-1\}^{k+1}F\_1. \end{aligned}$$

So we have shown that the identity holds for the first few values of n, and whenever we know it is true up to the k th identity, it also holds for the pk ` 1q th identity. Hence the identity holds for all n ě 1. QED

(b)(i) We suppose that

$$\frac{b}{a} < \frac{d}{c}$$

(if the inequality is reversed, the expression for the area is multiplied by "´1").

The lines x " 0, y " 0, x " a ` c, y " b ` d form a rectangle of area pa ` cqpb ` dq, which surrounds the parallelogram.

To get from this to the area of the parallelogram, we must subtract


$$\text{area}(OABC) = (a+c)(b+d) - 2bc - ab - cd = ad - bc.$$

(ii) 1

(iii) Half of the 2nd parallelogram is equal to half of the 1st – so both have the same area, namely 1.

Half of the 3rd parallelogram is equal to half of the 2nd – so they both have the same area, namely 1.

And so on. Hence the area of the n th parallelogram is equal to

$$|ad - bc| = |F\_{n-1}F\_{n+1} - F\_n^2| = 1.$$

Part (a)(ii) is more precise in that it says that F<sup>n</sup>´<sup>1</sup>F<sup>n</sup>`<sup>1</sup> ´ F 2 <sup>n</sup> " p´1q n : this says that the relative positions of the generators pa, bq, pc, dq for successive Fibonacci parallelograms alternate, with first <sup>b</sup> <sup>a</sup> ą d c , and then <sup>b</sup> <sup>a</sup> ă d c . (In fact the gradient of successive versions of the line OA, or the ratio of successive Fibonacci numbers, converges to the Golden Ratio τ , with successive Fibonacci points A " pF<sup>n</sup>´<sup>1</sup>, Fnq alternately above and below the line with equation y " τx.)

58.


$$F\_{n-1}^2 + F\_n^2 = F\_{2n-1}.$$

Note: When part (a) gives rise unexpectedly to "the odd-numbered terms of the Fibonacci sequence", it is almost impossible to believe that this is an accident. Yet the attempt to prove that this "Guess" is correct may well prove elusive – for it is hard to see how to relate the pn ´ 1q th and n th terms to the p2n ´ 1q th term.

One approach is to

"try to prove something stronger than what seems to be required".

Claim: For each n ě 1, both of the following are true:

$$F\_{n-1}^2 + F\_n^2 = F\_{2n-1} \quad \text{and} \quad F\_{n+1}^2 - F\_{n-1}^2 = F\_{2n}.$$

Proof: We have already checked that the first relation holds for n " 1, 2, 3, 4, 5. And it is easy to check that

$$\begin{aligned} F\_{1+1}^2 - F\_{1-1}^2 &= -1 - 0 = 1 = F\_2, \\ F\_{2+1}^2 - F\_{2-1}^2 &= -4 - 1 = 3 = F\_4, \\ F\_{3+1}^2 - F\_{3-1}^2 &= -9 - 1 = 8 = F\_6. \end{aligned}$$

So both identities hold for the first few values of n.

Now suppose we have checked that both relations hold all the way up to the k th pair of relations.

Then simply adding the two relations in the k th pair gives the first relation of the next pair:

$$\begin{aligned} \left(F\_k^2 + F\_{k+1}^2\right) &= \left(F\_{k-1}^2 + F\_k^2\right) + \left(F\_{k+1}^2 - F\_{k-1}^2\right) \\ &= \left(F\_{2k-1} + F\_{2k}\right) \\ &= \quad F\_{2k+1} \end{aligned}$$

To see that the second relation of the next pair also follows, consider

$$\begin{aligned} \begin{array}{rcl} F\_{k+2}^2 - F\_k^2 &=& \left( F\_k + F\_{k+1} \right)^2 - F\_k^2 \\ &=& F\_{k+1}^2 + 2F\_k F\_{k+1} \\ &=& \left( F\_{k+1}^2 - F\_{k-1}^2 \right) + F\_{k-1}^2 + 2F\_k F\_{k+1} \\ &=& F\_{2k} + \left( F\_{k+1} - F\_k \right)^2 + 2F\_k F\_{k+1} \\ &=& F\_{2k} + \left( F\_{k+1}^2 + F\_k^2 \right) \\ &=& F\_{2k} + F\_{2k+1} \\ &=& F\_{2k+2}. \end{array} \end{array}$$

So we have shown


Hence the two identities hold for all n ě 1. QED

#### 59.

	- (ii) Guess: F<sup>n</sup>´<sup>2</sup>F<sup>n</sup>`<sup>2</sup> " F 2 <sup>n</sup> ` p´1q n`1 . Proof: By part (i), this identity holds for n " 2, 3, 4, 5, 6. Suppose we have checked it as far as the k th instance:

$$F\_{k-2}F\_{k+2} = F\_k^2 + \left(-1\right)^{k+1}.$$

Then the next instance follows using 57, since

$$\begin{array}{rcl} F\_{\{k+1\}-2}F\_{\{k+1\}+2} & = & F\_{k-1}F\_{k+3} \\ & = & F\_{k-1}(F\_{k+1} + F\_{k+2}) \\ & = & F\_{k-1}F\_{k+1} + F\_{k-1}F\_{k+2} \\ & = & F\_k^2 + (-1)^k + (F\_{k+1} - F\_k)(F\_k + F\_{k+1}) \\ & = & (-1)^k + F\_{k+1}^2 .\end{array}$$

(b)(i) 0, 13, 21, 68, . . .

#### (ii) Guess:

$$F\_{n-3}F\_{n+3} = F\_n^2 + (-1)^{n+3-1}F\_3^2 \dots$$

This suggests that we should reinterpret our previous guesses, and that the "correction terms" on the RHS:


We leave the proof (or otherwise) of this conjecture as an exercise for the reader.

60.


$$(1.1a)(1.1b) = (1 + 0.1)^2ab = (1 + 0.2 + 0.01)ab = 1.21ab.$$

(iii) 0% – notice that

$$\frac{1.1a}{1.1b} = \frac{a}{b}.$$

61. If x is doubled in the expression "x", then the value of the expression doubles. If y is doubled in the expression x ˜ y, then the value of the expression is halved. If z is doubled in the expression x ˜ py ˜ zq, then the bracket is halved, and the expression is doubled.

Replacing "x, y, z" by "d, e, f" we see that, if the value of f is doubled, the value of the bracket pd ˜ pe ˜ fqq is also doubled.

If we now take x " b, y " c, z " pd ˜ pe ˜ fqq, then, when f is doubled, z is doubled, and the value of pb ˜ pc ˜ pd ˜ pe ˜ fqqqq is doubled.

Hence the value of the whole expression

$$a \div (b \div (c \div (d \div (e \div f))))$$

is halved.

62.

(a) The fact that one can add the entries in any order depends on the commutative and associative laws of addition. Expressing the subtotal in the second row as 2p1 ` 2 ` 3 ` 4q uses the distributive law. And expressing the overall sum

$$1 + (1+2+3+4) + 2(1+2+3+4) + 3(1+2+3+4) + 4(1+2+3+4)$$

as p1 ` 2 ` 3 ` 4q <sup>2</sup> uses the distributive law again.

	- (ii) p4`8`12`16q ` p12`8`4q " 4T<sup>4</sup> `4T3. Similarly, the k th reverse L-shape has sum

$$k \cdot T\_k + k \cdot T\_{k-1} = \frac{1}{2}k^2(k+1) + \frac{1}{2}k^2(k-1) = k^3.$$

Hence

$$C\_n = 1^3 + 2^3 + 3^3 + \dots + n^3 = \left(1 + 2 + 3 + \dots + n\right)^2 = \frac{1}{4} \cdot n^2 (n+1)^2 \dots$$

#### 63.

(a) The terms are 1 ˆ 2, 2 ˆ 3, 3 ˆ 4, etc,; so the r th term is r (r+1), and the last term is pn ´ 1qppn ´ 1q ` 1q.

The r th term can be expressed as "r <sup>2</sup> ` r", so the sum

$$1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + r(r+1) + \dots + (n-1)n$$

can be expressed as

$$\left(1^2 + 2^2 + 3^2 + \dots + (n-1)^2\right) + \left(1 + 2 + 3 + \dots + (n-1)\right) = S\_{n-1} + T\_{n-1}.$$

(b)(i) ∗ n " 2: 6 " 1 ˆ 2 ˆ 3.

∗ n " 3: 6 ` 18 " 24 " 2 ˆ 3 ˆ 4.

∗ n " 4: 6 ` 18 ` 36 " 60 " 3 ˆ 4 ˆ 5.

Guess: 3pS<sup>n</sup>´<sup>1</sup> ` T<sup>n</sup>´<sup>1</sup>q " pn ´ 1qnpn ` 1q.

Proof: This is true for n " 1, 2, 3, 4.

Suppose we have checked the claim for all values up to

$$3(S\_{k-1} + T\_{k-1}) = (k-1)k(k+1).$$

Then

$$\begin{aligned} 3(S\_k + T\_k) &=& 3([S\_{k-1} + k^2] + [T\_{k-1} + k]) \\ &=& (k-1)k(k+1) + 3k(k+1) \\ &=& k(k+1)(k+2). \end{aligned}$$

Hence our guess is true for all n ě 1.

(ii)

$$S\_n + T\_n = \frac{n(n+1)(n+2)}{3},$$

$$S\_n = \frac{n(n+1)(n+2)}{3} - T\_n = \frac{n(n+1)(2n+1)}{6}$$

.

so

\*\*64.\*\* If one tries to apply the usual algorithms for decimals, then one is likely to get in something of a mass. But if we re-interpret each decimal as a fraction, then  $\frac{1}{2}$  things are much easier.

$$\begin{aligned} \text{(a)} \quad &\frac{5}{9} + \frac{6}{9} = \frac{11}{9} = 1.22222\cdots. \\ \text{(b)} \quad &0.99999\cdots = \frac{9}{9} = 1; \, 1 + \frac{1}{9} = 1.11111\cdots. \\ \text{(c)} \quad &\frac{10}{9} - \frac{2}{9} = \frac{8}{9} = 0.88888\cdots \\ \text{(d)} \quad &\frac{1}{3} \times \frac{2}{3} = \frac{2}{9} = 0.22222\cdots. \\ \text{(e)} \quad &\frac{11}{9} \times \frac{9}{11} = 1. \end{aligned}$$

65.

(a) Such a decimal is by definition equal to the fraction with numerator

$$b\_n b\_{n-1} \cdot \cdots \cdot b\_1 b\_0 b\_{-1} b\_{-2} \cdot \cdots b\_{-k}$$

(an integer with n ` k ` 1 decimal digits) and with and denominator 10<sup>k</sup> .

(b) If <sup>p</sup> q is equivalent to a fraction with numerator

$$m = b\_n b\_{n-1} \cdots b\_1 b\_0 \,\_{\text{baso } 10}$$

and denominator 10<sup>k</sup> , then m has decimal representation

$$b\_n b\_{n-1} \cdots b\_k.b\_{k-1} \cdots b\_1 b\_0.$$


66.

(a) (i) <sup>1</sup> <sup>27</sup> ; (ii) <sup>10</sup> <sup>27</sup> ; (iii) <sup>19</sup> 27 (b) (i) <sup>a</sup> 9 ; (ii) ab <sup>99</sup> ; (iii) abc 999

67.

(a) 0.166666 ¨ ¨ ¨ (block length 1); 0.833333 ¨ ¨ ¨ (block length 1)

(b) All have block length 6:

.142857142857142857 ¨ ¨ ¨ ; .285714285714285714 ¨ ¨ ¨ ; .428571428571428571 ¨ ¨ ¨ ; .571428571428571428 ¨ ¨ ¨ ; .714285714285714285 ¨ ¨ ¨ ; .857142857142857142 ¨ ¨ ¨ .

Note: The repeating blocks are all cyclically related: e.g. the block for <sup>2</sup> 7 is the same as for <sup>1</sup> 7 , but starting at "2" instead of at "1".

(c) All have block length 2:

0.090909 ¨ ¨ ¨ ; 0.181818 ¨ ¨ ¨ ; 0.272727 ¨ ¨ ¨ ; 0.363636 ¨ ¨ ¨ ; 0.454545 ¨ ¨ ¨ ; 0.545454 ¨ ¨ ¨ ; 0.636363 ¨ ¨ ¨ ; 0.727272 ¨ ¨ ¨ ; 0.818181 ¨ ¨ ¨ ; 0.909090 ¨ ¨ ¨ .

Note: The repeating blocks are not all cyclically the same, but fall into five pairs:

– 1 <sup>11</sup> and <sup>10</sup> <sup>11</sup> are cyclically related;

$$- \text{ as are those for } \frac{2}{11} \text{ and } \frac{9}{11};$$


Note: They fall into two families of six, where each family is cyclically related :

1 <sup>13</sup> " 0.076923076923076923 ¨ ¨ ¨ , 3 <sup>13</sup> " 0.230769230769230769 ¨ ¨ ¨ , 4 <sup>13</sup> " 0.307692307692307692 ¨ ¨ ¨ , 9 <sup>13</sup> " 0.692307692307692307 ¨ ¨ ¨ , 10 <sup>13</sup> " 0.769230769230769230 ¨ ¨ ¨ , 12 <sup>13</sup> " 0.923076923076923076 ¨ ¨ ¨ ;

and

$$\begin{array}{l} \frac{2}{13} = 0.153846153846153846\cdots; \\ \frac{5}{13} = 0.384615384615384615\cdots; \\ \frac{6}{13} = 0.461538461538461538\cdots; \\ \frac{7}{13} = 0.538461538461538461\cdots; \\ \frac{8}{13} = 0.615384615384615384\cdots; \\ \frac{11}{13} = 0.846153846153846153\cdots. \end{array}$$

68.


69. Claim Decimal fractions have two decimal representations. All other numbers have exactly one decimal representation.

Proof: Every "decimal fraction" (that is, any fraction which can be written with denominator a power of 10) has two representations – one that terminates and one that ends with an endless string of 9s: if the last non-zero digit of the terminating decimal is k, then the second representation of the same number is obtained by changing the "k" to "k ´ 1" and following it with an endless string of 9s.

Consider an unknown number with two different decimal representations α and β. Since they are "different", α and β must differ in at least one position. Suppose the first, or left-most, position in which they differ is that in the k th decimal place (corresponding to 10´<sup>k</sup> ), and that the two digits in that position are a<sup>k</sup> (for α) and b<sup>k</sup> (for β).

We may suppose that a<sup>k</sup> ă bk. Then b<sup>k</sup> " a<sup>k</sup> ` 1 (otherwise b<sup>k</sup> ´ a<sup>k</sup> ą 1, and β ´ α ą 10´<sup>k</sup> , so α ‰ β).

Moreover, since β is not larger than α, the digits following b<sup>k</sup> must all be equal to 0, and the digits following a<sup>k</sup> must all be equal to 9. QED

70. In case (d), A only has to choose a recurring block (such as "55555 ¨ ¨ ¨ ", or "090909 ¨ ¨ ¨ ", or "123123123 ¨ ¨ ¨ ") for his/her positions – no matter where they are. B's control terminates at some stage, after which A's recurring block guarantees that the resulting number is rational.

The other parts all offer a guaranteed strategy for B. Let the positions chosen by B be numbered

$$n\_1, n\_2, n\_3, n\_4, \dots, n\_k, \dots$$

Now exploit the fact that the positive rationals are countable – that is, can be included in a single list. To see this we can use Cantor's (1845–1918) diagonal enumeration

0 1 ; 1 1 ; 1 2 , 2 1 ; 1 3 , 3 1 ; 1 4 , 2 3 , 3 2 , 4 1 ; 1 5 , 5 1 ; 1 6 , 2 5 , 3 4 , 4 3 , 5 2 , 6 1 ; 1 7 , . . . ,

which lists all rationals <sup>p</sup> <sup>q</sup> with HCFpp, qq " 1


#### and so on.

All B needs to do is to make sure that the resulting decimal is not the decimal of any number in this list, and s/he can do this by choosing a digit in the n th <sup>k</sup> position which is different from the digit which the k th rational in the above list has in that position. The resulting real number is then different from every number in the list – and hence must be irrational.

#### 71.


Note: Trying to do this should make it clear how easily we confound "the fourteenth positive integer" with its familiar base 10 representation. It takes time and effort to learn to see "14base 10" as "2 ˆ7", and "21base 10" as 3ˆ7, and hence to spot the common multiple "42base 10". In base 2 the same numbers evoke no such familiar echoes.

72. Let N " paka<sup>k</sup>´<sup>1</sup> ¨ ¨ ¨ a1a0qbase 2.


$$"{a\_0 - a\_1 + a\_2 - a\_3 + \dots \pm a\_k}",$$

is divisible by 3.

#### Proof

$$\begin{aligned} N &= \ (a\_k a\_{k-1} \cdots a\_1 a\_0)\_{\text{basa }2} \\ &= \ 2^k a\_k + 2^{k-1} a\_{k-1} + \cdots + 2a\_1 + a\_0. \end{aligned}$$

For each odd suffix m, increase the coefficient 2<sup>m</sup> by 1: then

$$2^m + 1 = (2+1)(2^{m-1} - 2^{m-2} + \dots - 2 + 1)$$

has 3 as a factor.

For each even suffix m " 2n, decrease the coefficient by 1: then

$$2^{2n} - 1 = (2^2 - 1)(2^{2n - 2} + 2^{2n - 4} + \dots + 2^2 + 1)!$$

has 3 as a factor.

Hence

$$\begin{aligned} N &= \ 2^k a\_k + 2^{k-1} a\_{k-1} + \dots + 2a\_1 + a\_0 \\ &= \ \text{(multiple of 3)} + (a\_0 - a\_1 + a\_2 - \dots \pm a\_k). \end{aligned}$$


$${"{a\_1 a\_0}" - {"{a\_3} a\_2"}} + {"{a\_5} a\_4" - 
\cdots}$$

is divisible by 5.

Proof:

$$\begin{array}{rcl} N &=& \langle a\_k a\_{k-1} \cdots a\_1 a\_0 \rangle\_{\text{base } 2} \\ &=& 2^k a\_k + 2^{k-1} a\_{k-1} + \cdots + 2a\_1 + a\_0 \\ &=& \langle 2a\_1 + a\_0 \rangle + 2^2 (2a\_3 + a\_2) + 2^4 (2a\_5 + a\_4) + \cdots \\ &=& \langle 2^2 + 1 \rangle (2a\_3 + a\_2) + (2^4 - 1)(2a\_5 + a\_4) + \cdots \\ & & \quad + [ (2a\_1 + a\_0) - (2a\_3 + a\_2) + (2a\_5 + a\_4) - \cdots ] \\ &=& \langle \text{a multiple of } 5 \rangle + [ \*a\_1 a\_0 \text{" $ } - \text{"} a\_3 a\_2 \text{"$  } + \text{" $a\_5} a\_4 \text{"$  } - \cdots ]. \end{array}$$

73.

(a) To weigh an object with weight 1, we must have w<sup>0</sup> " 1.

To weigh an object with weight 2, we must have w<sup>1</sup> " 2. We can then weigh any object of weight 3, but not one of weight 4.

(i) Now assume each positive weight w can be balanced in exactly one way. Then we cannot have w<sup>2</sup> " 3, so w<sup>2</sup> " 4.

Suppose that, continuing in this way, we have deduced that w<sup>i</sup> " 2 i for each i " 0, 1, 2, . . . , k.

Then the binary numeral system reveals precisely that every weight w from 0 up to

$$2^{k+1} - 1 = 1 + 2 + 2^2 + \dots + 2^k$$

can be uniquely represented, but 2<sup>k</sup>`<sup>1</sup> cannot. Hence

> w<sup>k</sup>`<sup>1</sup> " 2 k`1 .

The result follows by induction.

(ii) If the representation of each integer is not unique, then the sequence

$$w\_0, \ w\_1, \ w\_2, \ \dots$$

need not include the powers of 2. For example, it could begin

1, 2, 3, 5, . . .

(b) If each integer w is to be weighed in this way, then w has to be represented in the form

$$w = a\_1 w\_1 + a\_2 w\_2 + a\_3 w\_3 + \cdots$$

where each coefficient a<sup>i</sup> " 0 (if the weight w<sup>i</sup> is not used to weigh w), or " 1 (if the weight w<sup>i</sup> is used to balance w), or " ´1 (if the weight w<sup>i</sup> is used to supplement w).

If each representation is to be unique, then one can prove as in (a)(i) that the sequence of weights must be the successive powers of 3.

74. Write m in "base 2":

$$m = (a\_{n-1} \cdot \cdots \cdot a\_1 a\_0)\_{\text{base 2}},$$

where each a<sup>k</sup> " 0 or 1. Then

$$\frac{m}{2^n} = \frac{a\_0}{2^n} + \frac{a\_1}{2^{n-1}} + \dots + \frac{a\_{n-1}}{2}.$$

That is,

$$\frac{m}{2^n} = (0.a\_{n-1} \cdot \cdots \cdot a\_1a\_0)\_{\text{base 2}}.$$

75. We give an example, starting with N " 110 111 001base <sup>2</sup> .

Write N, and pair off the digits, starting at the units digit.

### 1 } 10 } 11 } 10 } 01

The left-most digit stands for 2<sup>8</sup> , so the square root is at least 2<sup>4</sup> (and less than 2 5 ). Hence the required square root has five digits (one for each "pair" of digits of N), and starts with a 1.

Root 1 } ? } ? } ? } ?

[We can also see that the final units digit will have to be a "1". But this is not the time to add such information.]

Let x " 10 000, and subtract x <sup>2</sup> " 100 000 000 from N:

$$\begin{array}{r|r|r|r} 1 & 00 & 00 & 00 & 00 \\ \parallel \text{ 10} \parallel \text{ 11} \parallel \text{ 10} \parallel \text{ 01} \end{array}$$

This residue has to be equal to "2xy ` y 2 ". However, as with long division, our immediate interest is in determining the next digit of our "partial square root". If the next digit is a 1 (contributing 2<sup>3</sup> ), then 2xy ě 2 8 , which would spill over and change the digit we have already determined. Hence the next digit is a 0.

Root 1 } 0 } ? } ? } ?

So we can again let x " 10 000 giving the same remainder, which has to be equal to "2xy ` y 2 ", but this time y ă 2 <sup>3</sup> has at most three digits.

The remainder

$$\begin{array}{c|c|c|c|c} \hline 1 & 10 & 11 & 10 & 01 \\ \hline \end{array}$$

is greater than 2<sup>7</sup> , so y ě 2 2 and the next digit must be a "1".

Root 1 } 0 } 1 } ? } ?

Now let x " 10 100, and subtract x <sup>2</sup> " 110 010 000 from N, leaving

$$\begin{array}{c|c} 10 & 10 \perp 01 \\ \end{array}$$

This residue has to equal 2xy ` y 2 , with x " 10 100. If the next digit in the square root is 1, then 2xy ě 2 <sup>6</sup> ą 101 001 " 2xy ` y 2 . Hence the next digit is 0, and the last digit is then 1.

Hence the required square root is equal to:

```
Root 1 } 0 } 1 } 0 } 1
```
76.

(b)(i) The fact that

n<sup>1</sup> " p<sup>1</sup> ` 1

says that

"n<sup>1</sup> is equal to a multiple of p<sup>1</sup> with remainder " 1".

(c)(i) The fact that

$$n\_2 = p\_1 \times p\_2 + 1$$

says that

"n<sup>2</sup> is equal to a multiple of p<sup>1</sup> with remainder " 1",

and that

"n<sup>2</sup> is equal to a multiple of p<sup>2</sup> with remainder " 1".

Hence neither p<sup>1</sup> nor p<sup>2</sup> are factors of n2.

(d) The fact that

n<sup>k</sup> " p<sup>1</sup> ˆ p<sup>2</sup> ˆ ¨ ¨ ¨ ˆ p<sup>k</sup> ` 1

says that

"n<sup>k</sup> is equal to a multiple of p<sup>i</sup> with remainder " 1"

for each suffix i, 1 ď i ď k. Hence none of the primes p1, p2, p3, . . . , p<sup>k</sup> is a factor of nk.

So the smallest prime factor of n<sup>k</sup> always gives us a new prime p<sup>k</sup>`<sup>1</sup>.

(e) If we start with p<sup>1</sup> " 2, then n<sup>1</sup> " p<sup>1</sup> ` 1 " 3, so p<sup>2</sup> " 3. Then n<sup>2</sup> " p<sup>1</sup> ˆ p<sup>2</sup> ` 1 " 7, so p<sup>3</sup> " 7. Then n<sup>3</sup> " p<sup>1</sup> ˆ p<sup>2</sup> ˆ p<sup>3</sup> ` 1 " 43, so p<sup>4</sup> " 43. Then n<sup>4</sup> " p<sup>1</sup> ˆ p<sup>2</sup> ˆ p<sup>3</sup> ˆ p<sup>4</sup> ` 1 " 1807 " 13 ˆ 139, so p<sup>5</sup> " 13.

77.

(a) We write rxs for the "first integer ě x". Then

πpre 1 sq " πp3q " 2; πpre 2 sq " πp8q " 4; πpre 3 sq " πp21q " 8; πpre 4 sq " πp55q " 16; πpre 5 sq " πp149q " 35; πpre 6 sq " πp404q " 79; πpre 7 sq " πp1097q " 184; πpre 8 sq " πp2981q " 429; πpre 9 sq " πp8104q " 1019.

(b) The initial "doubling" is an accident of small numbers, which soon turns into "slightly more than doubling".

The observation that should (eventually) jump out at you concerns the ratio e <sup>N</sup> : πpNq, which seems to be surprisingly close to N ´ 1. This suggests the possible

Conjecture: πpxq " <sup>x</sup> lnpxq´1 (where lnpxq " log<sup>e</sup> pxq).

All the evidence suggests that the shapes of reality are mathematical. George Steiner (1929– )

The previous chapter focused on aspects of the arithmetic of pure numbers – mostly without any surrounding context. However, our mathematical experience does not begin with pure numbers. At school level, mathematical concepts, and the reasoning we bring to understanding and using them, have their roots in language. And in real life, every application of mathematics starts out with a situation which is described in words, and which has to be reformulated mathematically before we can begin to calculate, and to draw meaningful mathematical conclusions. Word problems play an important, if limited, role in helping students to appreciate, and to handle the subtleties involved in

> the art of using the mathematics we know to solve problems given in words.

This art of using mathematics involves two distinct – but interacting – processes, which we refer to here as "simplifying" and "recognising structure".

• To identify the mathematical heart of a problem arising in the real world, one may first have to simplify – that is, to side-line details that seem unimportant or irrelevant, and then simplify as much as possible without changing the underlying problem (e.g. by replacing some awkward feature by a different quantity which is easier to measure, or by an approximation which is easier to work with).

This "simplifying" stage is well-illustrated by the tongue-in-cheek title of the classic textbook Consider a spherical cow . . . by John Harte (1985):

Milk production at a dairy farm was low, so [. . . ] a multidisciplinary team of professors was assembled. [. . . After] two weeks of intensive on-site investigation [. . . ] the farmer received the write up, and opened it to read [. . . ] "Consider a spherical cow . . . ".

The point to emphasise here is that the judgements needed when "simplifying" are subtle, depend on an understanding of the particular situation being modelled, and may lead to a model which at first sight seems to be counterintuitive, but which may not be as silly as it seems – and which therefore needs to be explained sensitively to non-mathematicians.

In contrast word problems by-pass the "simplifying" stage, and focus instead on "recognising structure": they present the solver with a problem which is already essentially mathematical, but where the inner structure is contextualised, and is described in words. All the solver has to do is to interpret the verbal description in a way that extracts the structure just beneath the surface, and to translate it into a familiar mathematical form. That is, word problems are designed to develop facility with the process of "recognising structure", while avoiding the complication of expecting students to make modelling judgements of the kind required by the subtler "simplifying" process.

Because word problems focus on the second process of "recognising structure", they tend to incorporate the relevant mathematical structure isomorphically. The underlying structure still needs to be identified and interpreted, but the interpretations are likely to be standard, with no need for imaginative assumptions and simplifications before the structure can be discerned. For example, if a problem in primary school refers to an unknown number of "sweets" to be "shared" between six children, then the collection of "sweets" is isomorphic to a pure number (the number of sweets); and the act of "sharing" is a thinly veiled reference to numerical division.

The story in a word problem may be a purely mathematical problem in disguise. But the art of identifying the correspondence between

the data given in the story line, and

the mathematical entities to which they correspond

and between

the actions in the story line, and

the corresponding mathematical operations on those mathematical entities

is non-trivial, and has to be learned the hard way. The first problem below illustrates the remarkable variety of instances of even the simplest subtraction, or difference.

As in Chapters 1 and 2 the "essence of mathematics" is to be found in the problems themselves. Some discussion of this "essence" is presented in the text between the problems; but most of the relevant observations are either to be found in the solutions (or in the Notes which follow many of the solutions), or are left for readers to extract for themselves.

### 3.1. Twenty problems which embody "3 ´ 1 " 2"

The answer to every one of the questions in Problem 78 is the same – at least, as a 'pure number'. The goal is therefore not to "solve" each problem, but to distinguish between, and to reflect upon, the different ways in which the very simple mathematical structure "3 ´ 1 " 2" turns out to be the relevant "model" in each case.

### Problem 78


<sup>4</sup> This question is historically correct. In 1946, in the Soviet Union, when these problems were formulated, Saturday was a working day.


### 3.2. Some classical examples

Problem 79 Katya and her friends stand in a circle in such a way that the two neighbours of each child are of the same gender. If there are five boys in the circle, how many girls are there? 4

Problem 80 How much pure water must be added to a vat containing 10 litres of 60% solution of acid to dilute it into a 20% solution of acid? 4

<sup>5</sup> Gorky (now the city of Nizhny Novgorod) lies to the east of Moscow.

Problem 81 A mother is 2 <sup>1</sup> 2 times as old as her daughter. Six years ago the mother was 4 times as old as her daughter. How old are mother and daughter? 4

### Problem 82


Problem 83 A team of mowers had to mow two fields, one twice as large as the other. The team spent half-a-day mowing the larger field. After that the team split: one half continued working on the big field and finished it by evening; the other half worked on the smaller field, and did not finish it that day – but the remaining part was mowed by one mower in one day. How many mowers were there? 4

### 3.3. Speed and acceleration

Problem 84 Jack and Jill went up the hill, and averaged 2 mph on the way up. They then turned round and went straight back down by the same route, this time averaging 4 mph. What was their average speed for the round trip (up and down)? 4

### Problem 85

	- (ii) I cycle for 3 hours round the track of a velodrome, averaging 40 km/h for the first hour, 30 km/h for the second hour, and 20 km/h for the final hour. What is my average speed over the whole 3 hours?
	- ∗ the arithmetic mean

$$\frac{u+v}{2}$$

of two positive quantities u, v, and

∗ the harmonic mean

$$\frac{2}{\frac{1}{u} + \frac{1}{v}}\text{.}$$

(ii) Give a purely algebraic proof of your inequality in (i). 4

Problem 86 A train started from a station and, moving with a constant acceleration, covered a distance of 4 km, finally reaching a speed of 72 km/hour. Find the acceleration of the train, and the time taken for the 4 km. 4

Problem 87 (Average speed of an accelerating car) A typical car (and maybe also a typical train!) does not move with constant acceleration. Starting from a standstill, a car moves through the gears and "accelerates more quickly" in lower gears, when travelling at lower speeds, than it does in higher gears, when travelling at higher speeds. Use this empirical fact to prove that the average speed of a car accelerating from rest is more than half of its final measured speed after the acceleration. 4

### 3.4. Hidden connections

Problem 88 Two old women set out at sunrise and each walked with a constant speed. One went from A to B, and the other went from B to A. They met at noon, and continuing without a stop, they arrived respectively at B at 4 pm and at A at 9 pm. At what time was sunrise on that day? 4 Problem 89 A paddle-steamer takes five days to travel from St Louis to New Orleans, and takes seven days for the return journey. Assuming that the rate of flow of the current is constant, calculate how long it takes for a raft to drift from St Louis to New Orleans. 4

Problem 90 [From Paolo dell'Abbaco's Trattato d'aritmetica] "From here to Florence is 60 miles, and there is one who walks it in 8 days [in one direction], another in five days [in the opposite direction]. It is asked: Departing at the same time, in how many days will they meet?" 4

Problem 91 Notice that in Problem 88 sunrise occurs t " ? 4 ˆ 9 hours before noon, and that ? 4 ˆ 9 is the geometric mean of 4 and 9. Once this is pointed out, can you reformulate your solution to Problem 88 to solve a more general problem? 4

### 3.5. Chapter 3: Comments and solutions

#### 78.


Note: This kind of "geometrical subtraction" is needed in many contexts (such as: proving the general formula

$$\frac{1}{2}(\text{base} \times \text{height})$$

for the area of a triangle, or showing that the area of the parallelogram spanned by the origin and vectors pa, bq, pc, dq is |ad ´ bc|, or in Euclid's Elements, Book I, Proposition 2). The idea can be strangely elusive.

(c) The situation here is significantly different. We start with Tanya's brothers and sisters, and finish with the related, but different, notion of "boys and girls in Tanya's family". The "3" does not represent anything specific: it is a numerical excess (of Tanya's brothers over her sisters). In contrast, the "1" seems to represent Tanya herself, who needs to be taken into account when we switch from the initial scenario (Tanya's brothers and sisters) to the final question about "boys and girls in Tanya's family".

(d) No doubt this can be solved by drawing a picture in which the underlying structure is only appreciated superficially. But beneath the surface, it seems to be a much more abstract representation of "3´1 " 2". The "3" certainly stands for the "three pieces". But the operation "´1" is not obviously subtracting anything.

The relevant observation is simply that, starting from one end, pieces and cuts alternate. So if we ignore the starting end, there must be the same number of pieces and cuts – except that if we start with a log (rather than a long tape from which we are cutting off pieces), the "last cut" is "the other end of the log", which has already been cut – so does not need to be cut again, and this obliges us to subtract 1 from the number of pieces to get the number of additional cuts.

Note: This idea arises in many settings, and is sometimes referred to as "Posts and gaps". Sometimes one has to "subtract 1" as here; at other times one has to "add 1" (e.g. when counting the number of "posts", if we are given the number of "gaps", or "fence panels").


Note: The impact of the extra step (switching from discrete counting number to continuous distance) can be seen more clearly in the number of errors made when students are faced with such variations as:

"There are ten lamp posts in my street, and they are 70 metres apart. How far is it from the first to the last?"

(h) One suspects that this superficially simple problem would prove inaccessible unless pupils have learned to represent word problems diagrammatically, or have already mastered simple algebra. The "3" and the "1" do not represent real-world entities; so one has to be prepared to mark a "3" on a number line, and to interpret "average" as indicating that the two unknown quantities lie equally spaced either side of it. "Half their difference" is then staring one in the face, and the smaller number (to the left) is clearly "3 ´ 1".

Note: This may look rather like the overdue train in part (e). We suggest that it is significantly different.

(i) The story line clearly adds layers of difficulty which we tend to overlook. Learning to "recognise structure" and to translate words into a form that allows one to calculate is clearly a non-trivial (and neglected) art. Distances in kilometres may convey something more active than the given "length of a barge pole" in part (b), or the reported times in part (e), even if the diagram – once constructed – is very similar (provided of course that "along the same road" is interpreted as meaning "in the same direction").

Note: Consider the following item from an authoritative international study TIMSS 2011<sup>6</sup> for pupils aged around 14:

"Points A, B, and C lie in a line and B is between A and C. If AB " 10 cm and BC " 5.2 cm, what is the distance between the midpoints of AB and BC?

A 2.4 cm B 2.6 cm C 5.0 cm D 7.6 cm"

The question is a multiple choice question, and the options represent different ways of failing to translate the words into a suitable diagram, or to interpret them correctly. The sampling (in around 50 countries) was done very carefully. So the different success rates in different countries (of which 5 are given below) suggest that some systems give far too little attention to helping pupils to learn the relevant underlying art:

Russia 60%, Hungary 41%, Australia 40%, England 38%, USA 29%


<sup>6</sup> Trends in International Mathematics and Science Study, https://timssandpirls.bc.edu/ timss2011/index.html


distances along the plane's path (measured in "plane's lengths"),

and on the other hand

the time taken by the anti-aircraft fire to reach the plane.

This comparison has to be made because of the added complication of the change in the relative velocity of the gun and the plane.

The given rule of thumb specifies the direction in which a stationary gunner should aim; and the reported (unrealistically fast, yet presumed to be steady) motion of the gun introduces a 2-dimensional (vector) version of "swimming upstream" – which suggests the expected answer "two thirds of 3 plane lengths", so that "1" of the "3 plane's lengths" is compensated by the gun's motion.

(q) A solution is again dependent on representing the given information in some form. Whether or not one uses symbols, the wording invites the solver to use "my present age (in years)" as a preferred unit, and to represent "my brother's present age" as "3" of these basic units. The "3 ´ 1" then represents how much older he is than I am – and hence how old he was when I was born, or "how many times my present age he was when I was born".

Note: The choice of unit may conceal the fact that the question and solution are rooted in ratio and proportion.


The "tram on the opposite track" is travelling in the opposite direction, is 1 km away, and is "3 km ahead" (or "3 km behind"); so one of these trams is 1 km from the end of the track, and the other is on the other track and 2 km from one end (travelling in the opposite direction). There are exactly two possible configurations – each arising from the other if we reverse the direction of travel. By choosing the direction of travel (or by allowing "negative speed") we may assume that tram A is 2 km from the same end of the track and that tram B in front of it is 1 km beyond the end of the track on the opposite side. Tram C is 3 km ahead of B, and hence 4 km down that 4.5 km stretch of track (so has not yet "turned the corner"). Hence it is 1 km closer to its nearest neighbour (A) than it is to B.

79. If we ignore the first sentence, then there could be zero girls (and five boys). But the first sentence guarantees that there is at least one girl ("Katya and her friends"). So boys and girls must alternate, giving rise to 5 girls.

80. The problem requires a degree of "modelling" in that "60% solution of acid" suggests that the initial ratio

$$\text{``acid : water''} = 60 \text{ : 40.}$$

Hence the initial 10 litres is made up of 4 litres of water and 6 litres of acid. Adding water does not change the amount of acid; so we want 6 litres to be 20% of the final mix – which must therefore be 30 litres. Hence we should add 20 litres.

81. The difference in ages is <sup>3</sup> <sup>2</sup> ˆ d, where d is the daughter's age in years. Six years ago the difference was three times the daughter's age, which was then d ´ 6 years. Hence

$$3(d-6) = \frac{3}{2} \times d,$$

so d " 12.

82.

Note: Underpinning all such problems is the "unitary method", which here comes into its own. It is an essential tool, which is scarcely taught, and not sufficiently practised. (As a result many students mindlessly translate "Tom takes 2 hours" as "T " 2", etc..)

(a) When they all work together we need to know not how long each takes to do the job, but at what rate each contributor works.

Tom does the job in 2 hours, so works at the rate of " <sup>1</sup> 2 of a job in 1 hour". Dick works at a rate of " <sup>1</sup> 3 of a job in 1 hour", and Harry works at the rate of " 1 4 of a job in 1 hour".

So working together, they can manage

$$\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{12}$$

of a job in 1 hour.

Hence, to complete 1 job they require <sup>12</sup> <sup>13</sup> of an hour.

(b) As in part (a), we need to know the rate at which each man works.

Suppose that Tom completes the fraction t of a job in 1 hour, that Dick completes the fraction d of a job in 1 hour, and that Harry completes the fraction h of a job in 1 hour.

Then in 1 hour, working together, they complete pt`d`hq jobs; so to complete 1 job takes them

$$\frac{1}{t+d+h} \quad \text{hours.}$$

We therefore need to find "t ` d ` h".

In 1 hour, Tom and Dick together complete t ` d jobs. And we are told that in 2 hours they complete 1 job, so t ` d " 1 2 . Similarly d ` h " 1 3 , and h ` t " 1 4 . Adding yields

$$2(t+d+h) = \frac{1}{2} + \frac{1}{3} + \frac{1}{4},$$

13

.

so

$$t + d + h = \frac{\gamma}{24}.$$
 Hence the time required for Tom, Dick and Happy to finish 1 job working together is 
$$\alpha.$$

$$\frac{1}{t+d+h} = \frac{24}{13}$$

hours.

Note: Alternatively, one might let Tom take T hours to complete 1 job, Dick take D hours to complete 1 job, and Harry take H hours to complete 1 job. Then

$$t = \frac{1}{T}, \quad d = \frac{1}{D}, \quad h = \frac{1}{H}.$$

83. Imagine the two fields as strips of equal width – with the larger field twice as long as the smaller one.

The large strip was completely mowed in two parts:


Hence the whole team mowed two thirds of the large field and the half team mowed the remaining one third.

So the half team, who worked on the smaller field, mowed the equivalent of one third of the larger field – that is, two thirds of the (half-size) smaller field. Therefore the remaining one third of the smaller field was mowed by a single man on the second day.

The previous two thirds of the smaller field (twice as much) was mowed in half a day (half the time), so must have required 4 (" four times as many) men. So the whole team contained 8 mowers.

Alternatively, we may suppose that there are 2n mowers (since the team is said to split into two halves), and that each mower mows at the rate of "r large fields per day".

The total work done in completing the larger field is then


where each part is equal to

pnumber of men ˆ rate of workingq ˆ plength of time workedq.

That is <sup>3</sup> 2 nr. So <sup>3</sup> 2 nr " 1.

The total work done on the smaller field is


That is <sup>n</sup>`<sup>2</sup> <sup>2</sup> ˆ r. So <sup>n</sup>`<sup>2</sup> <sup>2</sup> ˆ r " 1 2 (since the smaller field is half the larger field). Hence <sup>3</sup> 2 n " n ` 2.

84. The words "average speed" often provoke an unthinking assumption that one is simply being asked to find the average of the "speed numbers" given in the problem. A moment's thought should remind us that the "average speed" for a journey is not equal to the "average of the various speeds taken as pure numbers"; it is equal to

(the total distance travelled) ˜ (the total time taken).

If the distance up the hill is m miles, then the climb takes <sup>m</sup> 2 hours, and the descent takes <sup>m</sup> 4 hours. The total distance for the round trip is 2m miles, so Jack and Jill's average speed is

$$\frac{2m}{\frac{3m}{4}} = \frac{8}{3}\text{ mph.}$$

Note: We first meet averages for discrete quantities, or whole numbers, where the goal is to replace a collection of quantities, or numbers, by a single representative statistic. If n quantities contribute equally, then each contributes exactly p 1 n q th to the average.

One way of looking at this is to represent each of the quantities being averaged in a bar chart – as rectangles of width 1, and with height corresponding to the quantity represented. "Adding all the quantities and dividing by n" is then the same as "calculating the total area under the graph and then dividing by the total length of the interval". In other words, we have replaced the complicated bar chart by a constant function (or a single rectangle), which has the same domain as the bar chart, and which has the same area under it (or integral) as the more complicated bar chart.

More generally, given a function y " fpxq defined for values of x in the interval ra, bs, its average fra,b<sup>s</sup> (over the interval ra, bs) is defined to be

$$f\_{[a,b]} = \frac{\int\_a^b f(x)dx}{|b-a|}.$$

When we talk about "average speed", we are thinking of speed changing as a function of time; and the total distance covered in any given time interval ra, bs is equal to the area under the graph. We want a single "average speed" vra,b<sup>s</sup> (a constant function) that would cover the same distance in the same time as the more complicated reality of varying speed. That is,


$$v\_{[a,b]} = \frac{\int\_a^b v(t)dt}{|b-a|}.$$

In Problem 84 the walking speed is misleadingly given in terms of "up" and "down" – which represent the first half distance travelled, and the second half distance travelled. The careful solver knows that s/he has to find "total distance travelled" and divide by "total time taken"; but s/he may not notice that s/he has in fact reinterpreted the given information so that speed is seen as a function of time (rather than of distance).

85.

(a)(i) Let the distance covered on each lap be m km. Then the first lap takes me m <sup>40</sup> hours; the second lap takes me <sup>m</sup> <sup>30</sup> hours; the third lap takes me <sup>m</sup> <sup>20</sup> hours. So the total time taken for the three laps is

$$\frac{m}{40} + \frac{m}{30} + \frac{m}{20} = \frac{13m}{120} \text{ hours.}$$

Hence my average speed for the race covering 3m km is

$$\frac{3m}{\left(\frac{13m}{120}\right)} = \frac{360}{13} \text{ km/h.}$$

Note: Alternatively, because the two factors of m in the numerator and the denominator cancel each other, this answer may be formulated as the harmonic mean of the given speeds:

$$\frac{3}{\left[\frac{1}{40} + \frac{1}{30} + \frac{1}{20}\right]} \cdot$$

(ii) In the first hour I cycle 40 km; in the second hour I cycle 30 km; in the third hour I cycle 20 km. So in the three hours I cycle 40 ` 30 ` 20 " 90 km. So my average speed is 30 km/h.

Note: Alternatively, as long as the three time intervals t are equal, we land up with t as a factor in both the numerator and the denominator, so these common factors cancel out, and the answer is simply the arithmetic mean of the given speeds:

$$\frac{20+30+40}{3}$$

	- (ii) Again (unless u " v), the first cyclist spends more time cycling at the higher speed. Hence the first cyclist wins.

$$\frac{2}{\left[\frac{1}{u} + \frac{1}{v}\right]}\text{ km/h.}$$

Hence, part (b)(ii) shows that

$$\frac{u+v}{2} \gg \frac{2}{\left[\frac{1}{u} + \frac{1}{v}\right]} = \frac{2uv}{u+v}.$$

(ii) If we rearrange the required inequality

$$
\frac{u+v}{2} \geqslant \frac{2uv}{u+v},
$$

then we see that it is equivalent to proving that pu`vq <sup>2</sup> ě 4uv. This suggests that we should start with the universally true statement:

$$\left(\left(u-v\right)^{2}\geqslant 0\text{ for all }u,v\geqslant 0.$$

Adding 4uv to both sides yields pu ` vq <sup>2</sup> ě 4uv. Multiplying both sides by the non-negative quantity <sup>1</sup> 2pu`vq then gives the required inequality.

86. The only "modelling" required here is to translate the problem using the standard equations of kinematics. For motion from rest we have

(i) v " at, where t is the time, a is the uniform acceleration, and v the final speed, and

(ii) s " 1 2 at<sup>2</sup> , where s is the distance travelled.

There is a question as to what units we should use. For the moment we stick to measuring v in km/h as given, s in km, t in hours, and a in the (unfamiliar) units of km/h<sup>2</sup> : so 72 " at and 4 " 1 2 at<sup>2</sup> .

Dividing the second equation by the first gives <sup>1</sup> <sup>18</sup> " 1 2 t, so t " 1 9 hours (" 400 seconds).

Substituting in the first equation gives a " 72 ˆ 9 km/h<sup>2</sup> (" 1 <sup>20</sup> m/sec<sup>2</sup> ).

Note: Equations (i) and (ii) can be summarised as saying that, under uniform acceleration a, the distance travelled is s " p <sup>1</sup> 2 atq ˆ t. Hence the average speed for the complete journey is equal to exactly half of the final speed v " at.

In general, those tackling the problem may agree that the familiar units of speed and distance do not give us a very good gut feeling for the scale of acceleration. If we measure acceleration in km/h<sup>2</sup> , then we get huge numbers for acceleration which one cannot easily relate to. And if we switch to m (metres), m/sec, and m/sec<sup>2</sup> , then we get rather small numbers for the acceleration, which again convey relatively little.

[The original (Russian) version of this problem had the train travelling 2.1 km and reaching a speed of 54 km/h. This produces a nice answer for the time taken, but a relatively inscrutable answer for the acceleration. So we have changed the parameters.]

### 87.

''We explain why, when a vehicle accelerates from 0 to 20 mph, its average speed is more than 10 mph. In general, the average speed of an accelerating vehicle is more than half the final speed after the acceleration.

Consider first the case when the acceleration is constant: this means that the graph which represents the speed of the vehicle as a function of time is a straight line:

In that case, the distance travelled is equal to the area under the graph. But from the formula for the area of a triangle we know that this area equals the area of the rectangle with the same base and half the height of the triangle:

This means that the average speed in that case is exactly half of the final (maximum) speed.

But a car has higher acceleration in lower gears, that is, at smaller speeds. Therefore the graph of speed as a function of time is concave, and the area under the graph is greater than in the case of constant acceleration. Hence, while reaching the same speed, the car travels further and its average speed is higher:

We come to the conclusion that the average speed of an accelerating car is greater than half its speed at the end of acceleration."

Note: The text of this solution is reproduced from the appendix to a document prepared for, and submitted to, the Crown Prosecution Service in England. This may partly explain why it contains not a single formula. It was written by a student studying economics, and the mixture of language and graphs used illustrates the typical economist's way of thinking. Economists rarely have complete data, so they tend to rely on a combination of common sense and the basic patterns of economic variables – such as the "convexity" or "concavity" of functions. Indeed some chapters of mathematical economics could be described as outlining "the kinematics of money", and have surprising similarities to mechanics.

88. Suppose sunrise was t hours before noon – so that the first woman covers the total distance in t ` 4 hours, while the second covers the same distance in t ` 9 hours.

We know nothing about the distance from A to B, so it makes sense to choose this distance as our unit.

Then the first woman's speed is <sup>1</sup> t`4 , while the second woman's speed is <sup>1</sup> t`9 units per hour.

The relative speed of A and B (the speed with which the distance between them changes) is <sup>1</sup> <sup>t</sup>`<sup>4</sup> ` 1 t`9 .

They meet at noon, so in t hours, the distance between them reduces from 1 unit to 0.

Hence

$$1 = t \times \left(\frac{1}{t+4} + \frac{1}{t+9}\right);$$

that is, t <sup>2</sup> " 36, so t " 6, and sunrise was at p12 ´ 6q " 6 am.

89. Let us introduce a new measure of distance – which we call a league. (Readers may know from old documents or from poetry that this was an old measure of distance for journeys, without knowing exactly how far it was; so we feel free to use it as an abstract unit of unknown size.)

To mesh distance and time, the journey from St Louis to New Orleans needs to be some multiple of 7, and the journey from New Orleans to St Louis needs to be some multiple of 5. Hence we choose the distance to be equal to 5 ˆ 7 " 35 "leagues".

Then the speed of the paddle-steamer upstream is:

$$\frac{35}{7} = 5 \text{ \textquotedblleft leagues per day\textquotedblright}$$

and the speed downstream is:

$$\frac{35}{5} = 7 \text{ \textquotedblleft leagues per day\textquotedblright.}$$

The speed of the current gets subtracted from the speed of the paddle-steamer going upstream, and gets added to the speed of the paddle-steamer going downstream; so the speed of the current is:

$$\frac{7-5}{2} = 1 \text{ \textquotedblleft leguse per day\textquotedblright.}$$

Hence a raft will drift from St Louis to New Orleans in <sup>35</sup> <sup>1</sup> " 35 days.

Note: This elegant solution involves the introduction of a hidden intermediate parameter, an unknown quantity which helps us reason about the problem. The parameter is apparently the distance (from St Louis to New Orleans); but it is in fact a measure of distance chosen so as to be compatible with the time taken.

The art of identifying, and choosing, relevant "hidden parameters", and the analysis of their relation to the data, and their mutual relations, constitute an important and challenging part of the mathematical modelling process.

Notice that if we reformulate the problem in more general terms, with the paddle-steamer taking "a days" downstream and "b days" upstream, then the answer "d days" (for the time to drift downstream) happens to be the harmonic mean of the quantities "a" and "´b":

$$d = \frac{2}{\frac{1}{a} + \frac{1}{=b}}.$$

90. [This is "Problem 108" in Paolo dell'Abbaco's Trattato d'aritmetica (c.1370), with a rough translation of the solution procedure given there courtesy of Roy Wagner.]

"Do the following: multiply 5 by 8, which makes 40. Then say thus: in 40 days one will make the trip 8 times, and the other 5 times, so both together will make the trip 13 times.

Now say: if 40 days equals 13 trips, how many days are needed [on average] for one trip? And so multiply 1 times 40, which makes 40; then divide this by 13, which makes 3 days and <sup>1</sup> <sup>13</sup> of a day.

And so I say that in 3 days and <sup>1</sup> <sup>13</sup> of a day the two will come together.

And as this is done, so all similar problems are done."

Note: The problem as stated conveys an air of reality by giving the distance "from here to Florence" in miles; but this fact is not mentioned in the solution! Instead, the solution starts by introducing a hidden parameter, measured by a dimensionless unit: a trip.

This move (to invent a natural unit of measurement) also featured in Problem 89 above and has deep mathematical reasons. Problem 89 was borrowed from an interview with Vladimir Arnold (Notices of the AMS, vol. 44, no. 4), where we read:

Interviewer: Please tell us a little bit about your early education. Were you already interested in mathematics as a child?

Arnold: [. . . ] The first real mathematical experience I had was when our schoolteacher I.V. Morotzkin gave us the following problem [VA then formulated Problem 89].

I spent a whole day thinking on this oldie, and the solution (based on what are now called scaling arguments, dimensional analysis, or toric variety theory, depending on your taste) came as a revelation.

The feeling of discovery that I had then (1949 ) was exactly the same as in all the subsequent much more serious problems – be it the discovery of the relation between algebraic geometry of real plane curves and four-dimensional topology (1970 ), or between singularities of caustics and of wave fronts and simple Lie algebras and Coxeter groups (1972 ). It is the greed to experience such a wonderful feeling more and more times that was, and still is, my main motivation in mathematics.

Arnold refers here to scaling arguments or dimensional analysis: that is, the mathematical art of choosing and analysing the use of units of measurement. This has its origins in, and includes as an integral part, Euclid's classical theory of proportion.

91. Suppose as before that the sun rises t hours before noon; but replace 4 pm (the time the woman starting at A arrived at B) by a pm, and replace 9 pm (the time the woman starting at B arrived at A) by b pm. Let C be the point where they meet (at noon).

Then, since each woman walks at a constant speed, we have

$$\frac{t}{a} = \frac{|AC|}{|CB|} \text{ (for the woman starting from } A\text{)},$$

and

t b " |BC| |CA| (for the woman starting from Bq.

Hence

$$\frac{t}{a} = \frac{|AC|}{|CB|} = \frac{b}{t},$$

so t <sup>2</sup> " ab.

Note: This totally unexpected result validates the choice of the unknown t as the time in hours from sunrise to noon. Not knowing its significance in advance, this choice was motivated by the observation that "noon" occurs in the problem as the only common "origin", or reference point for time data.

The first rule of intelligent tinkering is to save all the parts. Paul R. Ehrlich (1932– )

Many important aspects of serious mathematics have their roots in the world of arithmetic. However, when we implement an arithmetical procedure by combining numbers with very different meanings to produce a single numerical output, it becomes almost impossible to see how the separate ingredients contribute to the final answer. In other words, calculating exclusively with numbers contravenes Paul Ehrlich's "first rule of intelligent tinkering". This is why in Chapters 1 and 2 we stressed the need to move beyond blind calculation, and to begin to think structurally – even when calculating purely with numbers. Algebra can be seen as a remarkable way of "tinkering with numbers", so that we not only "keep all the parts", but manage to keep them separate (by giving them different names), and hence can see clearly what contribution each ingredient variable makes to the final output. To benefit from this feature of algebra, we need to learn to "read" algebraic expressions, and to interpret what they are telling us – in much the same way that we learn to read numbers (so that, where appropriate, 100 is seen as 10<sup>2</sup> , and 10 is seen as 1 ` 2 ` 3 ` 4).

Before algebra proper was invented (around 1600), the ability to extract the general picture lying hidden inside each calculation was restricted to specialists. The ancient Babylonians (1700–1500 BC) described their general procedures as recipes, presented in the context of problems involving particular numbers. But they did this in such a way as to demonstrate convincingly that whoever formulated the procedure had managed to see "the general in the particular". The ancient Greeks used a geometrical setting to reveal generality, and encoded what we would see as "algebraic" methods in geometrical language. In the 9th century AD, Arabs such as Al-Khwarizmi (c.780–c.850), managed to encapsulate generality using a very limited kind of algebra, without the full symbolical language that would emerge later. The abacists, such as Paolo dell'Abbaco (1282–1374) who featured in Chapter 3, clearly saw that the power and spirit of mathematics was rooted in this generality. But modern algebraic symbolism – in particular, the idea that to express generality we need to use letters to represent not only variables, but also important parameters (such as the coefficients a, b, c in a general quadratic ax<sup>2</sup> ` bx ` c) – had to wait for the inscrutable writings of Vi`ete (1540–1603), and especially for Fermat (1601–1665) and Descartes (1596–1650) who simplified and extended Vi`ete's ideas in the 1630s.

Within a generation, the huge potential of this systematic use of symbols was revealed by the triumphs of Newton (1642–1727), Leibniz (1646–1716), and others in the years before 1700. Later, the refinements proposed by Euler (1707–1783) in his many writings throughout the 18th century, made this new language and its discoveries accessible to us all – much as Stevin's (1548–1620) version of place value for numbers made calculation accessible to Everyman.

Our coverage of algebra is necessarily selective. We focus on a few ideas that are needed in what follows, and which should ideally be familiar – but with an emphasis that may be less familiar. When working algebraically, the key mathematical messages are mostly implicit in the manipulations themselves. Hence many of the additional comments in this chapter are to be found as part of the solutions, rather than within the main text.

### 4.1. Simultaneous linear equations and symmetry

Problem 92 Dad took our new baby to the clinic to be weighed. But the baby would not stay still and caused the needle on the scales to wobble. So Dad held the baby still and stood on the scales, while nurse read off their combined weight: 78kg. Then nurse held the baby, while Dad read off their combined weight: 69kg. Finally Dad held the nurse, while the baby read off their combined weight: 137kg. How heavy was the baby? 4

The situation described in Problem 92 is representative of a whole class of problems, where the given information incorporates a certain symmetry, which the solver would be wise to respect. Hence one should hesitate before applying systematic brute force (as when using the information from one weighing to substitute for one of the three unknown weights – a move which effectively reduces the number of unknowns, but which fails to respect the symmetry in the data).

A similar situation arises in certain puzzles like the following.

Problem 93 Numbers are assigned (secretly) to the vertices of a polygon. Each edge of the polygon is then labelled with the sum of the numbers at its two end vertices.


In case any reader is inclined to dismiss such problems as "artificial puzzles", it may help to recall two familiar instances (Problems 94 and 96) which give rise to precisely the above situation.

Problem 94 In the triangle ABC with sides of lengths a (opposite A), b (opposite B), and c (opposite C), we want to locate the three points where the incircle touches the three sides – at point P (on BC), Q (on CA), and R (on AB). To this end, let the two tangents to the incircle from A (namely AQ and AR) have length x, the two tangents from B (namely BP and BR) have length y, and the two tangents from C (namely CP and CQ) have length z. Find the values of x, y, z in terms of a, b, c. 4

The second instance requires us first to review the basic properties of midpoints in terms of vectors.

### Problem 95


"MN is parallel to Y Z and half its length".

(c) Given any quadrilateral ABCD, let P be the midpoint of AB, let Q be the midpoint of BC, let R be the midpoint of CD, and let S be the midpoint of DA. Prove that P QRS is always a parallelogram. 4

### Problem 96


The previous five problems explore a common structural theme – namely the link between certain sums (or averages) and the original, possibly unknown, data. However this algebraic link was in every case embedded in some practical, or geometrical, context. The next few problems have been stripped of any context, leaving us free to focus on the underlying structure in a purely algebraic, or arithmetical, spirit.

Problem 97 Solve the following systems of simultaneous equations.

$$\begin{aligned} \text{(a)} \ (\text{i)} \ x+y=1, \quad y+z=2, \quad x+z=3\\ \text{(ii)} \ uv=2, \quad vw=4, \quad uw=8\\ \text{(b)} \ (\text{i)} \ x+y=2, \quad y+z=3, \quad x+z=4\\ \text{(ii)} \ uv=6, \quad vw=10, \quad uw=15\\ \text{(iii)} \ uv=6, \quad vw=10, \quad uw=30\\ \text{(iv)} \ uw=4, \quad vw=8, \quad uw=16 \end{aligned}$$

Problem 98 Use what you know about solving two simultaneous linear equations in two unknowns to construct the general positive solution to the system of equations:

$$u^a v^b = m, \quad u^c v^d = n.$$

Interpret your result in the language of Cramer's Rule. (Gabriel Cramer (1704–1752)). 4

### Problem 99

(a) For which values b, c does the following system of equations have a unique solution?

$$x + y + z = 3, \quad xy + yz + zx = b, \quad x^2 + y^2 + z^2 = c$$

(b) For which values a, b, c does the following system of equations have a unique solution?

$$x + y + z = a, \quad xy + yz + zx = b, \quad x^2 + y^2 + z^2 = c \qquad \triangle y$$

### 4.2. Inequalities and modulus

The transition from school to university mathematics is in many ways marked by a shift from simple variables, equations and functions, to conditions and analysis involving inequalities and modulus.

Problem 100 What is | ´ x| equal to: x or ´x? (What if x is negative?) 4

### 4.2.1 Geometrical interpretation of modulus, of inequalities, and of modulus inequalities

### Problem 101

(a) Mark on the coordinate line all those points x in the interval r0, 1q which have the digit "1" immediately after the decimal point in their decimal expansion. What fraction of the interval r0, 1q have you marked?

Note: "r0, 1q" denotes the set of all points between 0 and 1, together with 0, but not including 1. r0, 1s denotes the interval including both endpoints; and p0, 1q denotes the interval excluding both endpoints.


Problem 102 Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:

$$\forall x > 1, \ x > 2, \ x > 3, \ x > 4, \ x > 5, \ x > 6, \ x > 7. \tag{7.1}$$

Problem 103 Mark on the coordinate line all those points x for which

$$|x - 5| = 3. \tag{7}$$

### Problem 104

(a) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:

$$|x| \succ 1, \ |x| \succ 2, \ |x| \succ 3, \ |x| \succ 4, \ |x| \succ 5, \ |x| \succ 6, \ |x| \succ 7.$$

(b) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:

$$\begin{array}{c} \neg|x-1| > 1, \ |x-2| > 2, \ |x-3| > 3, \ |x-4| > 4, \ |x-5| > 5, \ |x-6| > 6, \ |x-7| > \frac{7}{\Delta} \\\\ \end{array}$$

Problem 105 Mark on the coordinate line all those points x for which

$$|x+1|+|x+2|=2. \tag{\Delta}$$

Problem 106 Find numbers a and b with the property that the set of solutions of the inequality

$$|x - a| < b$$

consists of the interval p´1, 2q. 4

### Problem 107

(a) Mark on the coordinate plane all points px, yq satisfying the inequality

$$|x - y| < 3.$$

(b) Mark on the coordinate plane all points px, yq satisfying the inequality

$$|x - y + 5| < 3.$$

(c) Mark on the coordinate plane all points px, yq satisfying the inequality

$$|x - y| < |x + y|. \tag{7}$$

#### 4.2.2 Inequalities

Problem 108 Suppose real numbers a, b, c, d satisfy <sup>a</sup> <sup>b</sup> ă c d .

(i) Prove that

$$\frac{a}{b} < \frac{\left(\frac{a}{b} + \frac{c}{d}\right)}{2} < \frac{c}{d}.$$

(ii) If b, d ą 0, prove that

$$
\frac{a}{b} < \frac{a+c}{b+d} < \frac{c}{d}.\tag{7}
$$

Problem 109 (Farey series) When the fully cancelled fractions in r0, 1s with denominator ď n are arranged in increasing order, the result is called the Farey series (or Farey sequence) of order n.



$$
\frac{1}{9}, \frac{2}{9}, \frac{3}{9}, \dots, \frac{8}{9}.
$$

Into which of the ten subintervals do they fall?

(ii) Imagine the n points

$$\frac{1}{n+1}, \frac{2}{n+1}, \frac{3}{n+1}, \dots, \frac{n}{n+1}$$

dividing the interval r0, 1s into n ` 1 subintervals of length <sup>1</sup> n`1 . Now insert the n ´ 1 points

$$\frac{1}{n}, \frac{2}{n}, \frac{3}{n}, \dots, \frac{n-1}{n}.$$

Into which of the n ` 1 subintervals do they fall?

	- (i) In the Farey series of order n the first two fractions are <sup>0</sup> <sup>1</sup> ă 1 n , and the last two fractions are <sup>n</sup>´<sup>1</sup> <sup>n</sup> ă 1 1 . Prove that every other adjacent pair of fractions <sup>a</sup> <sup>b</sup> ă c d in the Farey series of order n satisfies bd ą n.
	- (ii) Let <sup>a</sup> <sup>b</sup> ă c d be adjacent fractions in the Farey series of order n. Prove (by induction on n) that bc ´ ad " 1.

(d) Prove that if

$$\frac{a}{b} < \frac{c}{d} < \frac{e}{f}$$

are three successive terms in any Farey series, then

$$\frac{c}{d} = \frac{a+e}{b+f}.\tag{\Delta}$$

Problem 110 Solve the following inequalities.

$$\begin{aligned} \text{(a) } &x + \frac{1}{x} < 2\\ \text{(b) } &x \leqslant 1 + \frac{2}{x} \\ \text{(c) } &\sqrt{x} < x + \frac{1}{4} \end{aligned} $$

### Problem 111


2 " 1 <sup>a</sup> ` 1 b ‰ " 2ab a ` b ď ? ab ď a ` b 2 ď a <sup>2</sup> ` b 2 2 pHM ď GM ď AM ď QMq 4

c

Problem 112 The two hundred numbers

$$1, \ 2, \ 3, \ 4, \ 5, \ \dots, \ 200$$

are written on the board. Students take turns to replace two numbers a, b from the current list by their sum divided by ? 2. Eventually one number is left on the board. Prove that the final number must be less than 2000. 4

### 4.3. Factors, roots, polynomials and surds

### Problem 113

	- (ii) Find another such prime.
	- (ii) Find another such prime.
	- (ii) Find another such prime.
	- (ii) Find another such prime. 4

Problem 114 Factorise x <sup>4</sup> ` 1 as a product of two quadratic polynomials with real coefficients. 4

### 4.3.1 Standard factorisations

The challenge to factorise unfamiliar expressions, may at first leave us floundering. But if we assume that each such problem is solvable with the tools at our disposal, we then have no choice but to fall back on the standard tools we have available (in particular, the standard factorisation of a difference of two squares, in which "cross terms" cancel out). The next problem extends this basic repertoire of standard factorisations.

### Problem 115

	- (ii) Factorise a <sup>4</sup> ´ b <sup>4</sup> as a product of one linear factor and one factor of degree 3, and as a product of two linear factors and one quadratic factor.
	- (ii) Factorise a <sup>5</sup> ` b <sup>5</sup> as a product of one linear factor and one factor of degree 4.
	- (iii) Factorise a <sup>2</sup>n`<sup>1</sup> ` b <sup>2</sup>n`<sup>1</sup> as a product of one linear factor and one factor of degree 2n. 4

Problem 115 develops the ideas that were implicit in Problem 113. The clue lies in Problem 113(a), and in the comment made in the main text in Chapter 1 (after Problem 4 in Chapter 1), which we repeat here:

"The last part [of Problem 113(a)] is included to emphasise a frequently neglected message:

Words and images are part of the way we communicate. But most of us cannot calculate with words and images.

To make use of mathematics, we must routinely translate words into symbols. So "numbers" need to be represented by symbols, and points in a geometric diagram need to be properly labelled before we can begin to calculate, and to reason, effectively."

As soon as one reads the words "one less than a square", one should instinctively translate this into the form "x <sup>2</sup> ´ 1". Bells will then begin to ring; for it is impossible to forget the factorisation

$$x^2 - 1 = (x - 1)(x + 1).$$

And it follows that:

for a number that factorises in this way to be prime, the smaller factor x ´ 1 must be equal to 1;

6 x " 2, so there is only one such prime.

The integer factorisations in Problem 113(c) – namely

$$3^3 - 1 = 2 \times 13, \; 4^3 - 1 = 3 \times 21, \; 5^3 - 1 = 4 \times 31, \; 6^3 - 1 = 5 \times 43, \; \dots$$

may help one to remember (or to discover) the related factorisation

x <sup>3</sup> ´ 1 " px ´ 1qpx <sup>2</sup> ` x ` 1q.

6 For a number that factorises in this way to be prime, the smaller factor "x ´ 1" must be equal to 1;

6 x " 2, so there is only one such prime.

Problem 113 parts (a) and (c) highlight the completely general factorisation (Problem 115(a)(iii)):

$$x^n - 1 = (x - 1)(x^{n-1} + x^{n-2} + \dots + x^2 + x + 1).$$

This family of factorisations also shows that we should think about the factorisation of x <sup>2</sup> ´ 1 as px ´ 1qpx ` 1q, with the uniform factor px ´ 1q first (rather than as px ` 1qpx ´ 1qq. Similarly, the results of Problem 115 show that we should think of the familiar factorisation of a <sup>2</sup> ´ b <sup>2</sup> as pa ´ bqpa ` bq, (not as pa ` bqpa ´ bq, but always with the factor pa ´ bq first).

The integer factorisations in Problem 113(d) – namely

$$3^3 + 1 = 4 \times 7, 4^3 + 1 = 5 \times 13, 5^3 + 1 = 6 \times 21, 6^3 + 1 = 7 \times 31, 7^3 + 1 = 8 \times 43, \dots$$

may help one to remember (or to discover) the related factorisation

$$x^3 + 1 = (x+1)(x^2 - x + 1).$$

6 For such a number to be prime, one of the factors must be equal to 1.

This time one has to be more careful, because the first bracket may not be the "smaller factor" – so there are two cases to consider:


The factorisation for x <sup>3</sup> ` 1 works because "3 is odd", which allows the alternating `{´ signs to end in a "`" as required. Hence Problem 113(d)(iii) highlights the completely general factorisation for odd powers:

$$x^{2n+1} + 1 = (x+1)(x^{2n} - x^{2n-1} + x^{2n-2} - \dots + x^2 - x + 1).$$

You probably know that there is no standard factorisation of x <sup>2</sup> ` 1, or of x <sup>4</sup> ` 1 (but see Problem 114 above).

### Problem 116

(a) Derive a closed formula for the sum of the geometric series

$$1 + r + r^2 + r^3 + \dots + r^n.$$

(The meaning of closed formula was discussed in the Note to the solution to Problem 54(b) in Chapter 2.)

(b) Derive a closed formula for the sum of the geometric series

$$(a+ar+ar^2+ar^3+\cdots+ar^n.\,\qquad\qquad\qquad\Delta$$

We started this subsection by looking for prime numbers of the form x <sup>2</sup> ´ 1. A simple-minded approach to the distribution of prime numbers might look for formulae that generate primes – all the time, or infinitely often, or at least much of the time. In Chapter 1 (Problem 25) you showed that no prime of the form 4k ` 3 can be "represented" as a sum of two squares (i.e. in the form "x <sup>2</sup> ` y <sup>2</sup>"), and we remarked that every other prime can be so represented in exactly one way. It is true (but not obvious) that roughly half the primes fall into the second category; so it follows that substituting integers for the two variables in the polynomial x <sup>2</sup> ` y <sup>2</sup> produces a prime number infinitely often.

Problem 117 Experiment suggests, and Goldbach (1690–1764) showed in 1752 that no polynomial in one variable, and with integer coefficients, can give prime values for all integer values of the variable. But Euler (1707–1783) was delighted when he discovered the quadratic

$$f(x) = x^2 + x + 41.$$

Clearly fp0q " 41 is prime. And fp1q " 43 is also prime. What is the first positive integer n for which fpnq is not prime? 4

Problem 117 should be seen as a particular instance of the question as to whether prime numbers can be captured by a polynomial with integer coefficients, and in particular by a quadratic. The next two problems consider the simplest instances of representing prime numbers by expressions involving exponentials (that is, where the variable is in the exponent).

### Problem 118

	- (ii) How many primes are there among the first five such numbers

$$2^2 - 1, \; 2^3 - 1, \; 2^5 - 1, \; 2^7 - 1, \; 2^{11} - 1?$$

	- (ii) In the simplest case, where a " 2, how many primes are there among the first five such numbers

2 <sup>1</sup> ` 1, 2 <sup>2</sup> ` 1, 2 <sup>4</sup> ` 1, 2 <sup>8</sup> ` 1, 2 <sup>16</sup> ` 1? 4

Primes of the form 2<sup>p</sup>´1 are called Mersenne primes (after Marin Mersenne (1588–1648)). We now know at least fifty such primes (with the exponent p ranging up to around 80 million). Finding new primes is not in itself important, but the search for Mersenne primes has been used as a focus for many new developments in programming, and in number theory.

Primes of the form 2<sup>n</sup> ` 1 are called Fermat primes (after Pierre de Fermat (1601–1665)). The story here is very different. We now refer to the number 2 <sup>n</sup> ` 1 with n " 2 <sup>k</sup> as the k th Fermat number fk. You showed in Problem 118 (as Fermat did himself) that f0, f1, f2, f3, f<sup>4</sup> are all prime. Fermat then rather rashly claimed that f<sup>n</sup> is always prime. However, Euler showed (100 years later) that the very next Fermat number f<sup>5</sup> fails to be prime. And despite all the power of modern computers, we have still not found another Fermat number that is prime!

### 4.3.2 Quadratic equations

The general solution of quadratic equations dates back to the ancient Babylonians (« 1700 BC). Our modern understanding depends on two facts:


Problem 119 Solve the following quadratic equations:

(a) x <sup>2</sup> ´ 3x ` 2 " 0 (b) x <sup>2</sup> ´ 1 " 0 (c) x <sup>2</sup> ´ 2x ` 1 " 0 (d) x <sup>2</sup> ` ? 2x ´ 1 " 0 (e) x <sup>2</sup> ` x ´ ? 2 " 0 (f) x <sup>2</sup> ` 1 " 0 (g) x <sup>2</sup> ` ? 2x ` 1 " 0 4

### Problem 120 Let

$$p(x) = x^2 + \sqrt{2}x + 1.$$

Find a polynomial qpxq such that the product ppxqqpxq has integer coefficients. 4

### Problem 121

(a) I am thinking of two numbers, and am willing to tell you their sum s and their product p. Express the following procedure algebraically and explain why it will always determine my two unknown numbers.

Halve the sum s, and square the answer. Then subtract the product p and take the square root of the result, to get the answer. Add "the answer" to half the sum and you have one unknown number; subtract "the answer" from half the sum and you have the other unknown number.

(b) I am thinking of the length of one side of a square. All I am willing to tell you are two numbers b and c, where when I add b times the side length to the area I get the answer c. Express the following procedure algebraically and explain why it will always determine the side length of my square.

Take one half of b, square it and add the result to c. Then take the square root. Finally subtract half of b from the result.

	- (i) Prove that the diagonal AC is parallel to the side ED.
	- (ii) If AC and BD meet at X, explain why AXDE is a rhombus.
	- (iii) Prove that triangles ADX and CBX are similar.
	- (iv) If AC has length x, set up an equation and find the exact value of x.

4

Problem 121(a), (b) link to Problem 111(a) (and to Problem 129 below), in relating the roots and the coefficients of a quadratic. If we forget for the moment that the coefficients are usually known, while the roots are unknown, then we see that if α and β are the roots of the quadratic

$$x^2 + bx + c,$$

then

$$(x - \alpha)(x - \beta) = x^2 + bx + c, \quad (x + \alpha)(x - \beta) = 0$$

so

$$
\alpha + \beta = -b \text{ and } \alpha\beta = c.
$$

In other words, the two coefficients b, c are equal to the two simplest symmetric expressions in the two roots α and β. Part (a) of the next problem is meant to suggest that all other symmetric expressions in α and β (that is, any expression that is unchanged if we swap α and β) can then be written in terms of b and c. The full result proving this fact is generally attributed to Isaac Newton (1642–1727). Part (b) suggests that, provided one is willing to allow case distinctions, something similar may be true of anti-symmetric expressions (where the effect of swapping α and β is to multiply the expression by "´1").

Problem 122 Let α and β be the roots of the quadratic equation

$$x^2 + bx + c = 0.$$

	- (ii) Write α <sup>2</sup>β ` β <sup>2</sup>α in terms of b and c only.
	- (iii) Write α <sup>3</sup> ` β <sup>3</sup> ´ 3αβ in terms of b and c only.
	- (ii) Write α <sup>2</sup>β ´ β <sup>2</sup>α in terms of b and c only.
	- (iii) Write α <sup>3</sup> ´ β 3 in terms of b and c only. 4

#### Problem 123 (Nested surds, simplification of surds)

(a)(i) For any positive real numbers a, b, prove that

$$
\sqrt{a} + \sqrt{b} = \sqrt{a + b + \sqrt{4ab}}
$$

	- (ii) Simplify <sup>a</sup> 5 ´ 16 and <sup>a</sup> 6 ´ 20. 4

Problem 124 (Integer polynomials with a given root) We know that α " 1 is a root of the polynomial equation x <sup>2</sup> ´ 1 " 0; that α " ? 2 is a root of x <sup>2</sup> ´ 2 " 0; and that α " ? 3 is a root of x <sup>2</sup> ´ 3 " 0.

(a) Find a quadratic polynomial with integer coefficients which has

$$\alpha = 1 + \sqrt{2}$$

as a root.

(b) Find a quadratic polynomial with integer coefficients which has

$$\alpha = 1 + \sqrt{3}$$

as a root.

(c) Find a polynomial with integer coefficients which has

$$\alpha = \sqrt{2} + \sqrt{3}$$

as a root. What are the other roots of this polynomial?

(d) Find a polynomial with integer coefficients which has

$$\alpha = \sqrt{2} + \frac{1}{\sqrt{3}}$$

as a root. What are the other roots of this polynomial? 4

#### Problem 125


#### Problem 126 (Polynomial long division) Find


Problem 127 Find the remainder when we divide x <sup>2013</sup> ` 1 by x <sup>2</sup> ` x ` 1. 4

### 4.4. Complex numbers

Up to this point, the chapter and solutions have largely avoided mentioning complex numbers. However, the present chapter would be incomplete were we not to interpret some of the earlier material in terms of complex numbers. Readers who have already met complex numbers will probably still find much in the next two sections that is new. Those for whom complex numbers are as yet unfamiliar should muddle through as best they can, and may then be motivated to learn more in due course.

We already know that the square x <sup>2</sup> of any real number x is ě 0.


And that is where the matter would have rested.

From a modern perspective, we can see that complex numbers are implicit in the formula for the roots of a quadratic equation: complex numbers become explicit as soon as the coefficients of a quadratic ax<sup>2</sup> ` bx ` c give rise to a negative discriminant b <sup>2</sup> ´ 4ac ă 0.

But this may not have been quite how complex numbers were discovered. Contrary to oft-repeated myths, complex numbers may not have forced themselves on our attention by someone asking about "solutions to the quadratic equation x <sup>2</sup> " ´1". As long as we inhabit the domain of real numbers, we can be sure that no known number x could possibly have such a square, so we are unlikely to go in search of it.

New ideas in the history of mathematics tend to emerge when a fresh analysis of familiar entities forces us to consider the possible existence of some previously unsuspected universe. In the time from the ancient world up to the fifteenth century, the idea of "number", and of calculation, was restricted to the world of real (usually positive) numbers. In such a world, quadratic equations with non-real solutions simply could not arise.

However, in the Brave New World of the Renaissance, where novelty, exploration, and discovery were part of the Zeitgeist, a general method for solving cubic equations was part of the as-yet-undiscovered "wild west" of mathematics, part of the mathematical New World which invited exploration. Notice that this was not a wildly speculative venture (like trying to solve the meaningless equation "x <sup>2</sup> " ´1"), since a cubic polynomial always has at least one real root. After three thousand years in which little progress had been made, the first half of the sixteenth century witnessed an astonishing burst of progress, resulting in the solution not only of cubic equations, but also of quartic equations. We postpone the details until Section 4.5. All we note here is that,

the general method for solving cubic equations published in 1545, was given as a procedure, illustrated by examples, that showed how to find genuinely real solutions to equations of the third degree having genuinely real (and positive) coefficients.

The procedure clearly worked. And it proceeded as follows:

Construct the real solution x as the sum x " u ` v of two intermediate answers u and v – where the two summands u and v sometimes turned out to be what we would call "conjugate complex numbers", whose imaginary parts cancelled out, leaving a real result for the required root x.

Those who devised the procedure had no desire to leave the real domain: they were focused on a problem in the real domain (a cubic equation with real coefficients, having a real root), and devised a general procedure to find that genuinely real root. But the procedure they discovered led the solver on a journey that sometimes "passed into the complex domain", before returning to the real domain! (See Problem 129.)

Working with complex numbers depends on two skills – one very familiar, and one less so.

• The familiar skill is a willingness to work with a "number" in terms of its properties only, without wishing to evaluate it.

We are thoroughly familiar with this when we work with <sup>2</sup> 3 and other fractions: we know that <sup>2</sup> <sup>3</sup> " 2ˆ 1 3 ; and all we know about <sup>1</sup> 3 is that "whenever we have 3 copies of <sup>1</sup> 3 , we can simplify this to 1". Much the same happens when we first learn to work with ? 2, where we carry out such calculations as p1 ` ? 2q <sup>2</sup> " 3 ` 2 ? 2, based only on collecting up like terms and the fact that ? 2 ˆ ? 2 can always be replaced by 2.

	- First, these new solutions come in pairs: if i is one solution of x <sup>2</sup> " ´1 then ´i is another (because p´1q ˆ p´1q " 1 means that p´xq <sup>2</sup> " x 2 for all "numbers" x).
	- Second, the equation x <sup>2</sup> " ´1 has exactly two solutions – one the negative of the other (if x and y are both solutions, then x <sup>2</sup> " y 2 , so x <sup>2</sup> ´ y <sup>2</sup> " px ´ yqpx ` yq " 0, so either x " y, or x " ´y).
	- Third, we have no way of telling these two solutions apart: we know that each is the negative of the other, but there is no way of singling out one of them as "the main one" (as we could when defining the square root of a positive real such as 2). We can call them ˘i, but they are each as good as the other. This important fact is often undermined by referring to one of these roots as ? ´1 (as if it were the dominant partner), and to the other as ´ ? ´1 (as if it were somehow just the "negative" of the main root).

The truth is that "? ´1" is a serious abuse of notation, because there is no way to extend the definition of the function " ? " in the way that this implies: when we try to "take square roots" of negative (or complex) numbers, the output is inescapably "two-valued", so "? " is no longer a function. The two roots of x <sup>2</sup> " ´1 are like Tweedledum and Tweedledee: we know there are two of them, and we know how they are related; but we have no way of distinguishing them, or of singling one of them out.

Once we accept this, we can write complex numbers in the form a ` bi, where a and b are real numbers (just as we used to write numbers in the form a ` b ? 2, where a and b are rational numbers). And we can proceed to add, subtract, multiply, and divide such expressions, and then collect up the "real" and "imaginary" parts to tidy up the answer.

### Problem 128


Problem 129 Divide 10 into two parts, whose product is 40. 4

Problem 129 appears in Chapter XXXVII of Girolamo Cardano's (1501–1576) book Ars Magna (1545). Having previously presented the general methods for solving quadratic, cubic, and quartic equations, he honestly confronts the phenomenon that his method for solving cubic equations (see Problem 135) produces the required real (and positive) solution x as a sum of complex conjugates u and v – involving not only negative numbers, but square roots of negative numbers. After presenting the formal solution of Problem 129, and having shown that the calculation works exactly as it should, he adds the bemused remark:

"So progresses arithmetic subtlety, the end of which . . . is as refined as it is useless."

Arithmetic with complex numbers in the form a`bi is done by carrying out the required operations, and then collecting up the "real" and "imaginary" parts as separate components – just as with adding vectors pa, bq. We treat the two parts as Cartesian coordinates, and so identify the complex number a ` bi with the point pa, bq in the complex plane.

The "Cartesian" representation a ` bi is very convenient for addition. But the essential definition (and significance) of complex numbers is rooted in multiplication. And for multiplication it is often much better to work with complex numbers written in polar form. Suppose we mark the complex number w " a ` bi in the complex plane.

The ? modulus |w| of w (often denoted by r) is the distance r " a <sup>2</sup> ` b <sup>2</sup> of the complex number a ` bi from the origin in the complex plane.

The angle θ, measured anticlockwise from the positive real axis to the line joining the complex number w to the origin, is called the argument, Argpwq " θ, of w.

It is then easy to check that a " r cos θ, b " r sin θ, and that

$$w = r(\cos \theta + i \sin \theta).$$

This is the polar form for w. Instead of focusing on the Cartesian coordinates a, b, the polar form pinpoints w in terms of


### Problem 130

(a) Given two complex numbers in polar form:

w " rpcos θ ` isin θq, z " spcos φ ` isin φq,

show that their product is precisely

wz " rspcospθ ` φq ` isinpθ ` φqq.

(b) (de Moivre's Theorem: Abraham de Moivre (1667–1754)) Prove that

$$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta).$$

(c) Prove that, if

$$z = r(\cos \theta + i \sin \theta)$$

satisfies z <sup>n</sup> " 1 for some integer n, then r " 1. 4

The last three problems in this subsection look more closely at "roots of unity" – that is, roots of the polynomial equation x <sup>n</sup> " 1. In the real domain, we know that:


In contrast, in the complex domain, there are n "n th roots of unity". Problem 130(c) shows that these "roots of unity" all lie on the unit circle, centered at the origin. And if we put nθ " 2kπ in Problem 130(b) we see that the n nth roots of unity include the point "1 " cos 0 ` isin 0", and are then equally spaced around that circle with θ " 2kπ n p1 ď k ď n ´ 1q, and form the vertices of a regular n-gon.

### Problem 131


Problem 132 Use Problem 131(d) to factorise x <sup>4</sup> ` 1 as a product of four linear factors, and hence as a product of two quadratic polynomials with real coefficients. 4

### Problem 133


### 4.5. Cubic equations

The first recorded procedure for finding the positive roots of any given quadratic equation dates from around 1700 BC (ancient Babylonian). A corresponding procedure for cubic equations had to wait until the early sixteenth century AD. The story is a slightly complicated one – involving public contests, betrayal, and much else besides.

In Section 4.4 we saw that the cubic equation x <sup>3</sup> " 1 has three solutions – two of which are complex numbers. But in the sixteenth century, even negative numbers were viewed with suspicion, and complex numbers were still unknown. Moreover, symbolical algebra had not yet been invented, so everything was carried out in words: constants were "numbers"; a given multiple of the unknown was referred to as so many "things"; a given multiple of the square of the unknown was simply referred to as "squares"; and so on.

In short, we know that an improved method for sometimes finding a (positive) unknown which satisfied a cubic equation was devised by Scipione del Ferro (1465–1526) around 1515. He kept his method secret until just before his death, when he told his student Antonio del Fiore (1506–??). Niccol`o Tartaglia (1499–1557) then made some independent progress in solving cubic equations. At some stage (around 1535) Fiore challenged Tartaglia to a public "cubic solving contest". In preparing for this event, Tartaglia managed to improve on his method, and he seems to have triumphed in the contest. Tartaglia naturally hesitated to divulge his method in order to preserve his superiority, but was later persuaded to communicate what he knew to Girolamo Cardano (1501–1576) after Cardano promised not to publish it (either never, or not before Tartaglia himself had done so). Cardano improved the method, and his student Ferrari (1522–1565) extended the idea to give a method for solving quartic equations – all of which Cardano then published, contrary to his promise, but with full attribution to the rightful discoverers, in his groundbreaking book Ars Magna (1545 – just two years after Copernicus (1473–1543) published his De revolutionibus . . . ). Problem 134 illustrates the necessary first move in solving any cubic equation. Problem 135 then illustrates the general method in a relatively simple case.

### Problem 134


Problem 135 The equation x <sup>3</sup> ` 3x <sup>2</sup> ´ 4 " 0 clearly has "x " 1" as a positive solution. (The other two solutions are x " ´2, and x " ´2 – a repeated root; however negatives were viewed with suspicion in the sixteenth century, so this root might well have been ignored.) Try to understand how the following sequence of moves "finds the root x " 1":


$$(u+v)^3 = u^3 + 3uv(u+v) + v^3$$

as your cubic equation in y;


The simple method underlying Problem 135 is in fact completely general. Given any cubic equation

$$ax^3 + bx^2 + cx + d = 0 \quad \text{(with } a \neq 0\text{)}$$

we can divide through by a to reduce this to

$$x^3 + px^2 + qx + r = 0$$

with leading coefficient " 1. Then we can substitute y " x ` p 3 and reduce this to a cubic equation in y

$$y^3 - 3\left(\frac{p}{3}\right)^2 y + qy + \left[r + 2\left(\frac{p}{3}\right)^3 - q\left(\frac{p}{3}\right)\right] = 0$$

which we can treat as having the form

$$y^3 - my - n = 0.$$

So we can set y " u ` v (for some unknown u and v yet to be chosen), and treat the last equation as an instance of the identity

$$(u+v)^3 - 3uv(u+v) - (u^3 + v^3) = 0$$

which it will become if we simply choose u and v to solve the simultaneous equations

$$3uv = m, \quad u^3 + v^3 = n.$$

We can then solve these equations to find u, then v – and hence find y " u`v and x " y ´ p 3 .

### 4.6. An extra

Back in Chapter 1, Problem 6 we introduced the Euclidean algorithm for integers. The same idea was extended to polynomials with integer coefficients in Problem 126. In both these settings one starts with a domain (whether the set of integers, or the set of all polynomials with integer coefficients) where there is a notion of divisibility: given two elements m, n in the relevant domain, we say

"n divides m" if there exists an element q in the domain such that m " qn.

The next problem invites you to think how one might extend the Euclidean algorithm to a new domain, namely the Gaussian integers Zris – the set of all complex numbers a ` bi in which the real and imaginary "coordinates" a and b are integers.

Problem 136 Complex numbers a ` bi, where both a and b are integers, are called Gaussian integers. Try to formulate a version of the "division algorithm" for "division with remainder" (where the remainder is always "less than" the divisor in some sense) for pairs of Gaussian integers. Extend this to construct a version of the Euclidean algorithm to find the HCF of two given Gaussian integers. 4

> It is a profoundly erroneous truism . . . that we should cultivate the habit of thinking what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them.

> > Alfred North Whitehead (1861–1947)

### 4.7. Chapter 4: Comments and solutions

92. Answer: Humour aside, this is a common situation.

We know d ` b, n ` b, d ` n rather than the values of d, b, n.

The key is to exploit the symmetry in the given data, rather than solving blindly. Adding all three two-way totals gives 2pd ` b ` nq " 284, whence d ` b ` n " 142. We can then subtract the given value of d ` n " 137 to get the value of b " 5.

#### 93.

(a) Let the numbers at the three vertices be A, B, C. Adding shows that

$$a + b + c = 2(A + B + C)$$

so

$$A = \frac{a+b+c}{2} - (B+C) = \frac{b+c-a}{2}$$

etc.

(c) Note: We postpone the "solution" of part (b), and address part (c) first. Let the numbers at the five vertices be A, B, C, D, E. Adding shows that

$$d + e + a + b + c = 2(A + B + C + D + E)$$

so

$$\begin{aligned} A &=& \frac{d+e+a+b+c}{2} - (B+C) - (D+E) \\ &=& \frac{d-e+a-b+c}{2} \end{aligned}$$

etc.

(b) The second part is different. The four given edge-values do not determine the four unknown vertex-values. It may look as though four pieces of information should suffice to find four unknowns; but there is a catch: the sum of the numbers on the two opposite edges AB and CD is just the sum of the numbers at the four vertices, and so is equal to the sum of the numbers on the edges BC and DA. Hence one of the given edge-values is determined by the other three.

Note: This distinction between polygons with an odd and an even number of vertices would arise in exactly the same way if each edge was labelled with the average ("half the sum") of the numbers at its two end vertices.

$$\textbf{94.}\ a = BC = BP + PC = y + z; \ b = x + z; \ c = x + y. \ \text{Hence}\ a$$

$$a+b+c=2(x+y+z)$$

so

So

$$x + y + z = \frac{a + b + c}{2}.$$

$$\begin{aligned} x^2 &= \frac{a+b+c}{2} - (y+z)^2\\ &= \frac{b+c-a}{2} \end{aligned}$$

etc.

95.

(a) Let

$$M = \left(\frac{a+c}{2}, \frac{b+d}{2}\right).$$

The shift, or vector, from pa, bq to pc, dq goes

"along c ´ a in the x-direction" and "up d ´ b in the y-direction".

Draw the ordinate through Y and the abscissa through Z, to meet at P, so creating a right angled triangle with legs Y P of length |c ´ a| and P Z of length |d ´ b|. The midpoint of Y P clearly lies halfway along Y P at

$$S = \left(a + \frac{c - a}{2}, b\right)$$

and the midpoint of P Z clearly lies halfway up P Z at

$$T = \left(c, d - \frac{d-b}{2}\right).$$

Then 4Y SM and 4MT Z are both right-angled triangles and are congruent (by RHS congruence). Hence Y M " MZ, so M is the midpoint of Y Z.

(b)

$$M = \left(\frac{a}{2}, \frac{b}{2}\right), \quad N = \left(\frac{c}{2}, \frac{d}{2}\right)$$

so vector

$$\mathbf{MN} = \left(\frac{c-a}{2}, \frac{d-b}{2}\right) = \frac{1}{2}\mathbf{BC}.$$

(c) Note: We use the result from part (b), but not the method from part (b). By part (b) applied to 4BAC, P Q is half the length of AC and parallel to AC. By part (b) applied to 4DAC, SR is half the length of AC and parallel to AC. Hence P Q is parallel to SR.

Similarly one can prove (applying part (b) twice – first to 4ABD, and then to 4CBD) that P S is parallel to QR.

Hence P QRS is a parallelogram.

96.

 $\mathbf{r}(\mathbf{a})$   $\mathbf{p} = \frac{1}{2}(\mathbf{x} + \mathbf{y})$ ,  $\mathbf{q} = \frac{1}{2}(\mathbf{y} + \mathbf{z})$ ,  $\mathbf{r} = \frac{1}{2}(\mathbf{z} + \mathbf{x})$ , so 
$$\mathbf{p} + \mathbf{q} + \mathbf{r} = \mathbf{x} + \mathbf{y} + \mathbf{z}$$
.

Hence

$$\begin{array}{rcl} \mathbf{x} & = & (\mathbf{p} + \mathbf{q} + \mathbf{r}) - (\mathbf{y} + \mathbf{z}) = \mathbf{p} - \mathbf{q} + \mathbf{r} \\ \mathbf{y} & = & (\mathbf{p} + \mathbf{q} + \mathbf{r}) - (\mathbf{x} + \mathbf{z}) = \mathbf{p} + \mathbf{q} - \mathbf{r} \\ \mathbf{z} & = & (\mathbf{p} + \mathbf{q} + \mathbf{r}) - (\mathbf{x} + \mathbf{y}) = \mathbf{q} + \mathbf{r} - \mathbf{p}. \end{array}$$

(b)

$$\mathbf{p} = \frac{1}{2}(\mathbf{w} + \mathbf{x}), \ \mathbf{q} = \frac{1}{2}(\mathbf{x} + \mathbf{y}), \ \mathbf{r} = \frac{1}{2}(\mathbf{y} + \mathbf{z}), \ \mathbf{s} = \frac{1}{2}(\mathbf{z} + \mathbf{w})$$

so

$$
\mathbf{p} + \mathbf{q} + \mathbf{r} + \mathbf{s} = \mathbf{w} + \mathbf{x} + \mathbf{y} + \mathbf{z}.
$$

Hence

$$\mathbf{w} = (\mathbf{p} + \mathbf{q} + \mathbf{r} + \mathbf{s}) - (\mathbf{x} + \mathbf{y} + \mathbf{z});$$

but there is no obvious way to pin down px ` y ` zq.

In fact different quadrilaterals may give rise to the same four "midpoints". (It is an interesting exercise to identify the family of quadrilaterals corresponding to a given set of four midpoints.)

(c) As in parts (a) and (b),

$$\mathbf{p} = \frac{1}{2}(\mathbf{v} + \mathbf{w}),\\\mathbf{q} = \frac{1}{2}(\mathbf{w} + \mathbf{x}),\\\mathbf{r} = \frac{1}{2}(\mathbf{x} + \mathbf{y}),\\\mathbf{s} = \frac{1}{2}(\mathbf{y} + \mathbf{z}),\\\mathbf{t} = \frac{1}{2}(\mathbf{z} + \mathbf{v}).$$

Hence

$$\mathbf{p} + \mathbf{q} + \mathbf{r} + \mathbf{s} + \mathbf{t} = \mathbf{v} + \mathbf{w} + \mathbf{x} + \mathbf{y} + \mathbf{z}$$

so

v " pp ` q ` r ` s ` tq ´ pw ` xq ´ py ` zq " p ´ q ` r ´ s ` t w " pp ` q ` r ` s ` tq ´ px ` yq ´ pz ` vq " p ` q ´ r ` s ´ t x " pp ` q ` r ` s ` tq ´ pv ` wq ´ py ` zq " ´p ` q ` r ´ s ` t y " pp ` q ` r ` s ` tq ´ pw ` xq ´ pz ` vq " p ´ q ` r ` s ´ t z " pp ` q ` r ` s ` tq ´ pv ` wq ´ px ` yq " ´p ` q ´ r ` s ` t.

97.

(a)(i) As in Problems 93-95 we instinctively add to get

$$2(x+y+z) = 6$$

so

$$x + y + z = 3.$$

Hence

$$\begin{array}{rcl} x & = & 3 - (y + z) = 1 \\ y & = & 3 - (x + z) = 0 \\ z & = & 3 - (x + y) = 2. \end{array}$$

(ii) The same idea (replacing addition by multiplication) leads to

$$2 \times 4 \times 8 = 64 = uv \cdot vw \cdot wu = (uvw)^2$$

so uvw " ˘8. Hence

$$\begin{aligned} u &= \quad \frac{uvw}{vw} = \frac{\pm 8}{4} = \pm 2\\ v &= \quad \frac{uvw}{uw} = \frac{\pm 8}{8} = \pm 1\\ w &= \quad \frac{uvw}{uv} = \frac{\pm 8}{2} = \pm 4. \end{aligned}$$

6 pu, v, wq " p2, 1, 4q or p´2, ´1, ´4q.

Note: Alternatively, we may notice that u, v, w are either all positive, or all negative. If we restrict in the first instance to purely positive solutions, then we may set u " 2 x , v " 2 y , w " 2 z , translate (ii) into (i), and conclude that px, y, zq " p1, 0, 2q, so that pu, v, wq " p2, 1, 4q. We must then remember the negative solution pu, v, wq " p´2, ´1, ´4q.

(b)(i) As in (a)(i) we add to get 2px ` y ` zq " 9, so x ` y ` z " 9 2 . Hence

$$\begin{array}{rcl} x & = & \frac{9}{2} - (y+z) = \frac{3}{2} \\ y & = & \frac{9}{2} - (x+z) = \frac{1}{2} \\ z & = & \frac{9}{2} - (x+y) = \frac{5}{2} \end{array}$$

(ii) The same idea leads to

$$6 \times 10 \times 15 = 900 = uv \cdot vw \cdot wu = \left(uvw\right)^2,$$

so uvw " ˘30. Hence

$$\begin{array}{rcl} u & = & \frac{uvw}{vw} = \frac{\pm 30}{10} = \pm 3\\ v & = & \frac{uvw}{uw} = \frac{\pm 30}{15} = \pm 2\\ w & = & \frac{uvw}{uv} = \frac{\pm 30}{6} = \pm 5. \end{array}$$

Either u, v, w are all positive, or all negative. 6 pu, v, wq " p3, 2, 5q or p´3, ´2, ´5q.

(iii) The same idea leads to

$$6 \times 10 \times 30 = 2 \times 900 = vw \cdot wu = (uvw)^2,$$

so uvw " ˘30? 2. Hence

$$\begin{aligned} u &=& \frac{uvw}{vw} = \frac{\pm 30\sqrt{2}}{10} = \pm 3\sqrt{2} \\ v &=& \frac{uvw}{uw} = \frac{\pm 30\sqrt{2}}{15} = \pm 2\sqrt{2} \\ w &=& \frac{uvw}{uv} = \frac{\pm 30\sqrt{2}}{6} = \pm 5\sqrt{2} .\end{aligned}$$

Either u, v, w are all positive, or all negative. 6 pu, v, wq " p3 ? 2, 2 ? 2, 5 ? 2q or p´3 ? 2, ´2 ? 2, ´5 ? 2q.

(iv) We could of course repeat the same method.

Or we could again look in the first instance for positive solutions, notice that 4 " 2 2 , 8 " 2 3 , 16 " 2 4 , and take logs (to base 2). Then

$$\begin{aligned} \log\_2 u + \log\_2 v &=& 2\\ \log\_2 v + \log\_2 w &=& 3\\ \log\_2 u + \log\_2 w &=& 4, \end{aligned}$$

so (from part (i)) any positive solution satisfies

$$
\log\_2 u = \frac{3}{2}, \; \log\_2 v = \frac{1}{2}, \; \log\_2 w = \frac{5}{2},
$$

so

$$(u,v,w) = (2\sqrt{2}, \sqrt{2}, 4\sqrt{2}).$$

We must then remember to include

$$(u,v,w) = (-2\sqrt{2}, -\sqrt{2}, -4\sqrt{2}).$$

98. The simplest idea is to take logs, and reduce the system to a familiar linear system:

$$\begin{aligned} a \cdot \log u + b \cdot \log v &= \quad \log m \\ c \cdot \log u + d \cdot \log v &= \quad \log n \end{aligned}$$

Multiplying the first equation by c and subtracting it from the second equation multiplied by a gives:

$$
\log v \quad = \quad \frac{a \cdot \log n - c \cdot \log m}{ad - bc}.
$$

Multiplying the first equation by d and subtracting b times the second equation gives:

$$
\log u \quad = \quad \frac{d \cdot \log m - b \cdot \log n}{ad - bc}.
$$

If the numerators and denominators are expressed in determinant form, we get the 2 ˆ 2 version of Cramer's Rule. The original unknowns u, v can then be obtained by taking suitable powers.

What emerges looks interesting:

$$\begin{array}{rcl} u & = & m^{\frac{d}{ad-bc}} \cdot n^{-\frac{b}{ad-bc}}\\ v & = & m^{-\frac{c}{ad-bc}} \cdot n^{\frac{a}{ad-bc}} \end{array}$$

but it is not clear how it generalises.

#### 99.

(a) x`y`z " 3 is the equation of a plane through the three points p3, 0, 0q, p0, 3, 0q, p0, 0, 3q.

x <sup>2</sup> ` y <sup>2</sup> ` z <sup>2</sup> " c is the equation of a sphere, centered at the origin, with radius ? c. The sphere misses the plane completely when c ă 3, meets the plane in a single point when c " 3, and cuts the plane in a circle C when c ą 3 (the circle lying in the positive octant provided c ă 9).

If xy ` yz ` zx " b meets this intersection at all, then any permutation of the three coordinates x, y, z produces another point which also satisfies the other two equations (since they are both symmetrical). Hence for the system to have a unique solution, the circle C must contain a point with x " y " z. Hence c " 3, and b " 3, and the unique solution is

$$(x, y, z) = (1, 1, 1).$$

(b) We must have c ě 0 for any solution. If c " 0, then for a unique solution, we must have x " y " z " 0, so a " b " 0. If we exclude this case, then we may assume that c ą 0.

$$x + y + z = a$$

is the equation of a plane through the three points pa, 0, 0q, p0, a, 0q, p0, 0, aq.

$$x^2 + y^2 + z^2 = c$$

is the equation of a sphere, centre the origin, with radius ? c, which misses the plane completely when c ă a 2 3 , meets the plane in a single point when c " a 2 3 , and cuts the plane in a circle C when c ą a 2 3 (the circle lying in the positive octant provided c ă a 2 ). If

$$xy + yz + zx = b$$

meets this intersection at all, then any permutation of the three coordinates x, y, z produces another point which also satisfies the other two equations (since they are both symmetrical). Hence for the system to have a unique solution, the circle C must contain a point with x " y " z. Hence that point is x " y " z " a 3 , so c " a 2 <sup>3</sup> " b, and the unique solution is

$$(x, y, z) = \left(\frac{a}{3}, \frac{a}{3}, \frac{a}{3}\right).$$

100. | ´ x| is never negative. If x ě 0, then | ´ x| " x; if x is negative, then ´x is positive, so | ´ x| " ´x.

Note: We need to learn to see both x and ´x as algebraic entities, with x as a placeholder (which may well be negative, in which case ´x would be positive).

#### 101.

(a) The interval r0.1, 0.2q. We have marked exactly <sup>1</sup> <sup>10</sup> of the interval r0, 1q. (b) This needs a little thought. First we mark the interval r0.1, 0.2q, of length <sup>1</sup> 10 . Then we mark 9 smaller intervals

r0.01, 0.02q, r0.21, 0.22q, . . . , r0.91, 0.92q

of total length 9 ¨ ` 1 10 ˘2 . Then 9<sup>2</sup> smaller intervals

$$[0.001, 0.002), \ [0.021, 0.022), \ \dots, \ [0.991, 0.992)$$

of total length 9<sup>2</sup> ¨ ` 1 10 ˘3 . And so on.

(0.1, 0.2) 
$$\begin{aligned} \cup \quad & [0.01, 0.02) \cup [0.21, 0.22) \cup [0.31, 0.32) \cup [0.41, 0.42) \\ & \cup [0.51, 0.52) \cup [0.61, 0.62) \cup [0.71, 0.72) \\ & \cup [0.81, 0.82) \cup [0.91, 0.92) \\ \cup \quad & [0.001, 0.002) \cup [0.021, 0.022) \cup [0.031, 0.032) \cup \cdots \\ \cup \quad & \cdots \end{aligned}$$

It would seem that a vast number of points are left unmarked – namely, every point whose decimal representation uses only 0s, 2s, 3s, 4s, 5s, 6s, 7s, 8s, and 9s. However, the total length of the marked intervals is given by adding:

$$\frac{1}{10} + 9 \cdot \left(\frac{1}{10}\right)^2 + 9^2 \cdot \left(\frac{1}{10}\right)^3 + 9^3 \cdot \left(\frac{1}{10}\right)^4 + 9^4 \cdot \left(\frac{1}{10}\right)^5 + 9^5 \cdot \left(\frac{1}{10}\right)^6 + \dotsb$$

That is an infinite geometric series with first term a " 1 <sup>10</sup> and common ratio r " 9 <sup>10</sup> , and hence with sum " 1. In other words, the total length of what remains unmarked is zero.


r0.01, 0.02q, r0.21, 0.22q

of total length 2 ¨ ` 1 3 ˘2 . Then 2<sup>2</sup> smaller intervals

r0.001, 0.002q, r0.021, 0.022q, r0.201, 0.202q, r0.221, 0.222q

of total length 2<sup>2</sup> ¨ ` 1 3 ˘3 . And so on.

r0.1, 0.2q

Y r0.01, 0.02q Y r0.21, 0.22q Y r0.001, 0.002q Y r0.021, 0.022qq Y r0.201, 0.202q Y r0.2201, 0.02202q Y ¨ ¨ ¨

The set of marked points is the complement of the famous Cantor set (Georg Cantor (1845–1918)) and has total length

$$\frac{1}{3} + 2 \cdot \left(\frac{1}{3}\right)^2 + 2^2 \cdot \left(\frac{1}{3}\right)^3 + 2^3 \cdot \left(\frac{1}{3}\right)^4 + 2^4 \cdot \left(\frac{1}{3}\right)^5 + 2^5 \cdot \left(\frac{1}{3}\right)^6 + \dotsb$$

This is an infinite geometric series with first term a " 1 3 and common ratio r " 2 3 , and so has sum " 1.

Hence, the total length of what remains unmarked is zero.

Note: The set described in (d) leaves as its complement a collection of points – the Cantor set – which consists of the "endpoints" of the intervals that have been removed; these are points whose base 3 expansion involves only 0s and 2s. This complement:


102. (2, 3]. Each inequality implies all the ones before it. Hence the two which are true must be the first two. Hence x ď 3, and x ą 2.

103. If x ´ 5 ě 0, then we must solve x ´ 5 " 3; so x " 8; if x ´ 5 ă 0, we must solve x ´ 5 " ´3, so x " 2.

Note: The fact that |x| denotes the positive value of the pair tx, ´xu can be rephrased as: |x| is equal to the distance from x to 0.

In the same way, |x ´ 5| denotes the positive member of the pair

$$\{x - 5, -(x - 5)\}$$

so |x ´ 5| is equal to the distance from x ´ 5 to 0 (i.e. the distance from x to 5). This is a very important way to think about expressions like |x ´ 5|.

#### 104.


105. ´ 5 2 , ´ 1 2 ( . (We need all points x for which

"the distance from x to ´1" plus "the distance from x to ´2"

equals 2. This excludes all points between ´2 and ´1, for which the sum is equal to 1; for points between ´ 5 2 and ´ 1 2 the sum is <sup>ă</sup> 2; for points in ` ´8, ´ 5 2 ˘ or ` ´ 1 2 , 8 ˘ the sum is ą 2.)

106. a " 1 2 , b " 3 2 . (For solutions to exist, we must have b ą 0. The solutions of the given inequality then consist of all x such that

"the distance from x to a is less than b"

that is, all x in the interval pa ´ b, a ` bq. Hence a ´ b " ´1, a ` b " 2.)

#### 107.


$$
\left| 1 - \frac{2y}{x+y} \right| < 1.
$$

In other words:

$$0 < \frac{2y}{x+y} < 2.$$

If y ą 0, then x ` y ą 0, so 2x ` 2y ą 2y, whence x ą 0 (so "x ą 0 and y ą 0"). If y ă 0, then x ` y ă 0, so 2x ` 2y ă 2y, whence x ă 0 (so "x ă 0 and y ă 0"). If x ą 0 and y ą 0, or x ă 0 and y ă 0, then clearly |x ´ y| ă |x ` y|.)

108. Let

$$x = \frac{a}{b} < \frac{c}{d} = y.$$

(i) Since x ă y, it follows that

$$x - \frac{x}{2} = \frac{x}{2} < \frac{y}{2},$$

so x ă x`y 2 ; moreover <sup>x</sup> <sup>2</sup> ă y ´ y 2 , so <sup>x</sup>`<sup>y</sup> <sup>2</sup> ă y. (ii) Since <sup>a</sup> <sup>b</sup> ă c d and b, d ą 0, we can multiply both sides by bd to get ad ă bc. Therefore

$$a(b+d) = ab + ad \prec ba + bc = b(a+c),$$

and

$$(a+c)d = ad + cd < bc + dc = (b+d)c.$$

6 a <sup>b</sup> ă a`c b`d , and <sup>a</sup>`<sup>c</sup> <sup>b</sup>`<sup>d</sup> ă c d (since b, d, and b ` d are all ą 0, so we can divide the first inequality by bpb ` dq and the second by dpb ` dq).

#### 109.

(a)

$$\frac{10}{1} < \frac{1}{7} < \frac{1}{6} < \frac{1}{5} < \frac{1}{4} < \frac{2}{7} < \frac{1}{3} < \frac{2}{5} < \frac{3}{7} < \frac{1}{2} < \frac{4}{7} < \frac{3}{5} < \frac{2}{3} < \frac{5}{7} < \frac{3}{4} < \frac{4}{5} < \frac{5}{6} < \frac{6}{7} < \frac{1}{10}$$

(b)(i) It is tempting simply to consider the decimals

$$\frac{1}{9} = 0.111\dots, \frac{2}{9} = 0.222\dots, \frac{3}{9} = 0.333\dots, \dots, \dots, \frac{8}{9} = 0.888\dots$$

in order to conclude that these fractions miss the first and last subinterval, and then fall one in each of the remaining subintervals. In preparation for part (ii), it is better to observe that


$$\frac{1}{9} = \frac{1}{10} + \frac{1}{90}, \frac{2}{9} = \frac{2}{10} + \frac{2}{90}, \frac{3}{9} = \frac{3}{10} + \frac{3}{90}, \dots, \frac{8}{9} = \frac{8}{10} + \frac{8}{90}$$

and notice that

$$
\frac{0}{10} < \frac{1}{90} < \dots < \frac{8}{90} < \frac{1}{10},
$$

so that, for 1 ď m ď 9,

$$
\frac{m}{10} < \frac{m}{9} < \frac{m+1}{10};
$$

hence exactly one 9 th goes in each of the other eight subintervals.

	- ∗ 1 <sup>n</sup>`<sup>1</sup> ă 1 n and <sup>n</sup>´<sup>1</sup> <sup>n</sup> ă n n`1 , so none of the n ths land up in the first or last subintervals;
	- ∗ then rewrite

$$\frac{1}{n} = \frac{1}{n+1} + \frac{1}{n(n+1)},\\\frac{2}{n} = \frac{2}{n+1} + \frac{2}{n(n+1)},\\\dots,\frac{n-1}{n} = \frac{n-1}{n+1} + \frac{n-1}{n(n+1)}$$

and notice that

$$\frac{0}{n+1} < \frac{1}{n(n+1)} < \dots < \frac{n-1}{n(n+1)} < \frac{1}{n+1},$$

so, for 1 ď m ď n,

$$
\frac{m}{n+1} < \frac{m}{n} < \frac{m+1}{n+1};
$$

hence exactly one n th goes in each of the other n ´ 1 subintervals.


simple objects and "elementary" proofs can sometimes be more intricate than one anticipates.

Note 2: If

$$\frac{a}{b} < \frac{c}{d}$$

are consecutive terms in a Farey series, then "bc ´ ad" must be an integer ą 0. The fact that this difference is always equal to 1 is easily checked in any particular case, but it is unclear exactly why this is necessarily true (rather than an accident) – or even how one would go about proving it. Every treatment of Farey series has to find its own way round this difficulty. We give the simplest proof we can (in the sense that it assumes no more than we have already used: a little about numbers and some algebra). But it is not at all "easy". We indicate a different approach in the "Notes" at the end of part (d).

(i) [The fact that, except for the two end intervals, we have bd ą n will be needed in the proof of part (c)(ii).]

We proceed by mathematical induction on n (the "order" of the Farey series) – a technique which we have already met in Chapter 2 (Problems 54–59, 76) and which will be addressed more fully in Chapter 6.

∗ When n " 1, the Farey series of order n is just:

$$\frac{0}{1} < \frac{1}{1}$$

and this subinterval is both the first and the last, so the claim is "vacuously true" (because there is "nothing to check"). When n " 2, the Farey series of order n is:

$$
\frac{0}{1} < \frac{1}{2} < \frac{1}{1},
$$

and again the only subintervals are the first and the last, so again there is nothing to check.

∗ We now suppose that we know the claim is true for the Farey series of order k, for some k ą 1, and show that it must then also be true for the Farey series of order k ` 1. (Since we know it is true for n " 2, it will then be true for n " 3; and once we know it is true for n " 3, it must then be true for n " 4; and so on.)

To show that the claim is true for the Farey series of order k`1, we consider any adjacent pair of fractions

$$\frac{a}{b} < \frac{c}{d}$$

(other than the first pair and the last pair ) in the Farey series of order k`1.

Claim bd ą k ` 1.

Proof Note first that, since we are avoiding the two end subintervals, both b and d are ą 1.

Suppose first that the pair <sup>a</sup> <sup>b</sup> ă c d are not adjacent in the previous Farey series of order k. Then at least one of the two fractions has been inserted in creating the Farey series of order k`1, and so has denominator " k`1. (The fractions inserted are precisely those with denominator "k ` 1" which cannot be reduced by cancelling.) Hence the product

$$bd \geqslant 2(k+1) > k+1.$$

Thus we may assume that the pair <sup>a</sup> <sup>b</sup> ă c <sup>d</sup> were already adjacent in the Farey series of order k. But then by our "induction hypothesis" (namely that the desired result is already known to be true for the Farey series of order k), we know that bd ą k. If bd ą k ` 1, then the pair <sup>a</sup> <sup>b</sup> ă c d satisfies the required condition. Hence we only have to worry about the possibility that bd " k `1. Suppose that bd " k ` 1. Then the interval <sup>a</sup> <sup>b</sup> ă c d has length

$$\frac{bc - ad}{bd} = \frac{r}{k+1}$$

for some positive integer r " bc ´ ad.

If r ą 1, then the interval would have length ą 1 k`1 , so <sup>a</sup> <sup>b</sup> ă c <sup>d</sup> would not be successive terms in the series (for we would have inserted some additional term when moving from the Farey series of order k to the Farey series of order k ` 1).

Hence we can be sure that r " 1, that the subinterval <sup>a</sup> <sup>b</sup> ă c d has length exactly 1 k`1 . Now successive fractions with denominator k ` 1 differ by exactly <sup>1</sup> k`1 , so some fraction with denominator k+1 must lie in this subinterval. Since no additional fraction is inserted between them in passing from the series of order k to the series of order k`1, <sup>a</sup> b and <sup>c</sup> <sup>d</sup> must both be "cancelled versions" of two successive fractions with denominator k ` 1. But then, by part (b)(ii), there would have to be a fraction with denominator k in the interval <sup>a</sup> <sup>b</sup> ă c d – which is not the case.

Therefore the possibility bd " k ` 1 does not in fact occur.

So we can be sure that in every case, bd ą k ` 1.

Hence whenever the result is true for the Farey series of order k, it must then also be true for the Farey series of order k ` 1.

It follows that the result is true for the Farey series of order n, for all n ě 1. QED


Let <sup>a</sup> <sup>b</sup> ă c d be adjacent fractions in the Farey series of order k ` 1.

If <sup>a</sup> <sup>b</sup> ă c <sup>d</sup> were already adjacent fractions in the Farey series of order k (i.e. if no fraction has been inserted between <sup>a</sup> b and <sup>c</sup> d in passing from the series of order k to the series of order k ` 1), then we already know (by the induction hypothesis) that bc ´ ad " 1.

Thus we may concentrate on the case where <sup>a</sup> <sup>b</sup> ă c d are not adjacent fractions in the Farey series of order k. By (b)(iii), at most one fraction with denominator k ` 1 is inserted between any two adjacent fractions in the Farey series of order k, so we have either

$$
\frac{a}{b} < \frac{c}{d} < \frac{e}{f},
$$

with <sup>a</sup> <sup>b</sup> ă e f being adjacent fractions in the Farey series of order k (so be ´ af " 1), or

$$
\frac{e}{f} < \frac{a}{b} < \frac{c}{d},
$$

with <sup>e</sup> <sup>f</sup> ă c d being adjacent fractions in the Farey series of order k (so fc ´ ed " 1). We consider the first of these possibilities (the second is entirely similar).

Suppose

"

$$
\frac{a}{b} < \frac{c}{d} < \frac{e}{f},
$$

with <sup>a</sup> <sup>b</sup> ă e f being adjacent fractions in the Farey series of order k. By part (i) we know that bf ě k ` 1; and by induction we know that be ´ af " 1. Hence the interval <sup>a</sup> <sup>b</sup> ă e f has length at most <sup>1</sup> k`1 . We have to prove that bc ´ ad " 1.

$$\text{Let } bc - ad = r > 0 \text{, and } ed - fc = s > 0.$$

Then sa ` re " c, and sb ` rf " d.

In particular, HCFpr, sq " 1 (since HCFpc, dq " 1).

Hence <sup>c</sup> d belongs to the family

$$S = \left\{ \frac{xa+ye}{xb+yf} \,:\, \text{where } x, y \text{ are any positive integers with } HCF(x,y) = 1 \right\}.$$

\*

Since everything is positive, easy algebra shows that

$$
\frac{a}{b} < \frac{xa+ye}{xb+yf} < \frac{e}{f},
$$

so every element of S lies between <sup>a</sup> b and <sup>e</sup> f .

As long as we choose x, y such that HCFpx, yq " 1, any common factor of xa ` ye and xb ` yf would also divide both

$$(b(xa+ye)-a(xb+yf)=(be-af)y=y,$$

and

$$e(xb+yf) - f(xa+ye) = (be-af)x = x.$$

Hence

$$HCF(xa+ye, xb+yf) = 1$$

so each element of S is in lowest terms (i.e. no further cancelling is possible). We have shown that " <sup>c</sup> d belongs to the family S", and that all elements of S fit between <sup>a</sup> b and <sup>e</sup> f ; which are adjacent fractions in the Farey series of order k. So none of the elements of S can have arisen before the series of order k ` 1. But each fraction in S arises at some stage in a Farey series.

And the first to arise (because it has the smallest denominator) is " <sup>a</sup>`<sup>e</sup> b`f ". Hence

$$\frac{c}{d} = \frac{a+e}{b+f},$$

so r " s " 1, and bc ´ ad " 1 as required. QED

(d) Let

$$\frac{a}{b} < \frac{c}{d} < \frac{e}{f}$$

be three successive terms in any Farey sequence. By (c) we know that bc´ad " 1, and that de ´ cf " 1. In particular, bc ´ ad " de ´ cf, so

$$\frac{c}{d} = \frac{a+e}{b+f}.$$

Note 1: It may not be clear why we are proving this result "again" – since it appeared in the final line of the solution to part (c). However, in part (c) the statement that

$$\frac{c}{d} - \frac{a+e}{b+f}$$

was arrived at within the induction step, and so was subject to other assumptions. In contrast, now that the result in part (c) has been clearly established, we can use it to prove part (d) without any hidden assumptions.

Note 2: If we represent each fraction <sup>a</sup> b in the Farey series of order n by the point pb, aq, then each point lies in the right angled triangle joining p0, 0q, pn, 0q, and pn, nq, and each fraction in the Farey series is equal to the gradient of the line, or vector, joining the origin to the integer lattice point pb, aq. The ordering of the fractions in the Farey series corresponds to the sequence of increasing gradients, from <sup>0</sup> 1 up to <sup>1</sup> 1 . If <sup>a</sup> b and <sup>e</sup> f are adjacent fractions in some Farey

$$\text{QED}$$

series, then the result in (d) says that the next fraction to be inserted between them is <sup>a</sup>`<sup>e</sup> b`f corresponding to the vector sum of pb, aq and pf, eq. And the result in (c) says that the area of the parallelogram with vertices p0, 0q, pb, aq, pf, eq, pb ` f, a ` eq is equal to 1 (see Problem 57(b)). Hence the result in (c) reduces to the fact that

Theorem Any parallelogram, whose vertices are integer lattice points (i.e. points pb, aq where both coordinates are integers), and with no additional lattice points inside the parallelogram or on the four sides, has area 1.

#### 110.

	- (i) If x ą 0, then x <sup>2</sup> ´ x ´ 2 " px ´ 2qpx ` 1q ď 0. 6 ´1 ď x ď 2 (and x ą 0); hence 0 ă x ď 2, and all such x satisfy the original inequality.
	- (ii) If x ă 0, then x <sup>2</sup> ´ x ´ 2 ě 0, so px ´ 2qpx ` 1q ě 0. 6 either x ď ´1, or x ě 2 (and x ă 0); hence x ď ´1, and all such x satisfy the original inequality.

#### 111.

(a) If a ` b " 5 and ab " 7, then a, b are solutions of

$$(x-a)(x-b) = x^2 - 5x + 7 = 0.5$$

But the roots of this quadratic equation are

$$\frac{5 \pm \sqrt{25 - 28}}{2} = \frac{5 \pm \sqrt{-3}}{2},$$

so a and b cannot be "positive reals".

(b) We abbreviate the "arithmetic mean" by AM, the "geometric mean" by GM, the "harmonic mean" by HM, and the "quadratic mean" by QM.

$$\left(\sqrt{a} - \sqrt{b}\right)^2 \gg 0$$

so

$$a + b \geqslant 2\sqrt{ab}$$

therefore

$$
\sqrt{ab} \rhd \frac{2ab}{a+b} \qquad \text{(GM \gg HM)}
$$

and

$$
\sqrt{ab} \lessapprox \frac{a+b}{2} \qquad \text{(GM \lessapprox AM)}.
$$

Also

$$
\left(\frac{a-b}{2}\right)^2 \geqslant 0,
$$

so

$$\frac{a^2 + b^2}{4} \geqslant \frac{2ab}{4}$$

whence

$$\frac{a^2 + b^2}{2} \geqslant \frac{a^2 + b^2 + 2ab}{4} = \left(\frac{a+b}{2}\right)^2.$$

Therefore

$$
\sqrt{\frac{a^2 + b^2}{2}} \gg \frac{a+b}{2} \qquad \text{(QM \gg AM)}.\tag{\text{QED}}
$$

112. [This delightful problem was devised by Oleksiy Yevdokimov.] We need to find something which remains constant, or which does not increase, when we replace two terms a, b by <sup>a</sup>?`<sup>b</sup> 2 .

Idea: If the two terms a, b were replaced each time by their sum a ` b, then the sum of all the numbers in the list would be unchanged, so we could be sure that the final number after 199 such moves would have to be

$$1 + 2 + 3 + \dots + 200 = \frac{200 \times 201}{2}.$$

This doesn't work here. However, in the spirit of this section on inequalities, one may ask:

What happens to the sum of the squares of the terms in the list after each move?

When we move from one list to the next, only two terms are affected, and for these two terms, the previous sum of squares is replaced by ´ <sup>a</sup>?`<sup>b</sup> 2 ¯2 . How does this affect the sum of all squares on the list?

We know that a <sup>2</sup> `b <sup>2</sup> ě 2ab for all a, b. And it is easy to see that this is equivalent to: ˆ ˙2

$$a^2 + b^2 \geqslant \left(\frac{a+b}{\sqrt{2}}\right)^2 \cdot 1$$

So when we replace two terms a, b by <sup>a</sup>?`<sup>b</sup> 2 , the sum of the squares of all the terms in the list never increases. Hence the final term is less than or equal to the square root of the initial sum of squares

$$\begin{aligned} \left(1^2 + 2^2 + 3^2 + \dots + 200^2\right)^2 &= \frac{200 \times 201 \times 401}{6} \\ &< \frac{200 \times 300 \times 400}{6} \\ &= 4 \times 10^6. \end{aligned}$$

6 the final term is ă ? 4 ˆ 10<sup>6</sup> " 2000.

113.

	- (ii) It seems hard to find another.
	- (ii) 5 " 2 <sup>2</sup> ` 1 (or 17 " 4 <sup>2</sup> ` 1; or 37 " 6 <sup>2</sup> ` 1; or 101 " 10<sup>2</sup> ` 1; or . . . ). In other words, there seem to be lots.

Note: At first sight primes of this form "keep on coming". Given that we now know (see Problem 76) that the list of all prime numbers "goes on for ever", it is natural to ask: Are there infinitely many prime numbers "one more than a square"? Or does the list run out?

This is one of the simplest questions one can ask to which the answer is not yet known!

$$\text{(c)}\ (\text{i)}\ 7 = 2^3 - 1.$$

(ii) It seems hard to find another.

$$\text{(d)}\text{ (i)}\text{ 2 } = 1^3 + 1.$$

(ii) It seems hard to find another.

Note: Parts (a), (c) and (d) should make one suspicious – provided one notices that:


This problem is so instructive that its solution is discussed in the main text following Problem 115.

114.

$$x^4 + 1 = \left(x^2 + \sqrt{2} \cdot x + 1\right)\left(x^2 - \sqrt{2} \cdot x + 1\right)\dots$$

(Suppose

$$x^4 + 1 = \left(x^2 + ax + b\right)\left(x^2 + cx + d\right)\dots$$

It is natural to try b " d " 1 in order to make the constant term bd " 1, and then to try c " ´a (so that the coefficients of x 3 and of x are both 0). It then remains to choose the value of a so that the total coefficient "2 ´ a 2 " of all terms in x 2 is equal to 0: that is, a " ? 2.)

#### 115.

(a)(i)

$$a^3 - b^3 = (a - b)(a^2 + ab + b^2).$$

(ii)

$$\begin{aligned} \label{eq:1} a^4 - b^4 &=& (a-b) \left(a^3 + a^2b + ab^2 + b^3\right) \\ &=& \left(a^2 - b^2\right) \left(a^2 + b^2\right) \\ &=& (a-b)(a+b) \left(a^2 + b^2\right) .\end{aligned}$$

(iii)

$$a^n - b^n = (a - b) \left( a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \dots + ab^{n-2} + b^{n-1} \right) \dots$$

Note: The general factorisation

$$x^n - 1 = (x - 1) \left( x^{n-1} + x^{n-2} + \dots + x^2 + x + 1 \right)$$

provides a fresh slant on the test for divisibility by 9 in base 10, or in general for divisibility by b ´ 1 in base b (see Problem 51):

"an integer written in base b is divisible by b ´ 1 precisely when its digit sum is divisible by b ´ 1".

$$\begin{array}{c}\text{(b)}\quad\text{(b)}\quad\text{(c)}\quad\text{(d)}\quad\text{(e)}$$

$$a^3 + b^3 = (a+b)\left(a^2 - ab + b^2\right).$$

(ii)

$$a^5 + b^5 = (a+b)(a^4 - a^3b + a^2b^2 - ab^3 + b^4).$$

(iii)

$$a^{2n+1} + b^{2n+1} = (a+b)\left(a^{2n} - a^{2n-1}b + a^{2n-2}b^2 - a^{2n-3}b^3 + \cdots - ab^{2n-1} + b^{2n}\right).$$

#### 116.

(a) Replace a by 1, b by r, and n by n ` 1 in the answer to 115(a)(iii), to see that:

$$1 + r + r^2 + r^3 + \dots + r^n = \frac{1 - r^{n+1}}{1 - r}.$$

(b) Multiply the closed formula in (a) by "a" to see that:

$$a + ar + ar^2 + ar^3 + \dots + ar^n = a \cdot \frac{1 - r^{n+1}}{1 - r}.$$

117. When x " 40,

$$f(x) = x^2 + (x + 40) + 1 = 40^2 + 2 \times 40 + 1 = 41^2$$

is not prime. So the sequence of prime outputs must stop some time before fp40q. But it in fact keeps going as long as it possibly could, so that

$$f(0), \ f(1), \ f(2), \ \dots, \ f(39)$$

are all prime. (This may explain Euler's delight.)

Note: The links between polynomials with integer coefficients (even lowly quadratics) and prime numbers are still not fully understood. For example, you might like to look up Ulam's spiral. (Ulam (1909–1984) plotted the positive integers in a square spiral and found the primes arranging themselves in curious patterns that we still do not fully understand.)

Interest in the connections between polynomials and primes was revived in the second half of the 20th century. It was eventually proved that there exists a polynomial in 10 variables, with integer coefficients, which takes both positive and negative values when the variables run through all possible non-negative integer values, but which does so in such a way that it generates all the primes as the set of positive outputs.

#### 118.

(a)(i) For

$$a^n - 1 = (a - 1) \left( a^{n-1} + a^{n-2} + \dots + a + 1 \right)$$

to be prime, the smaller factor must be " 1, so a " 2. If n is not prime, we can factorise n " rs, with r, s ą 1. Then

$$2^n - 1 = 2^{rs} - 1 = (2^r)^s - 1 = (2^r - 1) \left( 2^{r\{s1\}} + 2^{r\{s2\}} + \dots + 2 + 1 \right);$$

Hence 2<sup>n</sup> 1 also factorises, so could not be prime. Hence n must be prime.

(ii) 2 <sup>2</sup> ´ 1 " 3, 2<sup>3</sup> ´ 1 " 7, 2<sup>5</sup> ´ 1 " 31, 2<sup>7</sup> ´ 1 " 127 are all prime; 2<sup>11</sup> ´ 1 " 2047 " 23 ˆ 89 is not.

Note: This is a simple example of the need to distinguish carefully between the statement

"if 2<sup>n</sup> ´ 1 is prime, then n is prime" (which is true),

and its converse

"if n is prime, then 2<sup>n</sup> ´ 1 is prime" (which is false).

(b)(i) Suppose that a ą 1. Then a <sup>n</sup> ` 1 ą 2; so for a <sup>n</sup> ` 1 to be prime, it must be odd, so a must be even.

If n has an odd factor m ą 1, we can write n " km. Then

$$\begin{aligned} \left(a^n + 1\right) &= \quad a^{km} + 1\\ &= \quad (a^k)^m + 1\\ &= \quad (a^k + 1)\left(a^{k\{m-1\}} - a^{k\{m-2\}} + \dots - a^k + 1\right). \end{aligned}$$

Since m is odd and ą 1, we have m ě 3. It is then easy to show that

a <sup>k</sup> ` 1 ď a <sup>k</sup>pm´1<sup>q</sup> ´ a <sup>k</sup>pm´2<sup>q</sup> ` ¨ ¨ ¨ ´ a <sup>k</sup> ` 1.

And since a ą 1, neither factor " 1, so a <sup>n</sup> ` 1 can never be prime. Hence n can have no odd factor ą 1, which is the same as saying that n " 2 <sup>r</sup> must be a power of 2.

(ii)  $2^1 + 1 = 3$ ,  $2^2 + 1 = 5$ ,  $2^4 + 1 = 17$ ,  $2^8 + 1 = 257$ ,  $2^{16} + 1 = 65$  537 are all prime. (The very next such expression

$$2^{32} + 1 = 4 \, 294 \, 967 \, 297 = 641 \times 6 \, 700 \, 417$$

is not prime.)

Note: The sad tale of Fermat's claim that "all Fermat numbers are prime" shows that mathematicians are not exempt from the obligation to distinguish carefully between a statement and its converse!

#### 119.


– to complete the square

$$x^2 + \sqrt{2}x - 1 = \left(x + \frac{\sqrt{2}}{2}\right)^2 - 1 - \frac{1}{2},$$

so

$$x + \frac{\sqrt{2}}{2} - \pm \sqrt{\frac{3}{2}},$$

– or to use the quadratic formula:

$$x = \frac{-\sqrt{2} \pm \sqrt{2+4}}{2}.$$

(e) x <sup>2</sup> ` x ´ ? 2 " 0 requires us – to complete the square

so

$$\begin{aligned} x^2 + x - \sqrt{2} &= \left( x + \frac{1}{2} \right)^2 - \sqrt{2} - \frac{1}{4}, \\\\ x + \frac{1}{2} &= \pm \sqrt{\sqrt{2} + \frac{1}{4}} \end{aligned}$$

– or to use the quadratic formula:

$$x = \frac{-1 \pm \sqrt{1 + 4\sqrt{2}}}{2}.$$

(f) x <sup>2</sup> ` <sup>1</sup> " 0 yields <sup>x</sup> " ˘? ´1. (g) x <sup>2</sup> ` ? 2x ` 1 " 0 yields

$$x = \frac{-\sqrt{2} \pm \sqrt{2-4}}{2} = \frac{-\sqrt{2} \pm \sqrt{-2}}{2}.$$

120. qpxq " x <sup>2</sup> ´ ? 2 ¨ x ` 1. (There is no obvious magic method here. However, it should be natural to try to insert a term ? ? 2 ¨ x in qpxq to "resolve" the term 2 ¨ x in ppxq; and the familiar cancelling of cross terms in pa ` bqpa ´ bq should then suggest the possible benefit of trying qpxq " x <sup>2</sup> ´ ? 2 ¨ x ` 1.)

Note: ppxqqpxq " x <sup>4</sup> ` 1 (see Problem 114).

#### 121.

(a) Let the two unknown numbers be α and β. Then s " α ` β, and p " αβ. "The square of half the sum" ` s 2 ˘2 " ` <sup>α</sup>`<sup>β</sup> 2 ˘2 . Subtracting <sup>p</sup> " αβ produces ` <sup>α</sup>´<sup>β</sup> 2 ˘2 whose "square root" will be either <sup>α</sup>´<sup>β</sup> 2 , or ´ ` <sup>α</sup>´<sup>β</sup> 2 ˘ – whichever is positive.

Adding this to "half the sum" gives one root; subtracting gives the other root.

(b) Let the length of one side be x. We are told that x <sup>2</sup> ` bx " c.

"Take half of b, square it, and add the result to c"

translates as:

"Rewrite the equation as: ` x ` b 2 ˘<sup>2</sup> " <sup>c</sup> ` ` b 2 ˘2 ." ˘2

That is, we have "completed the square" ` x ` b 2 . If we now take the (positive) square root and subtract <sup>b</sup> 2 , we get a single value for x, which determines the side length of my square as required.

If the same method is applied to the general quadratic equation

$$ax^2 + bx + c = 0,$$

with the extra initial step of "multiply through by <sup>1</sup> a ", we produce first

$$x^2 + \frac{b}{a}x + \frac{c}{a} = 0,$$

then

$$
\left(x + \frac{b}{2a}\right)^2 + \left(\frac{c}{a} - \left(\frac{b}{2a}\right)^2\right) = 0,
$$

then

$$x + \frac{b}{2a} = \pm \sqrt{\left(\frac{b}{2a}\right)^2 - \frac{c}{a}} = \frac{\pm \sqrt{b^2 - 4ac}}{2a},$$

which leads to the familiar quadratic formula.

(c) See Problem 3(c)(iv). AD : CB " DX : BX, so x : 1 " 1 : px ´ 1q. Hence x <sup>2</sup> ´ x ´ 1 " 0. If we use the quadratic formula derived in the answer to part (b) above, and realise that x ą 1, then we obtain x " 1` ? 5 2 .

Note: The procedure given in (a) dates back to the ancient Babylonians (" 1700 BC) and later to the ancient Greeks (" 300 BC). Both cultures worked without algebra. The Babylonians gave their verbal procedures as recipes in words, in the context of particular examples. The Greeks expressed everything geometrically. In modern language, if we denote the unknown numbers by α and β, then

$$(x - \alpha)(x - \beta) = x^2 - (\alpha + \beta)x + \alpha\beta.$$

Being told the sum and product is therefore the same as being given the coefficients of a quadratic equation, and being asked to find the two roots.

Our method for factorizing a quadratic involves a mental process of 'inverse arithmetic', where we juggle possibilities in search of α and β, when all we know are the coefficients (that is, the sum α ` β, and the product αβ).

The procedure in (b) also dates back to the ancient Babylonians, and is essentially our process of completing the square. It was given as a procedure, without our algebraic notation. The Babylonians seem not to have been hampered (as the Greeks were) by the fact that it makes no sense to add a length and an area! They worded things geometrically, but seem to have understood that they were really playing numerical games (an idea which European mathematicians found elusive right up to the time of Descartes (1590–1656)).

Similarly, the modern use of symbols – allowing one to represent either positive or negative quantities – was widely resisted right into the nineteenth century. What we would write as a single family of quadratic equations, ax<sup>2</sup>`bx`c " 0, had to be split into separate cases where two positive quantities were equated. For example, the groundbreaking book Ars Magna in which Cardano (1501–1576) explained how to solve cubic and quartic equations begins with quadratics – where his procedure distinguishes four different cases: "squares equal to numbers", "squares equal to things", "squares and things equal to numbers", "squares and numbers equal to things".

122.

	- (ii) α <sup>2</sup>β ` β <sup>2</sup>α " αβpα ` βq " c ¨ p´bq " ´bc.
	- (iii) We rearrange

$$\begin{aligned} \alpha^3 + \beta^3 - 3\alpha\beta &= \quad \left(\alpha + \beta\right)\left(\alpha^2 - \alpha\beta + \beta^2\right) - 3\alpha\beta \\ &= \quad \left(-b\right) \cdot \left(b^2 - 3c\right) - 3c \\ &= \quad -b^3 + 3bc - 3c. \end{aligned}$$

[Alternatively: α <sup>3</sup> ` β <sup>3</sup> " pα ` βq <sup>3</sup> ´ 3αβpα ` βq, etc.]

(b)(i) [Cf 121(a).] pα ´ βq <sup>2</sup> " pα ` βq <sup>2</sup> ´ 4αβ. Therefore α ´ β " ? b <sup>2</sup> ´ 4c if α ě β,

and

$$\alpha - \beta = -\sqrt{b^2 - 4c} \text{ if } \,\alpha < \beta.$$

(ii)

$$
\alpha^2 \beta - \beta^2 \alpha = -\alpha \beta (\alpha - \beta) = -c\sqrt{b^2 - 4c} \text{ if } \; \alpha \ni \beta,
$$

and

$$
\alpha^2 \beta - \beta^2 \alpha = -\alpha \beta (\alpha - \beta) = c\sqrt{b^2 - 4c} \text{ if } \,\, \alpha < \beta.
$$

(ii)  $\alpha^3 - \beta^3 = (\alpha - \beta)(\alpha^2 + \alpha\beta + \beta^2)$ 

Therefore 
$$\alpha^3 - \beta^3 = \left[\sqrt{b^2 - 4c}\right](b^2 - c) \quad \text{if} \quad \alpha \gg \beta,$$

and 
$$\alpha^3 - \beta^3 = \left[-\sqrt{b^2 - 4c}\right](b^2 - c) \quad \text{if} \quad \alpha < \beta.$$

#### 123.

(a)(i) ? a ` ? <sup>b</sup> and <sup>a</sup> a ` b ` ? 4ab are both positive. And it is easy to check that they have the same square:

$$
\left(\sqrt{a} + \sqrt{b}\right)^2 = a + b + 2\sqrt{ab},
$$

and

$$\left(\sqrt{a+b+\sqrt{4ab}}\right)^2 = a+b+\sqrt{4ab}.$$

$$\boxed{\phantom{\frac{\pi}{4}}}$$

Hence

$$
\sqrt{a} + \sqrt{b} = \sqrt{a + b + \sqrt{4ab}}.
$$

(ii) 5 " 2 ` 3, and 24 " 4 ˆ 2 ˆ 3; Therefore b

$$
\sqrt{2+3+\sqrt{4\times2\times3}} = \sqrt{2}+\sqrt{3}
$$

(which is easy to check).

(b)(i) Claim If a ě b (‰ 0), then

$$
\sqrt{a} - \sqrt{b} = \sqrt{a + b - \sqrt{4ab}}.
$$

Proof ? a ´ ? <sup>b</sup> and <sup>a</sup> a ` b ´ ? 4ab are both ě 0 (Why?). And it is easy to check that ´? ¯2

$$
\left(\sqrt{a} - \sqrt{b}\right)^2 = a + b - 2\sqrt{ab},
$$

and

$$\left(\sqrt{a+b-\sqrt{4ab}}\right)^2 = a+b-\sqrt{4ab}.\tag{QED}$$

$$\begin{pmatrix} 1 & \sqrt{a-\sqrt{20}} \end{pmatrix}$$

(ii) Simplify <sup>a</sup> 5 ´ ? 16 and <sup>a</sup> 6 ´ 20. <sup>5</sup> " <sup>4</sup> ` 1 and 16 " <sup>4</sup> <sup>ˆ</sup> <sup>4</sup> <sup>ˆ</sup> 1, so <sup>a</sup> 5 ´ ? 16 " ? 4 ´ ? 1 " 1. Actually, there is a simpler solution: b

$$
\sqrt{5-\sqrt{16}} = \sqrt{5-4} = \sqrt{1} = 1.
$$

<sup>6</sup> " <sup>5</sup> ` 1 and 20 " <sup>4</sup> <sup>ˆ</sup> <sup>5</sup> <sup>ˆ</sup> 1, so <sup>a</sup> 6 ´ ? 20 " ? 5 ´ ? 1 " ? 5 ´ 1.

#### 124.

(a) Let α " 1 ` ? 2. Then α <sup>2</sup> " 3 ` 2 ? 2. Hence α <sup>2</sup> ´ 2α " 1, so α satisfies the quadratic polynomial equation x <sup>2</sup> ´ 2x ´ 1 " 0.

Note: Observe that the resulting polynomial is equal to

$$
\left(x - \left(1 + \sqrt{2}\right)\right)\left(x - \left(1 - \sqrt{2}\right)\right)\dots
$$

In other words, to rationalize the coefficients, we need a polynomial which has both α " 1 ` ? 2 and its "conjugate" 1 ´ ? 2 as roots.

(b) Let α " 1 ` ? 3. Then α <sup>2</sup> " 4 ` 2 ? 3. Hence α <sup>2</sup> ´ 2α " 2, so α satisfies the quadratic polynomial equation x <sup>2</sup> ´ 2x ´ 2 " 0.

Note: Observe that the resulting polynomial is equal to

$$
\left(x - \left(1 + \sqrt{3}\right)\right)\left(x - \left(1 - \sqrt{3}\right)\right)\dots
$$

In other words, to rationalize the coefficients, we need a polynomial which has both α " 1 ` ? 3 and its "conjugate" 1 ´ ? 3 as roots.

(c) Let α " ? 2 ` ? 3. Then α <sup>2</sup> " 5 ` 2 ? 6, so α <sup>2</sup> ´ 5 " 2 ? 6, and ` α <sup>2</sup> ´ 5 ˘<sup>2</sup> " 24. Hence α satisfies the quartic polynomial equation x <sup>4</sup> ´ 10x <sup>2</sup> ` 1 " 0.

Note: Observe that the resulting polynomial is equal to

$$
\left(x - \left(\sqrt{2} + \sqrt{3}\right)\right)\left(x - \left(\sqrt{2} - \sqrt{3}\right)\right)\left(x - \left(-\sqrt{2} + \sqrt{3}\right)\right)\left(x - \left(-\sqrt{2} - \sqrt{3}\right)\right)\dots
$$

In other words, the roots are: ? 2` 3 (as required), and also ? 2´ 3, ´ 2´ 3, and ´ ? 2 ` ? 3.

(d) Let α " ? 2 ` ?<sup>1</sup> 3 . Then

so ˆ

$$
\alpha^2 = \frac{7}{3} + 2\sqrt{\frac{2}{3}},
$$

$$
\left(\alpha^2 - \frac{7}{3}\right)^2 = \frac{8}{3},
$$

$$
\dots
$$

c

and α satisfies the quartic polynomial equation

$$x^4 - \frac{14}{3} \cdot x^2 + \frac{25}{9} = 0.$$

Note:

$$\begin{array}{rcl} x^4 - \frac{14}{3} \cdot x^2 + \frac{25}{9} & = & \left( x - \left[ \sqrt{2} + \frac{1}{\sqrt{3}} \right] \right) \left( x - \left[ \sqrt{2} - \frac{1}{\sqrt{3}} \right] \right) \\ & \cdot \left( x + \left[ \sqrt{2} + \frac{1}{\sqrt{3}} \right] \right) \left( x + \left[ \sqrt{2} - \frac{1}{\sqrt{3}} \right] \right), \end{array}$$

so the roots are:

$$x = \sqrt{2} + \frac{1}{\sqrt{3}}, \ \sqrt{2} - \frac{1}{\sqrt{3}}, \ -\sqrt{2} - \frac{1}{\sqrt{3}}, \ -\sqrt{2} + \frac{1}{\sqrt{3}}.$$

125. A direct approach can be made to work in both cases (but see the Notes).

(a) Suppose to the contrary that ? 2 ` ? 3 " p q , for some integers p, q with HCFpp, qq " 1. Then ` 5 ` 2 ? 6 ˘ q <sup>2</sup> " p 2 , so ? 6 is rational, and we can write ? 6 " r <sup>s</sup> with HCFpr, sq " 1. But then 6s <sup>2</sup> " r 2 ; hence r " 2t must be even; so 3s <sup>2</sup> " 2t 2 ? , but then s must be even – contradicting HCFpr, sq " 1. Hence 2 ` ? 3 cannot be rational.

Note: It is slightly easier to rewrite the initial equation in the form

$$
\sqrt{3} = \frac{p}{q} - \sqrt{2}
$$

before squaring to get

$$
\left(\frac{p}{q}\right)^2 - 1 = \frac{2p}{q}\sqrt{2},
$$

which would imply that ? 2 is rational.

(b) Suppose to the contrary that ? 2 ` ? 3 ` ? 5 " p q , for some integers p, q with HCFpp, qq " 1. Then

$$10 + 2\left(\sqrt{6} + \sqrt{10} + \sqrt{15}\right) = \left(\frac{p}{q}\right)^2,$$

so ? 6 ` ? 10 ` ? 15 is rational. Squaring ? 6 ` ? 10 ` ? 15 then gives that

$$
\sqrt{60} + \sqrt{90} + \sqrt{150} = 5\sqrt{6} + 3\sqrt{10} + 2\sqrt{15}
$$

is rational. Subtracting 2p ? 6 ` ? 10 ` a <sup>15</sup><sup>q</sup> then shows that 3? 6 ` ? 10 is rational, and we can proceed as in part (a) to obtain a contradiction. Hence ? 2 ` ? 3 ` ? 5 cannot be rational.

Note: It is simpler to rewrite the original equation in the form

$$
\sqrt{2} + \sqrt{3} = \frac{p}{q} - \sqrt{5}
$$

before squaring to obtain

$$5 + 2\sqrt{6} = \left(5 + \left(\frac{p}{q}\right)^2\right) - \frac{2p}{q}\sqrt{5},$$

whence 2? 6 ` 2p q ? 5 is rational, and we may proceed as in part (a).

#### 126.

(i) We just have to fill in the missing bits of the partial factorisation

$$x^{10} + 1 = \left(x^3 - 1\right)\left(x^7 + x^4 + \cdots\right) + \text{ remainder.}$$

To produce the required term x <sup>10</sup> we first insert x 7 . This then creates an unwanted term "´x 7 ", so we add `x 4 to cancel this out. This in turn creates an unwanted term "´x 4 ", so we add `x to cancel this out. Hence the quotient is x <sup>7</sup> ` x <sup>4</sup> ` x, and the remainder is "x ` 1":

$$x^{10} + 1 = \left(x^3 - 1\right)\left(x^7 + x^4 + x\right) + (x + 1).$$

Note: ` It is worth noting a short cut. The factorised term of the form x <sup>3</sup> ´ 1 ˘ `x <sup>7</sup> ` ¨ ¨ ¨ ˘ is equal to zero when x <sup>3</sup> " 1.

So one way to get the remainder is to "treat x 3 as if it were equal to 1". Then

$$x^{10} = \left(x^3\right)^3 \cdot x$$

is just like 1 ¨ x, and x <sup>10</sup> ` 1 behaves as if it were equal to x ` 1, which is the remainder.

(ii)

$$x^{2013+1} = \left(x^2 - 1\right)\left(x^{2011} + x^{2009} + x^{2007} + \dots + x\right) + \left(x + 1\right),$$

so the remainder " x ` 1.

Note: If we treat x 2 "as if it were equal to 1", then

$$x^{2013} + 1 = \left(x^2\right)^{1006} \cdot x + 1$$

behaves as if it were equal to 1 ¨ x ` 1.

(iii) Apply the Euclidean algorithm to m and n in order to write m " qn ` r, where 0 ď r ă n:

$$x^m = x^{qn+r} = (x^n)^q \cdot x^r.$$

Then

$$\begin{aligned} \;x^m + 1 &= \quad x^{qn+r} + 1\\ &= \quad (x^n - 1) \left( x^{n\{q-1\}+r} + x^{n\{q-2\}+r} + x^{n\{q-3\}+r} + \dots + x^r \right) + x^r + 1. \end{aligned}$$

So the remainder is x <sup>r</sup> ` 1.

Note: If we treat x <sup>n</sup> ´ 1 as if were 0 – that is, if we treat x n as if it were equal to 1 – then

$$x^m + 1 = x^{qn+r} + 1 = (x^n)^q \cdot x^r + 1$$

which behaves like 1<sup>q</sup> ¨ x <sup>r</sup> ` 1.

$$\textbf{127. Suppose } x^{2013} + 1 = \left(x^2 + x + 1\right)q(x) + r(x)\text{, where } \text{deg}(r(x)) < 2. \text{ Then}$$

$$\begin{aligned} \left(x^{2013} + 1\right)\left(x - 1\right) &=& x^{2014} - x^{2013} + x - 1\\ &=& \left(x^3 - 1\right)q(x) + (x - 1)r(x). \end{aligned}$$

Now

$$x^{2014} - x^{2013} + x - 1 \quad = \begin{array}{c} \left(x^3 - 1\right) \left(x^{2011} - x^{2010} + x^{2008} - x^{2007} + x^{2004} - x^{2003} + \cdots + x \cdot 2000 + \cdots \right) \\ + 2x - 2 \end{array}$$

˘

so the remainder rpxq " 2.

Note: If x satisfies x <sup>2</sup> ` x ` 1 " 0, then x <sup>3</sup> ´ 1 " 0 and x ‰ 1. 6 x <sup>2013</sup> ` 1 " ` x 3 ˘<sup>671</sup> ` 1 behaves just like 1<sup>671</sup> ` <sup>1</sup> " 2, so <sup>r</sup>pxq " 2.

#### 128.

(a)

$$(a+bi)^{-1} = \frac{a}{a^2+b^2} - \left[\frac{b}{a^2+b^2}\right]i.$$

(b)

$$p(x) = x^2 - 2ax + \left(a^2 + b^2\right).$$

(Suppose that the quadratic equation

$$p(x) = x^2 + cx + d = 0,$$

with real coefficients c, d, has x " a ` ib as a root. Then take the complex conjugate of the equation ppxq " 0 to see that x " a ´ ib is also a rooti of

$$p(x) = x^2 + cx + d = 0.$$

Therefore

$$\begin{aligned} p(x) &= \quad x^2 + cx + d\\ &= \quad (x - (a + ib))(x - (a - ib)), \end{aligned}$$

$$\text{so } c = -2a \text{, and } d = a^2 + b^2 \text{.} )$$

129. Let the two unknown numbers be α and β. Then 10 " α ` β, and 40 " αβ, so α and β are roots of the quadratic equation x <sup>2</sup> ´ 10x ` 40 " 0. Hence

$$\alpha, \beta = \frac{10 \pm \sqrt{100 - 160}}{2} = 5 \pm \sqrt{-15}.$$

#### 130.

(a) Applying a simple rearrangement:

$$\begin{aligned} \begin{array}{rcl} wz &=& r(\cos\theta + i\sin\theta) \cdot s(\cos\phi + i\sin\phi) \\ &=& rs[(\cos\theta \cdot \cos\phi - \sin\theta\sin\phi) + i(\cos\theta \cdot \sin\phi + \sin\theta \cdot \cos\phi)] \\ &=& rs[\cos(\theta + \phi) + i\sin(\theta + \phi)] \end{array} \end{aligned}$$

(by the usual addition formula: Problem 35)

(b) By part (a),

$$\left(\cos\theta + i\sin\theta\right)^2 = \cos(2\theta) + i\sin(2\theta).$$

Hence

$$\begin{aligned} \left(\cos\theta + i\sin\theta\right)^3 &=& \left[\cos\theta + i\sin\theta\right)^2 \left(\cos\theta + i\sin\theta\right) \\ &=& \left[\cos(2\theta) + i\sin(2\theta)\right] \cdot \left(\cos\theta + i\sin\theta\right) \\ &=& \cos(3\theta) + i\sin(3\theta) .\end{aligned}$$

Etc.

Note: This should really be presented as a "proof by mathematical induction", where (having established the initial cases) we "suppose the result holds for powers n " 1, 2, 3, . . . , k", and then conclude that

$$\begin{aligned} \left(\cos\theta + i\sin\theta\right)^{k+1} &=& \left[\cos\theta + i\sin\theta\right)^{k}\left(\cos\theta + i\sin\theta\right) \\ &=& \left[\cos(k\theta) + i\sin(k\theta)\right]\left(\cos\theta + i\sin\theta\right) \\ &=& \cos((k+1)\theta) + i\sin((k+1)\theta) .\end{aligned}$$

(c) z <sup>n</sup> " r n pcospnθq ` i sinpnθqq. Hence if z <sup>n</sup> " 1, then |z n | " r <sup>n</sup> " 1, so r " 1 (since r ě 0).

#### 131.

(a) We factorise: x <sup>3</sup> ´ 1 " px ´ 1q ` x <sup>2</sup> ` x ` 1 ˘ , so the roots are x " 1; and

$$x = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$$

that is, the other two roots are

$$x = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)$$

and

$$x = \cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right).$$

(b) We factorise:

$$x^4 - 1 = \left(x^2 - 1\right)\left(x^2 + 1\right) = (x - 1)(x + 1)\left(x^2 + 1\right),$$

so the roots are x " 1, x " ´1, x " i, x " ´i.

(c) We factorise:

$$\begin{aligned} \label{eq:1} x^6 - 1 &=& \left[ \left( x^2 \right)^3 - 1 \right] \\ &=& \left( x^2 - 1 \right) \left( x^4 + x^2 + 1 \right) \\ &=& (x - 1)(x + 1) \left[ \left( x^2 \right)^2 + x^2 + 1 \right], \end{aligned}$$

so the roots are

– x " 1, x " ´1, and

– four further values of x satisfying x <sup>2</sup> " ´ 1 <sup>2</sup> ˘ ? 3 2 i : that is,

$$x = \cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\pi}{3}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2}i$$

and

$$x = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$$

and

$$x = \cos\left(\frac{-\pi}{3}\right) + i\sin\left(\frac{-\pi}{3}\right) = \frac{1}{2} - \frac{\sqrt{3}}{2}i$$

and

$$x = \cos\left(\frac{-2\pi}{3}\right) + i\sin\left(\frac{-2\pi}{3}\right) = -\frac{1}{2} - \frac{\sqrt{3}}{2}i.$$

(d) We factorise:

$$\begin{aligned} \left(x^{8}-1\right) &= \quad \left(x^{4}-1\right)\left(x^{4}+1\right) \\ &= \quad \left(x^{2}-1\right)\left(x^{2}+1\right)\left(x^{2}+\sqrt{2}\cdot x+1\right)\left(x^{2}-\sqrt{2}\cdot x+1\right) \end{aligned}$$

so the roots are


$$x = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$$

and

$$x = \cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$$

and

$$x = \cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$$

and

$$x = \cos\left(-\frac{3\pi}{4}\right) + i\sin\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$$

132. [In Problem 114 you were left to work out the required factorisation with your bare hands – and a bit of inspired guesswork. The suggested approach here is more systematic.]

The roots of x <sup>4</sup> ` 1 " 0 are complex numbers whose fourth power is equal to ´1: that is, ¯ ¯ ? ?

$$x = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$$

and

$$x = \cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$$

$$\therefore \quad \dots \quad \dots \quad \dots \quad \dots$$

and

$$x = \cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\pi$$

and

$$x = \cos\left(-\frac{3\pi}{4}\right) + i\sin\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i.$$

The first two are complex conjugates and give rise to two linear factors whose product is x <sup>2</sup> ` ? 2 ¨ x ` 1; the other two are complex conjugates and give rise to two linear factors whose product is x <sup>2</sup> ´ ? 2 ¨ x ` 1. Hence

$$x^4 + 1 = \left(x^2 + \sqrt{2} \cdot x + 1\right)\left(x^2 - \sqrt{2} \cdot x + 1\right).$$

#### 133.

(a) The roots of x <sup>5</sup> ´ 1 " 0 are precisely the five complex numbers of the form

$$\cos\left(\frac{2k\pi}{5}\right) + i\sin\left(\frac{2k\pi}{5}\right), \text{ for } k = 0, 1, 2, 3, 4;$$

that is,

$$\begin{aligned} x &= -1 \\ x &= -\cos\left(\frac{2\pi}{5}\right) + i\sin\left(\frac{2\pi}{5}\right) \\ x &= -\cos\left(\frac{4\pi}{5}\right) + i\sin\left(\frac{4\pi}{5}\right) \\ x &= -\cos\left(\frac{6\pi}{5}\right) + i\sin\left(\frac{6\pi}{5}\right) \\ x &= -\cos\left(\frac{8\pi}{5}\right) + i\sin\left(\frac{8\pi}{5}\right) \end{aligned}$$

From Problem 3(c) we know that

$$\begin{aligned} \cos\left(\frac{2\pi}{5}\right) &= \quad \frac{\sqrt{5}-1}{4} &= \cos\left(\frac{8\pi}{5}\right) \\ \sin\left(\frac{2\pi}{5}\right) &= \quad \frac{\sqrt{10+2\sqrt{5}}}{4} = -\sin\left(\frac{8\pi}{5}\right) \\ \cos\left(\frac{4\pi}{5}\right) &= \quad -\cos\left(\frac{\pi}{5}\right) &= -\frac{\sqrt{5}+1}{4} = \cos\left(\frac{6\pi}{5}\right) \\ \sin\left(\frac{4\pi}{5}\right) &= \quad \frac{\sqrt{10-2\sqrt{5}}}{4} = -\sin\left(\frac{6\pi}{5}\right) \end{aligned}$$

(b) The linear factor is clearly px ´ 1q. Each quadratic factor arises as the product of two conjugate linear factors. We saw in Problem 128(b) that two linear factors corresponding to roots a ` bi and a ´ bi produce the quadratic factor x <sup>2</sup> ´ 2ax ` ` a <sup>2</sup> ` b 2 ˘ . Hence the two quadratic factors are:

$$x^2 - \frac{\sqrt{5} - 1}{2} \cdot x + 1, \quad \text{and} \quad x^2 + \frac{\sqrt{5} + 1}{2} \cdot x + 1$$

(whose product is equal to x <sup>4</sup> ` x <sup>3</sup> ` x <sup>2</sup> ` x ` 1).

134.


$$x^3 + px^2 + qx + r = 0.$$

If we now put y " x ` p 3 , then y 3 incorporates both the x 3 and the x 2 terms, and the equation reduces to:

$$y^3 + \left[q - 3\left(\frac{p}{3}\right)^2\right]y + \left[r + 2\left(\frac{p}{3}\right)^3 - q\left(\frac{p}{3}\right)\right] = 0.5$$

135. Given the equation x <sup>3</sup> ` 3x <sup>2</sup> ´ 4 " 0. Let y " x ` 1.

(i) Then y <sup>3</sup> " x <sup>3</sup> ` 3x <sup>2</sup> ` 3x ` 1, so 0 " x <sup>3</sup> ` 3x <sup>2</sup> ´ 4 " y <sup>3</sup> ´ 3y ´ 2.

(ii) Set y " u ` v and use the fact that

$$(u+v)^3 = u^3 + 3uv(u+v) + v^3$$

is an identity, and so holds for all u and v.


136. The Euclidean algorithm for ordinary integers arises by repeating the division algorithm:

given integers m, n (‰ 0), there exists unique integers q, r such that m " qn ` r where 0 ď r ă n.

Here q is the quotient (the integer part of the division m˜n), and r is the remainder. If we then replace the initial pair pm, nq by the new pair pn, rq and repeat until we obtain the remainder 0, then the last non-zero remainder is equal to HCFpm, nq (see Problem 6). The same idea also works for polynomials with integer coefficients (see Problem 126).

We start by clarifying what we mean by divisibility for Gaussian integers. Given two Gaussian integers, m " a ` bi and n " c ` di, we say that n " c ` di divides m " a ` bi (exactly) precisely

when there exists some other Gaussian integer q " e ` f i such that m " qn: that is, a ` bi " pe ` f iqpc ` diq.

For example, 2 ` 3i divides ´4 ` 7i because p1 ` 2iqp2 ` 3iq " ´4 ` 7i.

If m " a ` bi and n " c ` di are any old Gaussian integers, then it will not in general be true that "n divides m", but we can imitate the division algorithm. The important idea here when carrying out particular calculations is to realize that "divide by c ` di" is the same as "multiply by <sup>c</sup>´di c <sup>2</sup>`d<sup>2</sup> "

• first carry out the division

$$m \div n = \frac{(a+bi)(c-di)}{c^2 + d^2};$$

• then take the "nearest" Gaussian integer q " e ` f i, and let the difference m ´ qn " r be the remainder.

As for ordinary integers, any Gaussian integer that is a "common factor of m and n" is then automatically a common factor of n and of r " m ´ qn, and conversely. That is, the common factors of m and n are precisely the same as the common factors of n and r. So we can repeat the process replacing m, n by n, r. Provided the "remainder" r is in some sense "smaller" than n, we can continue until we reach a stage where the remainder r " 0 – at which point, the last non-zero remainder is equal to the HCFpm, nq (that is, the Gaussian integer which is the HCF of the two initial Gaussian integers m, n).

The feature of the remainders that gets progressively smaller is their norm (see Problem 25, and Problem 54). As so often, this becomes clearer when we look at an example.

Let us try to find the HCF of the two Gaussian integers m " 14´42i and n " 4´7i.

• First do the division

$$m \div n = \frac{(14 - 42i)(4 + 7i)}{4^2 + 7^2} = \frac{350}{65} - \frac{70}{65}i.$$


$$n \div r = \frac{(4 - 7i)(1 + 3i)}{1^2 + 3^2} = \frac{5}{2} + \frac{1}{2}i.$$


$$1 - 3i = -(1 + i)(1 + 2i)$$

with remainder 0. Hence

$$1 + 2i = HCF(14 - 42i, 4 - 7i).$$

Note: One way to picture the process is to learn to "see" the Gaussian integers geometrically. Every Gaussian integer (such as a`bi) can be written as an integer combination of the two basic Gaussian integers "1" and "i" – namely

$$a + bi = a \times 1 + b \times i.$$

Since 1 and i are both of length 1 and perpendicular to each other, this represents the set of all Gaussian integers as the dots in a "square dot lattice" generated by translations in the x- and y- directions of the basic unit square spanned by 0, 1, i, and 1 ` i.

Any other given Gaussian integer, such as n " c ` di, then generates a "stretched and rotated" square lattice, which consists of all "Gaussian multiples" of c ` di – generated by the basic square which is spanned by

$$0, \ (c+di) \times 1, \ (c+di) \times i, \ \text{ and } \ (c+di) \times (1+i).$$

Every Gaussian integer (or rather the point, or dot, which corresponds to it) lies either on the boundary, or inside, one of these larger "stretched and rotated" squares: if the diagonal of one of these larger squares has length 2k, then any other Gaussian integer m " a ` bi lies inside one of these larger squares, and so lies within distance k (that is, half a diagonal) of some (Gaussian) multiple qn of n " c ` di. And the difference m ´ qn is precisely the required remainder r.

Extra: We interpret ?<sup>3</sup> 8 " 8 1 <sup>3</sup> " 2. Prove that

$$
\sqrt{-3}\sqrt{-1} \approx 23\frac{1}{7}
$$

(where « denotes "approximately equal to").

# V. Geometry

Those who fear to experiment with their hands will never know anything. George Sarton (1884–1956)

Mathematical truth is not determined arbitrarily by the rules of some 'man-made' formal system, but has an absolute nature and lies beyond any such system of specifiable rules. Roger Penrose (1930– )

Geometry is in many ways the most natural branch of elementary mathematics through which to convey "the essence" of the discipline.

	- a universe that is bursting with surprising facts, whose statements can be easily understood; and
	- which has a clear logical structure, in terms of which the proofs of these facts are accessible, if sometimes tantalisingly elusive.

This combination of elusive problems to be solved and the steady accumulation of proven results has provided generations of students with their first glimpse of serious mathematics. All readers can imagine the kind of experiences which lie behind the first bullet point above: many of the problems we have already met (such as Problems 4, 19, 20, 26, 27, 28, 29, 30, 31, 37, 38, 39) do not depend on the "semi-formal treatment" referred to in the second bullet point, so can be tackled by anyone who is interested – provided they accept the importance of learning to construct their own diagrams (in the spirit of the George Sarton quotation).

> The hand is the cutting edge of the mind. Jacob Bronowski (1908–1974)

But there is a catch – which explains why the present chapter appears so late in the collection. For many problems to successfully convey "the essence of mathematics" there has to be some shared understanding of what constitutes a solution. And in geometry, many solutions require the construction of a proof. Yet many readers will never have experienced a coherent "semi-formal treatment" of elementary geometry in the spirit of the second bullet point. Hence in Problems 3(c), 18, 21, 32, 34, 36 we committed the cardinal sin of leading the reader by the nose – breaking each problem into steps in order to impose a logical structure. This may have been excusable in Chapter 1; but in a chapter explicitly devoted to geometry, the underlying challenge has to be faced head on: that is, the raw experience of the hand has to be refined to provide a deductive structure for the mind.

As in Chapter 1, some of the problems listed from Section 5.3 onwards can be tackled without worrying too much about the logical structure of elementary geometry. But in many instances, the "essence" that is captured by a problem requires that the problem be seen within an agreed logical hierarchy – a sequencing of properties, results, and methods, which establishes what is a consequence of what – and hence, what can be used as part of a solution. In particular, we need to construct proofs that avoid circular reasoning.

If B is a consequence of A, or if B is equivalent to A, then a 'proof' of A which makes use of B is at best dubious, and may well be a delusion.

The need to avoid such circular reasoning arose already in Problem 21 (the converse of Pythagoras' Theorem), where we felt the need to state explicitly that it would be inappropriate to use the Cosine Rule: (see Problem 192 below).

Such concerns may explain why this chapter on geometry is the last of the chapters relating to elementary 'school mathematics', and why we begin the chapter with


Those with a strong background in geometry may choose to skip these sections on a first reading, and move straight on to the problems which start in Section 5.3. But they may then fail to see how the cumulative architecture of Section 5.2 conveys a rather different aspect of the "essence of mathematics", deriving not just from the individual problems, but from the way a carefully crafted, systematic arrangement of simple "bricks" can create a much more significant mathematical structure.

### 5.1. Comparing geometry and arithmetic

The opening quotations remind us that the mental universe of formal mathematics draws much of its initial inspiration from human perception and activity – activity which starts with infants observing, moving around, and operating with objects in time and in space. Many of our earliest pre-mathematical experiences are quintessentially proto-geometrical. We make sense of visual inputs; we learn to recognise faces and objects; we crawl around; we learn to look 'behind' and 'underneath' obstructions in search of hidden toys; we sort and we build; we draw and we make; etc.. However, for this experience to develop into mathematics, we then need to


Too little attention has been given to achieving a consensus as to how this transition (from informal experience, to formal reasoning) can best be established for beginners in elementary geometry. In contrast, number and arithmetic move much more naturally


Counting is rooted in the idea of a repeated unit – a notion that may stem from the ever-present, regular heartbeat that envelops every embryo (where the beat is presumably felt long before it is heard). Later we encounter repeated units with longer time scales (such as the cycles of day and night, and the routines of feeding and sleeping). The first months and years of life are peppered with instances of numerosity, of continuous quantity, of systematic ordering, of sequences, of combinations and partitions, of grouping and replicating, and of relations between quantities and operations – experiences which provide the raw material for the mathematics of number, of place value, of arithmetic, and later of 'internal structure' (or algebra).

The need for political communities to construct a formal school curriculum linking early infant experience and elementary formal mathematics is a recent development. Nevertheless, in the domain of number, quantity, and arithmetic (and later algebra), there is a surprising level of agreement about the steps that need to be incorporated – even though the details may differ in different educational systems and in different classrooms. For example:

	- relate these ideas to quantities,
	- require pupils to interpret and solve word problems, and
	- cultivate both mental arithmetic and standard written algorithms.

Our early geometrical experience is just as natural as that relating to number; but it is more subtle. And there is as yet no comparable consensus about the path that needs to be followed if our primitive geometrical experience is to be formalised in a useable way.

The 1960s saw a drive to modernise school mathematics, and at the same time to make it accessible to all. Elementary geometry certainly needed a re-think. But the reformers in most countries simply dismissed the traditional mix (e.g. in England, where one found a blend of technical drawing, Euclidean, and coordinate geometry in different proportions for different groups of students) in favour of more modern-sounding alternatives. Some countries favoured a more abstract, deductive framework; some tried to exploit motion and transformations; some used matrices and groups; some used vectors and linear algebra; some even toyed with topology. More recently we have heard similarly ambitious claims on behalf of dynamic geometry software. And although each approach has its attractions,

none of the alternatives has succeeded in helping more students to visualise, to reason, and to calculate effectively in geometrical settings.

At a much more advanced level, geometry combines


However, these subtle formalisms are totally irrelevant for beginners, who need an approach


The subtlety and flexibility of dynamic geometry software may be hugely impressive; but if students are to harness this power, they need prior mastery of some simple, semi-formal framework, together with the associated language and modes of reasoning. Despite the lack of an accepted consensus, the experience of the last 50 years would seem to suggest that the most relevant framework for beginners at secondary level involves some combination of:


### 5.2. Euclidean geometry: a brief summary

Philosophy is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and to interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth. Galileo Galilei (1564–1642)

This section provides a detailed, but compressed, outline of an initial formalisation of school geometry – of a kind that one would like good students and all teachers to appreciate. It is unashamedly a semi-formal approach for beginners, not a strictly formal treatment (such as that provided by David Hilbert (1862–1943) in his 1899 book Foundations of Geometry, or in the more detailed exposition by Edwin Moise (1918–1998) Elementary Geometry from an Advanced Standpoint, published in 1963). In particular:


We begin with the intuitive idea of points and lines in the plane. Two points A, B determine


Three points A, B, C determine an angle =ABC (between the two line segments BA and BC).

We can then begin to build more complicated figures, such as


And so on. Two given points A, B also allow us to construct the circle with centre A, and passing through B (that is, with radius AB).

This very limited beginning already opens up the world of ruler and compasses constructions. In particular, given a line segment AB, one can draw:


If the two circles meet at C,

• then AB " AC (radii of the first circle), and BA " BC (radii of the second circle).

Hence we have constructed the equilateral triangle 4ABC on the given segment AB. This construction is the very first proposition in Book 1 of the Elements of Euclid (flourished c. 300 BC). Euclid's second proposition is presented next as a problem.

Problem 137 Given three points A, B, C, show how to construct – without measuring – a point D such that the segments AB and CD are equal (in length). 4

Problem 137 looks like a simple starter (where the only available construction is to produce the third vertex of an equilateral triangle on a given line segment). However, to produce a valid solution requires a clear head and a degree of ingenuity.

Given two points A, B, the process of constructing an equilateral triangle 4ABC illustrates how we are allowed to construct new points from old.

• Whenever we construct two lines or circles that cross, the points where they cross (such as the point C in the above construction of the equilateral triangle 4ABC) become available for further constructions. So, if points A and B are given, then once C has been constructed, we may proceed to draw the lines AC and BC.

However, the fact that we can construct a line segment AB does not allow us to 'measure' the segment with a ruler, and then to use the resulting measurement to 'copy' the segment AB to the point C in order to construct the required point D such that AB " CD. The "ruler" in ruler and compasses constructions is used only to draw the line through two known points – not to measure. (Measuring is an approximate physical action, rather than an exact "mental construction", and so is not really part of mathematics.) Hence in Problem 137 we have to find another way to produce a copy CD of the segment AB starting at the point C. Similarly, we can construct the circle with centre A and passing through B, but this does not allow us to use the pair of compasses to transfer distances physically (e.g. by picking up the compasses from AB and placing the compass point at C, like using the old geometrical drawing instrument that was called a pair of dividers). In seeking the construction required in Problem 137, we are restricted to "exact mental constructions" which may be described in terms of:


If on the line AB, the point X lies between A and B, then we obtain a straight angle =AXB at X (or rather two straight angles at X – one on each side of the line AB). If we assume that all straight angles are equal, then it follows easily that "vertically opposite angles are always equal".

Problem 138 Two lines AB and CD cross at X, where X lies between A and B and between C and D. Prove that =AXC " =BXD. 4

Define a right angle to be 'half a straight angle'. Then we say that two lines which cross at a point X are perpendicular if an angle at X is a right angle (or equivalently, if all four angles at X are equal). The next step requires us to notice two things – partly motivated by experience when coordinating hand, eye and brain to construct, and to think about, physical structures.


The first of these two bullet points has an important consequence – namely that solving any problem in 2- or in 3-dimensions generally reduces to working with triangles. In particular, solving problems in 3-dimensions reduces to working in some 2-dimensional cross-section of the given figure (since three points not only determine a triangle, but also determine the plane in which that triangle lies). It follows that 2-dimensional geometry holds the key to solving problems in 3-dimensions, and that working with triangles is central in all geometry.

The second bullet point forces us to think carefully about:


A triangle 4ABC incorporates six pieces of data, or information: the three sides AB, BC, CA and the three angles =ABC, =BCA, =CAB. We say that two (ordered) triangles 4ABC and 4A1B1C <sup>1</sup> are congruent (which we write as

$$
\triangle ABC \equiv \triangle A'B'C',
$$

where the order in which the vertices are listed matters) if their sides and angles "match up" in pairs, so that

$$
\angle ABC = \frac{AB}{\angle A'B'C'}, \frac{BC}{\angle BCA} = \frac{B'C'}{\angle B'C'A'}, \frac{CA}{\angle CAB} = \frac{C'A'}{\angle C'B'A},
$$

As a result of drawing and experimenting with our hands, our minds may realise that certain subsets of these six conditions suffice to imply the others. In particular:

### SAS-congruence criterion: if

$$
\underline{AB} = \underline{A'B'}, \quad \angle ABC = \angle A'B'C', \quad \underline{BC} = \underline{B'C'},
$$

then

$$
\triangle ABC \equiv \triangle A'B'C'
$$

(where the name "SAS" indicates that the three listed match-ups occur in the specified order S (side), A (angle), S (side) as one goes round each triangle).

### SSS-congruence criterion: if

$$\underline{AB} = \underline{A'B'}, \quad \underline{BC} = \underline{B'C'}, \quad \underline{CA} = \underline{C'A'},$$

then

$$
\triangle ABC \equiv \triangle A'B'C'.
$$

### ASA-congruence criterion: if

$$
\angle ABC = \angle A'B'C', \quad \underline{BC} = \underline{B'C'}, \quad \angle BCA = \angle B'C'A',
$$

then

$$
\triangle ABC \equiv \triangle A'B'C'.
$$

If in a given triangle 4ABC we have AB " AC, then we say that 4ABC is isosceles with apex A, and base BC (iso = same, or equal; sceles = legs). Problem 139 Let 4ABC be an isosceles triangle with apex A. Let M be the midpoint of the base BC. Prove that 4AMB " 4AMC and conclude that AM is perpendicular to the base BC. 4

Problem 140 Construct two non-congruent triangles, 4ABC and 4A<sup>1</sup>B1C 1 , where =BCA " =B1C <sup>1</sup>A<sup>1</sup> " 30˝ , |CA| " |C <sup>1</sup>A<sup>1</sup> | " ? 3, |AB| " |A1B<sup>1</sup> | " 1.

Conclude that there is in general no "ASS-congruence criterion". 4

The congruence criteria allow one to prove basic results such as:

Claim In any isosceles triangle 4ABC with apex A (i.e. with AB " AC), the two base angles =B and =C are equal.

Proof 1 Let M be the midpoint of BC. Then 4AMB " 4AMC (by the SSS-congruence criterion, since AM " AM, MB " MC (by construction of M as the midpoint) BA " CA (given)). 6 =B " =ABM " =ACM " =C. QED

Proof 2 4BAC " 4CAB (by the SAS-congruence criterion, since BA " CA (given), =BAC " =CAB (same angle), AC " AB (given)). 6 =B " =ABC " =ACB " =C. QED

We also have the converse result:

Claim In any triangle 4ABC, if the base angles =B and =C are equal, then the triangle is isosceles with apex A (i.e. AB " AC).

Proof 4ABC " 4ACB (by the ASA-congruence criterion, since =ABC " =ACB (given), BC " CB, and =BCA " =CBA (given)). 6 AB " AC. QED

### Problem 141

(i) A circle with centre O passes through the point A. The line AO meets the circle again at B. If C is a third point on the circle, prove that =ACB is equal to =CAB ` =CBA.

(ii) Conclude that, if the angles in 4ABC add to a straight angle, then =ACB is a right angle. 4

Once we introduce the parallel criterion, and hence can prove that the three angles in any triangle add to a straight angle, Problem 141 will guarantee that "the angle subtended on the circumference by a diameter is always a right angle".

Problem 142 Show how to implement the basic ruler and compasses constructions:


Prove that your constructions do what you claim. 4

Problem 143 Given two points A and B.


Problem 143 shows that, given a line segment AB, the perpendicular bisector of AB is the locus of all points X which are equidistant from A and from B. This observation is what lay behind the construction of the circumcentre of a triangle (back in Chapter 1, Problem 32(a)):

Given any 4ABC. Let O be the point where the perpendicular bisectors of AB and BC meet. Then OA " OB and OB " OC. 6 OA " OB " OC. Hence O is the centre of a circle passing through all three vertices A, B, C. Moreover O also lies on the perpendicular bisector of CA.

This circle is called the circumcircle of 4ABC, and O is called the circumcentre of 4ABC. As indicated back in Problem 32, the radius of the circumcircle of 4ABC (called the circumradius of the triangle) is generally denoted by R. Later we will meet other circles and "centres" associated with a given triangle 4ABC.

Before moving on it is worth extending Problem 143 to three dimensions.

Problem 144 Given any two points N, S in 3D space, prove that the locus of all points X which are equidistant from N and from S form the plane perpendicular to the line NS and passing through the midpoint M of NS. 4

The next two fundamental results are often neglected.

Problem 145 Given any 4ABC, if we extend the side BC beyond C to a point X, then the "exterior angle" =ACX at C is greater than each of the "two interior opposite angles" =A and =B. 4

### Problem 146


$$
\underline{AB} + \underline{BC} \rhd \underline{AC}.\tag{7}
$$

The results in Problems 145 and 146 have surprisingly many consequences. For example, they allow one to prove the converse of the result in Problem 141.

Problem 147 Suppose that in 4ABC, =C " =A`=B. Prove that C lies on the circle with diameter AB.

(In particular, if the angles of 4ABC add to a straight angle, and =ACB is a right angle, then C lies on the circle with diameter AB.) 4

We come next to a result whose justification is often fudged. At first sight it is unclear how to begin: there seems to be so little information to work with – just two points and a line through one of the points.

Problem 148 A circle with centre O passes through the point P. Prove that the tangent to the circle at P is perpendicular to the radius OP. 4

Problem 148 is an example of a result which implies its own converse – though in a backhanded way. Suppose a circle with centre O passes through the point P. If OP is perpendicular to a line m passing through P, then m must be tangent to the circle (because we know that the tangent at P is perpendicular to OP, so the angle between m and the tangent is "zero", which forces m to be equal to the tangent). This converse will be needed later, when we meet the incircle.

Problem 149 Let P be a point and m a line not passing through P. Prove that, among all possible line segments P X with X on the line m, a perpendicular from P to the line m is the shortest. 4

The result in Problem 149 allows us to define the "distance" from P to the line m to be the length of any perpendicular from P to m. (As far as we know at this stage of the development, there could be more than one perpendicular from P to m.)

Note that all the results mentioned so far have avoided using the Euclidean "parallel criterion" (or – equivalently – the fact that the three angles in any triangle add to a straight angle). So results proved up to this point should still be "true" in any geometry where we have points, lines, triangles, and circles satisfying the congruence criteria – whether or not the geometry satisfies the Euclidean "parallel criterion".

The idea that there is only one "shortest" distance from a point to a line may seem "obvious"; but it is patently false on the sphere, where every line (i.e. 'great circle') from the North pole P to the equator is perpendicular to the equator (and all these lines have the same "length"). The proof that there is just one such perpendicular from P to m depends on the parallel criterion (see below) – a criterion which fails to hold for geometry on the sphere.

Euclid's Elements started with a few basic axioms that formalised the idea of ruler and compasses constructions. He then added a simple axiom that allowed one to compare angles in different locations. He made the forgivable mistake of omitting an axiom for congruence of triangles – imagining that it can be proved. (It can't.) However he then stated, and carefully developed the consequences of, a much more subtle axiom about parallel lines (two lines m, n in the plane are said to be parallel if they never meet, no matter how far they are extended). For reasons that remain unclear, instead of appreciating that Euclid's "parallel postulate" constituted a profound insight into the foundations of geometry, mathematicians in later ages saw the complexity of Euclid's postulate as some kind of flaw, and so tried to show that it could be derived from the other, simpler postulates. The attempt to "correct" this perceived flaw became a kind of Holy Grail.

The story is instructive, but too complicated to summarise accurately here. The situation was eventually clarified by two nineteenth century mathematicians (more-or-less at the same time, but working independently). In the revolutionary, romantic spirit of the nineteenth century, J´anos Bolyai (Hungarian: 1802–1860) and Nikolai Lobachevski (Russian: 1792–1856) each allowed himself to consider what would happen if one adopted a different assumption about how "parallel lines" behave. Both discovered that one can then derive an apparently coherent theory of a completely novel kind, with its own beautiful results: that is, a geometry which seemed to be internally "consistent" – but different from Euclidean geometry. Lobachevski published brief notes of his work in 1829–30 (in Kazan); Bolyai knew nothing of this and published incomplete notes of his researches in 1832. Lobachevski published a more detailed booklet in 1840.

Neither mathematician got the recognition he might have anticipated, and it was only much later (largely after their deaths) that others realised how to show that the fantasy world they had each dreamt up was just as "internally consistent" as traditional Euclidean geometry. The story is further complicated by the fact that the dominant mathematician of the time – namely Gauss (1777–1855) – claimed to have proved something similar (and he may well have done so, but exactly what he knew has to be inferred from cryptic remarks in occasional letters, since he published nothing on the subject). If there is a moral to the story, it could be that success in mathematics may not be recognised, or may only be recognised after one's death: so those who spend their lives exploring the mathematical universe had better appreciate the delights of the mathematical journey, rather than being primarily motivated by a desire for immediate recognition and acclaim!

Two lines m, n in the plane are said to be parallel if they never meet – no matter how far they are extended. We sometimes write this as "m k n".

Given two lines m, n in the plane, a third line p which crosses both m and n is called a transversal of m and n.

Parallel criterion: Given two lines m and n, if some transversal p is such that the two "internal" angles on one side of the line p (that is the two angles that p makes with m and with n, and which lie between the two lines m and n) add to less than a straight angle, then the lines m and n must meet on that side of the line p.

If the internal angles on one side of p add to more than a straight angle, then internal angles on the other side of p add to less than a straight angle, so the lines m and n must meet on the other side of p. It follows

• that two lines m and n are parallel precisely when the two internal angles on one side of a transversal add to exactly a straight angle.

Parallel lines can be thought of as "all having the same direction"; so it is convenient to insist that "every line is parallel to itself" (even though it has lots of points in common with itself). It then follows


All this then allows one


"given a line AB and a point P, construct the line through P which is parallel to AB"

(namely, by first constructing the line P X through P, perpendicular to AB, and then the line through P, perpendicular to P X).

One can then prove the standard result about the angles in any triangle.

Claim The angles in any triangle 4ABC add to a straight angle.

Proof Construct the line m through A that is parallel to BC. Then AB and AC are transversals, which cross both the line m and the line BC, and which make three angles at the point A on m:


The three angles at A clearly add to a straight angle, so the three angles =A, =B, =C also add to a straight angle. QED

Once we know that the angles in any triangle add to a straight angle, we can prove all sorts of other useful facts. One is a simple reformulation of the above Claim.

Problem 150 Given any triangle 4ABC, extend BC beyond C to a point X. Then the exterior angle

$$
\angle XCA = \angle A + \angle B.
$$

("In any triangle, each exterior angle is equal to the sum of the two interior opposite angles.") 4

Another important consequence is the result which underpins the sequence of "circle theorems".

Problem 151 Let O be the circumcentre of 4ABC. Prove that

$$
\angle AOB = 2 \cdot \angle ACB. \tag{7}
$$

Problem 151 implies that

"the angles subtended by any chord AB on a given arc of the circle are all equal",

and are equal to exactly one half of the angle subtended by AB at the centre O of the circle. This leads naturally to the familiar property of cyclic quadrilaterals.

Problem 152 Let ABCD be a quadrilateral inscribed in a circle (such a quadrilateral is said to be cyclic, and the four vertices are said to be concyclic – that is, they lie together on the same circle). Prove that opposite angles (e.g. =B and =D) must add to a straight angle. (Two angles which add to a straight angle are said to be supplementary.) 4

These results have lots of lovely consequences: we shall see one especially striking example in Problem 164. Meantime we round up our summary of the "circle theorems".

Problem 153 Suppose that the line XAY is tangent to the circumcircle of 4ABC at the point A, and that X and C lie on opposite sides of the line AB. Prove that =XAB " =ACB. 4

### Problem 154

(a) Suppose C, D lie on the same side of the line AB.

(i) If D lies inside the circumcircle of 4ABC, then =ADB ą =ACB.


Another result which follows now that we know that the angles of a triangle add to a straight angle is a useful additional congruence criterion – namely the RHS-congruence criterion. This is a 'limiting case' of the failed ASS-congruence criterion (see the example in Problem 140). In the failed ASS criterion the given data correspond to two different triangles – one in which the angle opposite the first specified side (the first "S" in "ASS") is acute, and one in which the angle opposite the first specified side is obtuse. In the RHS-congruence criterion, the angle opposite the first specified side is a right angle, and the two possible triangles are in fact congruent.

RHS-congruence criterion: If =ABC and =A1B1C <sup>1</sup> are both right angles, and BC " B1C 1 , CA " C <sup>1</sup>A<sup>1</sup> , then

$$
\triangle ABC \equiv \triangle A'B'C'.
$$

Proof Suppose that AB " A1B<sup>1</sup> . Then

$$\begin{array}{l} \underline{AB} = \underline{A'B'},\\ \angle ABC = \angle A'B'C',\\ \underline{BC} = \underline{B'C'}. \end{array}$$

Hence we may apply the SAS-congruence criterion to conclude that 4ABC " 4A<sup>1</sup>B<sup>1</sup>C 1 .

If on the other hand AB ‰ A<sup>1</sup>B<sup>1</sup> , we may suppose that BA ą B<sup>1</sup>A<sup>1</sup> . Now construct A<sup>2</sup> on BA such that BA<sup>2</sup> " B<sup>1</sup>A<sup>1</sup> . Then

$$\begin{array}{l} A''B = \underline{A'B'},\\ \angle A''BC = \angle A'B'C',\\ \underline{BC} = \underline{B'C'},\\ \therefore \triangle A''BC \equiv \triangle A'B'C' \text{ (by SAS-configurece)}.\\ \text{Hence } \underline{A''C} = \underline{A'C'} = \underline{AC}, \text{ so } \triangle CAA'' \text{ is isosceles.}\\ \therefore \angle CA''A = \angle CAA''. \end{array}$$

However, =CA<sup>2</sup>A ą =CBA (since the exterior angle =CA<sup>2</sup>A in 4CBA<sup>2</sup> must be greater than the interior opposite angle =CBA, by Problem 145).

But then the two base angles in the isosceles triangle 4CAA<sup>2</sup> are each greater than a right angle – so the angle sum of 4CAA<sup>2</sup> is greater than a straight angle, which is impossible. Hence this case cannot occur. QED RHS-congruence seems to be needed to prove the basic result (Problem 161 below) about the area of parallelograms, and this is then needed in the proof of Pythagoras' Theorem (Problem 18). In one sense RHS-congruence looks like a special case of SSS-congruence (as soon as two pairs of sides in two right angled triangles are equal, Pythagoras' Theorem guarantees that the third pair of sides are also equal). However this observation cannot be used to justify RHS-congruence if RHS-congruence is needed to justify Pythagoras' Theorem.

Problem 155 Given a circle with centre O, let Q be a point outside the circle, and let QP, QP<sup>1</sup> be the two tangents from Q, touching the circle at P and at P 1 . Prove that QP " QP<sup>1</sup> , and that the line OQ bisects the angle =P QP<sup>1</sup> . 4

Problem 156 You are given two lines m and n crossing at the point B.


Problem 156 shows that, given two lines m and n that cross at B, the bisectors of the two pairs of vertically opposite angles formed at B form the locus of all points X which are equidistant from the two lines m and n. This allows us to mimic the comments following Problem 143 and so to construct the incentre of a triangle.

Given any 4ABC, let I be the point where the angle bisectors of =ABC and =BCA meet.

Let the perpendiculars from I to the three sides AB, BC, CA meet the sides at P, Q, R respectively. Then

IP " IQ (since I lies on the bisector of =ABC) and IQ " IR (since I lies on the bisector or =BCA).

Hence the circle which has centre I and which passes through P also passes through Q and R.

Moreover, I also lies on the bisector of =CAB; and since the radii IP, IQ, IR are perpendicular to the sides of the triangle, the circle is tangent to the three sides of the triangle (by the comments following Problem 148).

This circle is called the incircle of 4ABC, and I is called the incentre of 4ABC. The radius of the incircle of 4ABC is called the inradius, and is generally denoted by r.

A quadrilateral ABCD in which AB k DC and BC k AD is called a parallelogram. A parallelogram ABCD with a right angle is a rectangle. A parallelogram ABCD with AB " AD is called a rhombus. A rectangle which is also a rhombus is called a square.

Problem 157 Let ABCD be a parallelogram.


Problem 158 Let ABCD be a parallelogram with centre X (where the two main diagonals AC and BD meet), and let m be any straight line passing through the centre. Prove that m divides the parallelogram into two parts of equal area. 4

We defined a parallelogram to be "a quadrilateral ABCD in which AB k DC and BC k AD"; however, in practice, we need to be able to recognise a parallelogram even if it is not presented in this form. The next result hints at the variety of other conditions which allow us to recognise a given quadrilateral as being a parallelogram "in mild disguise".

### Problem 159


The next problem presents a single illustrative example of the kinds of things which we know in our bones must be true, but where the reason, or proof, may need a little thought.

Problem 160 Let ABCD be a parallelogram. Let M be the midpoint of AD and N be the midpoint of BC. Prove that MN k AB, and that MN passes through the centre of the parallelogram (where the two diagonals meet). 4

Problem 161 Prove that any parallelogram ABCD has the same area as the rectangle on the same base DC and "with the same height" (i.e. lying between the same two parallel lines AB and DC). 4

The ideas and results we have summarised up to this point provide exactly what is needed in the proof of Pythagoras' Theorem outlined back in Chapter 1, Problem 18. They also allow us to identify two more "centres" of a given triangle 4ABC.

Problem 162 Given any triangle 4ABC, draw the line through A which is parallel to BC, the line through B which is parallel to AC, and the line through C which is parallel to AB. Let the first two constructed lines meet at C 1 , the second and third lines meet at A<sup>1</sup> , and the first and third lines meet at B<sup>1</sup> .


Let the foot of the perpendicular from A to BC be P, the foot of the perpendicular from B to CA be Q, and the foot of the perpendicular from C to AB be R. Then 4P QR is called the orthic triangle of 4ABC. The "circle theorems" (especially Problems 151 and 154(c)) lead us to discover that this triangle has two quite unexpected properties. As a partial preparation for one of the properties we digress slightly to introduce a classic problem.

Problem 163 My horse is tethered at H some distance away from my village V . Both H and V are on the same side of a straight river. How should I choose the shortest route to lead the horse from H to V , if I want to water the horse at the river en route? 4 Problem 164 Let 4ABC be an acute angled triangle.


We come next to the fourth among the standard "centres of a triangle".

Problem 165 Given 4ABC, let L be the midpoint of the side BC. The line AL is called a median of 4ABC. (It is not at all obvious, but if we imagine the triangle as a lamina, having a uniform thickness, then 4ABC would exactly balance if placed on a knife-edge running along the line AL.) Let M be the midpoint of the side CA, so that BM is another median of 4ABC. Let G be the point where AL and BM meet.

	- (ii) Prove that 4BCM and 4BAM have equal area. Conclude that 4BCG and 4BAG have equal area.

The point where all three medians meet is called the centroid of the triangle. For the geometry of the triangle, this is all you need to know. However, it is worth noting that the centroid is the point that would be the 'centre of gravity' of the triangle if the triangle is thought of as a thin lamina with a uniform distribution of mass.

Next we revisit, and reprove in the Euclidean spirit, a result that you proved in Problem 95 using coordinates – namely the Midpoint Theorem.

Problem 166 (The Midpoint Theorem) Given any triangle 4ABC, let M be the midpoint of the side AC, and let N be the midpoint of the side AB. Draw in MN and extend it beyond N to a point M<sup>1</sup> such that MN " NM<sup>1</sup> .

(a) Prove that 4ANM " 4BNM<sup>1</sup> .


The Midpoint Theorem can be reworded as follows:

Given 4AMN. Extend AM to C such that AM " MC and extend AN to B such that AN " NB. Then CB k MN and CB " 2 ¨ MN.

This rewording generalizes SAS-congruence in a highly suggestive way, and points us in the direction of "SAS-similarity".

SAS-similarity (ˆ2): if A1B<sup>1</sup> " 2 ¨ AB, =BAC " =B1A1C 1 , and A1C <sup>1</sup> " 2 ¨ AC, then B1C <sup>1</sup> " 2 ¨ BC, =ABC " =A1B1C 1 , and =BCA " =B1C <sup>1</sup>A<sup>1</sup> .

Proof Extend AB to the point B<sup>2</sup> such that AB<sup>2</sup> " A1B<sup>1</sup> , and extend AC to the point C 2 such that AC<sup>2</sup> " A1C 1 . Then 4B2AC<sup>2</sup> " 4B1A1C 1 (by SAS-congruence), so B2C <sup>2</sup> " B1C 1 , =B2C <sup>2</sup>A " =B1C <sup>1</sup>A<sup>1</sup> , =C <sup>2</sup>B2A " =C <sup>1</sup>B1A<sup>1</sup> . By construction we have AB<sup>2</sup> " 2 ¨ AB, and AC<sup>2</sup> " 2 ¨ AC. Hence (by the Midpoint Theorem): B2C <sup>2</sup> " 2 ¨ BC (so B1C <sup>1</sup> " 2 ¨ BC), and BC k B2C 2 (so =BCA " =B2C <sup>2</sup>A and =CBA " =C <sup>2</sup>B2A). 6 =B2C <sup>2</sup>A " =B1C <sup>1</sup>A<sup>1</sup> " =BCA, and =C <sup>2</sup>B<sup>2</sup>A " =C <sup>1</sup>B<sup>1</sup>A<sup>1</sup> " =CBA. QED

The SAS-similarity (ˆ2) interpretation of the Midpoint Theorem is like the SAS-congruence criterion in that one pair of corresponding angles in 4BAC and 4B<sup>1</sup>A<sup>1</sup>C <sup>1</sup> are equal, while the sides on either side of this angle in the two triangles are related; but instead of the two pairs of corresponding sides being equal, the sides of 4B<sup>1</sup>A<sup>1</sup>C <sup>1</sup> are double the corresponding sides of 4BAC.

In general we say that

4ABC is similar to 4A<sup>1</sup>B<sup>1</sup>C 1 (written as 4ABC " 4A<sup>1</sup>B<sup>1</sup>C 1 ) with scale-factor m if each angle of 4ABC is equal to the corresponding angle of 4A<sup>1</sup>B<sup>1</sup>C 1 , and if corresponding sides are all in the same ratio:

$$\underline{A'B'} : \underline{AB} = \underline{B'C'} : \underline{BC} = \underline{C'A'} : \underline{CA} = m : 1.$$

If two triangles 4A1B1C <sup>1</sup> and 4ABC are similar, with (linear) scale factor A1B<sup>1</sup> : AB " m : 1, then the ratio between their areas is

> areap4A 1B 1C 1 q : areap4ABCq " m<sup>2</sup> : 1.

Two similar triangles 4ABC and 4A1B1C <sup>1</sup> give rise to six matching pairs:


In the case of congruence, the congruence criteria tell us that we do not need to check all six pairs to guarantee that two triangles are congruent: these criteria guarantee that certain triples suffice. The similarity criteria guarantee much the same for similarity.

Suppose we are given triangles 4ABC, 4A1B1C 1 .

### AAA-similarity: If

$$
\angle ABC = \angle A'B'C', \; \angle BCA = \angle B'C'A', \; \angle CAB = \angle C'A'B',
$$

then

$$
\underline{A'B'} : \underline{AB} = \underline{B'C'} : \underline{BC} = \underline{C'A'} : \underline{CA},
$$

so the two triangles are similar.

### SSS-similarity: If

$$
\underline{A'B'} : \underline{AB} = \underline{B'C'} : \underline{BC} = \underline{C'A'} : \underline{CA},
$$

then

$$
\angle ABC = \angle A'B'C', \; \angle BCA = \angle B'C'A', \; \angle CAB = \angle C'A'B',
$$

so the two triangles are similar.

#### SAS-similarity: If

$$\underline{A'B'} : \underline{AB} = \underline{A'C'} : \underline{AC} = m : 1$$

and

$$
\angle B'A'C' = \angle BAC,
$$

then

$$\underline{B'C'} : \underline{BC} = \underline{A'B}' : \underline{AB} = \underline{A'C'} : \underline{AC}'$$

and

$$
\angle A'B'C' = \angle ABC, \quad \angle B'C'A' = \angle BCA,
$$

so the two triangles are similar.

Our rewording of the Midpoint Theorem gave rise to a version of the third of these criteria, with m " 2.

AAA-similarity in right angled triangles is what makes trigonometry possible. Suppose that two triangles 4ABC, 4A1B1C <sup>1</sup> have right angles at A and at A<sup>1</sup> . If =ABC " =A1B1C 1 , then (since the angles in each triangle add to two right angles) we also have =BCA " =B1C <sup>1</sup>A<sup>1</sup> . It then follows (from AAA-similarity) that

$$
\underline{A'B'} : \underline{AB} = \underline{B'C'} : \underline{BC} = \underline{C'A'} : \underline{CA},
$$

so the trig ratio in 4ABC

$$
\sin B = \frac{\underline{AC}}{\underline{BC}}
$$

has the same value as the corresponding ratio in 4A1B1C 1

$$
\sin B' = \frac{\underline{A'C'}}{\underline{B'C'}}.
$$

Hence this ratio depends only on the angle B, and not on the triangle in which it occurs. The same holds for cos =B and for tan =B.

The art of solving geometry problems often depends on looking for, and identifying, similar triangles hidden in a complicated configuration. As an introduction to this, we focus on three classic properties involving circles, where the figures are sufficiently simple that similar triangles should be fairly easy to find.

Problem 167 The point P lies outside a circle. The tangent from P touches the circle at T, and a secant from P cuts the circle at A and at B. Prove that P A ˆ P B " P T<sup>2</sup> . 4

Problem 168 The point P lies outside a circle. Two secants from P meet the circle at A, B and at C, D respectively. Prove in two different ways that

$$
\underline{PA} \times \underline{PB} = \underline{PC} \times \underline{PD}.\tag{7}
$$

Problem 169 The point P lies inside a circle. Two secants from P meet the circle at A, B and at C, D respectively. Prove that

$$
\underline{PA} \times \underline{PB} = \underline{PC} \times \underline{PD}.\tag{7}
$$

We end our summary of the foundations of Euclidean geometry by deriving the familiar formula for the area of a trapezium and its 3-dimensional analogue, and a formulation of the similarity criteria which is often attributed to Thales (Greek 6th century BC).

Problem 170 Let ABCD be a trapezium with AB k DC, in which AB has length a and DC has length b.


Problem 171 A pyramid ABCDE, with apex A and square base BCDE of side length b, is cut parallel to the base at height d above the base, leaving a frustum of a pyramid, with square upper face of side length a. Find a formula for the volume of the resulting solid (in terms of a, b, and d). 4

The following general result allows us to use "equality of ratios of line segments" whenever we have three parallel lines (without first having to conjure up similar triangles).

Problem 172 (Thales' Theorem) The lines AA<sup>1</sup> and BB<sup>1</sup> are parallel. The point C lies on the line AB, and C 1 lies on the line A<sup>1</sup>B<sup>1</sup> such that CC<sup>1</sup> k BB<sup>1</sup> . Prove that AB : BC " A<sup>1</sup>B<sup>1</sup> : B<sup>1</sup>C 1 . 4

Under certain conditions, the similarity criteria guarantee the equality of ratios of sides of two triangles. Thales' Theorem extends this "equality of ratios" to line segments which arise whenever two lines cross three parallel lines. One of the simplest, but most far-reaching, applications of this result is the tie-up between geometry and algebra which lies behind ruler and compasses constructions, and which underpins Descartes' (1596–1650) re-formulation of geometry in terms of coordinates (see Problem 173).

Thales (c. 620–c. 546 BC) was part of the flowering of Greek thought having its roots in Milesia (in the south west of Asia Minor, or modern Turkey). Thales seems to have been interested in almost everything – philosophy, astronomy, politics, and also geometry. In Britain his name is usually attached to the fact that the angle subtended by a diameter is a right angle. On the continent, his name is more strongly attached to the result in Problem 172. His precise contribution to geometry is unclear – but he seems to have played a significant role in kick-starting what became (300 years later) the polished version of Greek mathematics that we know today.

Thales' contributions in other spheres were perhaps even more significant than in geometry. He seems to have been among the first to try to "explain" phenomena in reductionist terms – identifying "water" as the single "element", or first principle, from which all substances are derived. Anaximenes (c. 586–c. 526 BC) later argued in favour of "air" as the first principle. These two elements, together with "fire" and "earth", were generally accepted as the four Greek "elements" – each of which was supposed to contribute to the construction of observed matter and change in different ways.

Problem 173 To define "length", we must first decide which line segment is deemed to have unit length. So suppose we are given line segments XY of length 1, AB of length a, (i.e. AB : XY " a : 1), and CD of length b.


### 5.3. Areas, lengths and angles

Problem 174 A rectangular piece of fruitcake has a layer of icing on top and down one side to form a larger rectangular slab of cake (as shown in Figure 3).

Figure 3: Icing on the cake

Describe how to make a single straight cut so as to divide both the fruitcake and the icing exactly in half. (The thickness of the icing on top is not necessarily the same as the thickness down the side.) 4

### Problem 175


Problem 176 The twelve hour marks for a clock are marked on the circumference of a unit circle to form the vertices of a regular dodecagon ABCDEF GHIJKL. Calculate exactly (i.e. using Pythagoras' Theorem rather than trigonometry) the lengths of all the possible line segments joining two vertices of the dodecagon. 4

Problem 177 Consider the lattice of all points pk, m, nq in 3-dimensions with integer coordinates k, m, n. Which of the following distances can be realised between lattice points?

```
?
  1,
     ?
       2,
          ?
            3,
               ?
                  4,
                    ?
                       5,
                          ?
                            6,
                               ?
                                 7,
                                    ?
                                       8,
                                         ?
                                            9,
                                               ?
                                                 10,
                                                     ?
                                                        11,
                                                            ?
                                                              12,
                                                                   ?
                                                                     13,
                                                                         ?
                                                                            14,
                                                                                ?
                                                                                  15,
                                                                                       ?
                                                                                         16,
                                                                                             ?
                                                                                                17.
                                                                                                   4
```
### Problem 178


### Problem 179

(a) A regular pentagon ABCDE with edges of length 1 is surrounded in the plane by five new regular pentagons – ABLMN joined to AB, BCOP Q joined to BC, and so on.

	- (i) Prove that ABCDE is a regular pentagon.
	- (ii) Prove that A, B, and M are three vertices of a regular pentagon ABLMN, where L lies on MP and N lies on MY .
	- (iii) Find the edge length of the regular pentagon ABCDE. 4

### 5.4. Regular and semi-regular tilings in the plane

In Problem 36 we saw that a regular n-gon has a circumcentre O. If we join each vertex to the point O, we get n triangles, each with angle sum π. Hence the total angle sum in all n triangles is πn. Since the n angles around the point O add to 2π, the angles of the regular n-gon itself have sum <sup>p</sup><sup>n</sup> ´ <sup>2</sup>qπ. Hence each angle of the regular <sup>n</sup>-gon has size ` 1 ´ 2 n ˘ π. (In the next chapter you will prove the general result that the sum of the angles in any n-gon is equal to pn ´ 2qπ radians.)

Problem 180 A regular tiling of the plane is an arrangement of identical regular polygons, which fit together edge-to-edge so as to cover the plane with no overlaps.


We refer to the arrangement of tiles around a vertex as the vertex figure. In a regular tiling all vertex figures are automatically identical, so it is natural to refer to the tiling in terms of its vertex figure. When p " 3, exactly q " 6 tiles fit together at each vertex, and we abbreviate "six equilateral triangles" as 3<sup>6</sup> . In the same way we denote the tiling whose vertex figure consists of "four squares" as 4<sup>4</sup> , and the tiling whose vertex figure consists of "three regular hexagons" as 6<sup>3</sup> .

The natural approach in part (a) of Problem 180 is first to identify which vertex figures have no gaps or overlaps – giving a necessary condition for a regular tiling to exist. It is tempting to stop there, and to assume that this obvious necessary condition is also sufficient. The temptation arises in part because 2-dimensional regular tilings are so familiar. But it is important to recognize the distinction between a necessary and a sufficient condition; so the temptation should be resisted, and a construction given.

The procedure hidden in the solution to Problem 180 illustrates a key strategy, which dates back to the ancient Greeks, and which is called the method of analysis.


This is what we did in a very simple way in the solution to Problem 180: the condition on vertex figures gave an evident necessary condition, which turned out to be sufficient to guarantee that such a tiling exists. The same general strategy guided our classification of primitive Pythagorean triples back in Problem 23.

In the seventeenth century, this ancient Greek strategy was further developed by Fermat (1601–1665), and by Descartes (1596–1650). For example, Fermat left very few proofs; but his proof that the equation

$$x^4 + y^4 = z^4$$

has no solutions in positive integers x, y, z illustrated the method:


Descartes developed a "method", whereby hard geometry problems could be solved by translating them into algebra – essentially using the method of analysis.


The importance of the final step in this process (checking that the list of necessary constraints is also sufficient) is underlined in the next problem where we try to classify certain "almost regular" tilings.

Problem 181 A semi-regular tiling of the plane is an arrangement of regular polygons (not necessarily all identical), which fit together edge-to-edge so as to cover the plane without overlaps, and such that the arrangements of tiles around any two vertices are congruent.

	- (ii) Try to find additional necessary conditions to eliminate vertex figures which cannot be realized, until your list of necessary conditions seems likely to be sufficient.

Semi-regular tilings are often called Archimedean tilings. The reason for this name remains unclear. Pappus (c. 290–c. 350 AD), writing more than 500 years after the death of Archimedes (d. 212 BC), stated that Archimedes classified the semi-regular polyhedra. Now the classification of semi-regular polyhedra (Problem 190) uses a similar approach to the classification of planar tilings, except that the sum of the angles at each vertex has sum less than (rather than exactly equal to) 360˝ . So it may be that the semi-regular tilings are named after Archimedes simply because he did something similar for polyhedra; or it may be that, since inequalities are harder to control than

equalities, someone inferred (perhaps dodgily) that Archimedes must have known about semi-regular tilings as well as about semi-regular polyhedra.

Whatever the reason, tilings and polyhedra have fascinated mathematicians, artists and craftsmen for all sorts of unexpected reasons – as indicated by:


### 5.5. Ruler and compasses constructions for regular polygons

Euclid's Elements include methods for constructing the regular polygons that are required for the construction of the regular polyhedra (see Section 5.6). In one sense, Euclid is thoroughly modern: he is reluctant to work with entities that cannot be constructed. And for him, geometrical construction means construction "using ruler and compasses" only.

For each regular polygon, there are two related (and sometimes very different) construction problems:


Before Problem 137 we saw how to construct an equilateral triangle ABC given the points A, B. And in Problem 36 we saw that every regular polygon has a circumcentre O.

Problem 182 Given points O, A, show how to construct the regular 3-gon ABC with circumcentre O. 4

### Problem 183


### Problem 184

	- (ii) Given two points O, A, show how to construct a regular 8-gon ABCDEF GH with circumcentre O.
	- (ii) Given points A, B, show how to construct a regular 8-gon ABCDEF GH. 4

### Problem 185

	- (ii) Given points O, A, show how to construct a regular 10-gon ABCDEF GHIJ with circumcentre O.
	- (ii) Given points A, B, show how to construct a regular 10-gon ABCDEF GHIJ. 4

We shall not prove it here, but it is impossible to construct a regular 7-gon, or a regular 9-gon, or a regular 11-gon using ruler and compasses. All constructions with ruler and compasses come down to two moves:


Put slightly differently, all ruler and compasses constructions involve solving linear or quadratic equations, so the only new points, or lengths we can construct are those which involve iterated square roots of expressions or lengths which were previously known.

This iterated extraction of square roots is linked to a fact first proved by Gauss (1777–1855), namely that the only regular p-gons (where p is a prime) that can be constructed are those where p is a Fermat prime – that is, a prime of the form p " 2 <sup>k</sup> ` 1 (in which case k has to be a power of 2: see Problem 118). Gauss proved (as a teenager, though it was first published in his book Disquisitiones arithmeticae in 1801):

a regular n-gon can be constructed with ruler and compasses if and only if n has the form

> 2 <sup>m</sup> ¨ p<sup>1</sup> ¨ p<sup>2</sup> ¨ p<sup>3</sup> ¨ ¨ ¨ pk,

where p1, p2, p3, . . . , p<sup>k</sup> are distinct Fermat primes.

As we noted in Chapter 2, the only known Fermat primes are the five discovered by Fermat himself, namely 3, 5, 17, 257, and 65 537.

### 5.6. Regular and semi-regular polyhedra

We have seen how regular polygons sometimes fit together edge-to-edge in the plane to create tilings of the whole plane. When tiling the plane, the angles of polygons meeting edge-to-edge around each vertex must add to 360˝ , or two straight angles. If the angles at a vertex add to less than 360˝ , then we are left with an empty gap and two free edges; and when these two free edges are joined, or glued together, the vertex figure rises out of the plane and becomes a 3-dimensional corner, or solid angle.

To form such a corner we need at least three polygons, or faces – and hence at least three edges and three faces meet around each vertex. For example, three squares fit nicely together in the plane, but leave a 90˝ gap. When the two spare edges are glued together, the result is to form a corner of a cube, where we have a vertex figure consisting of three regular 4-gons: so we refer to this vertex figure as 4<sup>3</sup> .

Given a 3-dimensional corner, it may be possible to extend the construction, repeating the same vertex figure at every vertex. The resulting shape may then 'close up' to form a convex polyhedron. The assumption that in each vertex figure, the angles meeting at that vertex add to less than 360˝ , means that all the corners then project outwards – which is roughly what we mean when we say that the polyhedron is "convex".

A regular polygon is an arrangement of finitely many congruent line segments, with two line segments meeting at each vertex (and never crossing, or meeting internally), and with all vertices alike; a regular polygon can be inscribed in a circle (Problem 36), and so encloses a convex subset of the plane. In the same spirit, a regular polyhedron is an arrangement of finitely many congruent regular polygons, with two polygons meeting at each edge, and with the same number of polygons in a single cycle around every vertex, enclosing a convex subset of 3-dimensional space (i.e. the polyhedron separates the remaining points of 3D into those that lie 'inside' and those that lie 'outside', and the line segment joining any two points of the polyhedral surface contains no points lying outside the polyhedron).

The important constraints here are the assumptions: that the polygons meet edge-to-edge with exactly two polygons meeting at each edge; that the same number of polygons meet around every vertex; and that the overall number of polygons, or faces, is finite. The assumption that the figure is convex should be seen as a temporary additional constraint, which means that the angles in polygons meeting at each vertex have sum less than 360˝ .

Problem 186 A vertex figure is to be formed by fitting regular p-gons together, edge-to-edge, for a fixed p. If there are q of these p-gons at a vertex, we denote the vertex figure by p q . If the angles at each vertex add to less than 360˝ , prove that the only possible vertex figures are 3<sup>3</sup> , 3<sup>4</sup> , 3<sup>5</sup> , 4 3 , 5<sup>3</sup> . 4

The vertex figure 4<sup>3</sup> is realized by the way the positive axes meet at the vertex p0, 0, 0q, where


If we include an eighth vertex p1, 1, 1q, and


we see that all eight vertices have the same vertex figure 4<sup>3</sup> . Hence the possible vertex figure 4<sup>3</sup> in Problem 186 arises as the vertex figure of a regular polyhedron – namely the cube.

If we select the four vertices whose coordinates have odd sum A " p1, 0, 0q, B " p0, 1, 0q, C " p0, 0, 1q, D " p1, 1, 1), then the distance between any two of these vertices is equal to ? 2, so each triple of vertices (such as p1, 0, 0q, p0, 1, 0q, p0, 0, 1q) defines a regular 3-gon ABC, with three such 3-gons meeting at each vertex of ABCD. Hence the possible vertex figure 3 3 in Problem 186 arises as the vertex figure of a regular polyhedron – namely the regular tetrahedron (tetra = four; hedra = faces).

Problem 187 With A " p1, 0, 0q etc. as above, write down the coordinates of the six midpoints of the edges of the regular tetrahedron ABCD (or equivalently, the six centres of the faces of the original cube). Each edge of the regular tetrahedron meets four other edges of the regular tetrahedron (e.g. AB meets AC and AD at one end, and BC and BD at the other end). Choose an edge AB and its midpoint P. Calculate the distance from P to the midpoints Q, R, S, T of the four edges which AB meets (namely the midpoints of AC, AD, BD, BC respectively). Confirm that the triangles 4P QR, 4P RS, 4P ST, 4P T Q are all regular 3-gons, and that the vertex figure at P is of type 3<sup>4</sup> . Conclude that the possible vertex figure 3<sup>4</sup> in Problem 186 arises as the vertex figure of a regular polyhedron P QRST U – namely the regular octahedron (octa = eight; hedra = faces). 4

### Problem 188

	- (i) Let the four triangles which meet at A be ABC, ACD, ADE, AEB. Prove that BCDE must be a square.
	- (ii) Suppose that all the triangles have edges of length 2, and that the octahedron sits with one face BCF on the table – next to the regular tetrahedron from part (a). Which of these two solids is the taller? 4

Problem 189 Let O " p0, 0, 0q, A " p1, 0, 0q, B " p0, 1, 0q, C " p0, 0, 1q be four vertices of the cube as described after Problem 186 above. Draw equal and parallel line segments (initially of unknown length 1´2a) through the centres of each pair of opposite faces – running in the three directions parallel to OA, or to OB, or to OC,


Figure 4: Construction of the regular icosahedron.

These are to form all 12 vertices and six of the 30 edges (of length 1 ´2a) of a polyhedron, see Figure 4. The other 24 edges join each of these 12 vertices to its four natural neighbours on adjacent faces of the cube – to form the 20 triangular faces of the polyhedron: for example,

N joins: to S; to W; to X; and to U.


The polyhedron is called the regular icosahedron (icosa = twenty, hedra = faces).

In the paragraph before Problem 187 we constructed the dual of the cube by marking the circumcentre of each of the six square faces of the cube, and then joining each circumcentre to its four natural neighbours. We now construct the dual of the regular icosahedron in exactly the same way. Each of the 20 circumcentres of the 20 triangular faces of a regular icosahedron has three natural neighbours (namely the circumcentres of the three neighbouring triangular faces). If we construct the 30 edges joining these 20 circumcentres, the five circumcentres of the five triangles in each vertex figure of the regular icosahedron form a regular pentagon, which becomes a face of the dual polyhedron – so we get 12 regular pentagons (one for each vertex of the regular icosahedron), with three pentagons meeting at each vertex of the dual polyhedron to give a vertex figure 5<sup>3</sup> at each of the 20 vertices, which form a regular dodecahedron.

Hence each of the five possible vertex figures in Problem 186 can be realised by a regular polyhedron. These are sometimes called the Platonic solids because Plato (c. 428–347 BC) often used them as illustrative examples in his writings on philosophy.

Constructing the five regular polyhedra is part of the essence of mathematics for everyone. In contrast, what comes next (in Problem 190) may be viewed as "optional" at this stage. The ideas are worth noting, but the details may be best postponed for a rainy day.

Just as you classified semi-regular tilings in Section 5.4, so one can look for semi-regular polyhedra. A polyhedron is semi-regular if all of its faces are regular polygons (possibly with differing numbers of edges), fitting together edge-to-edge, with exactly the same ring of polygons around each vertex – the vertex figure of the polyhedron. Problem 190 uses "the method of analysis" - combining simple arithmetic, inequalities, and a little geometric insight – to achieve a remarkable complete classification of semi-regular polyhedra. There are usually said to be thirteen individual semi-regular polyhedra (excluding the five regular polyhedra); but one of these has a vertex figure that extends to a polyhedron in two different ways – each being the reflection of the other. There are in addition two infinite families – namely


Notice that the cube can also be interpreted as being a "4-gonal prism", and the regular octahedron can be interpreted as being a "3-gonal antiprism". Those interested in regular and semi-regular polyhedra are referred to the classic book Mathematical models by H.M. Cundy and A.P. Rollett.

Problem 190 Find possible combinations of three or more regular polygons whose angles add to less than 360˝ , and hence derive a complete list of possible vertex figures for a (convex) semi-regular polyhedron. Try to eliminate those putative vertex figures that cannot be extended to a semi-regular polyhedron. 4

### 5.7. The Sine Rule and the Cosine Rule

Where given information, or a specified geometrical construction, determines an angle or length uniquely, it is sometimes – but not always – possible to find this angle or length using simple-minded angle-chasing and congruence.

### Problem 191

	- (i) Calculate (exactly) =ADB and =CBD.
	- (ii) Calculate =BDC and =ACD.
	- (i) Calculate (exactly) the size of =BDC ` =ACD.
	- (ii) Explain how we can be sure that =BDC and =ACD are uniquely determined, even though we cannot calculate them immediately. 4

If it turns out that the simplest tools do not allow us to determine angles and lengths, this is usually because we are only using the most basic properties: the congruence criteria, and the parallel criterion. The general art of 'solving triangles' depends on the similarity criterion (usually via trigonometry). And the two standard techniques for 'solving triangles' that go beyond "angle-chasing" and congruence are the Sine Rule, which was established back in Problem 32 (and its consequences, such as the area formula <sup>1</sup> 2 ab sin C – see Problem 33), and the Cosine Rule.

The next problem invites you to use Pythagoras' Theorem to prove the Cosine Rule – an extension of Pythagoras' Theorem which applies to all triangles ABC (including those where the angle at C may not be a right angle).

Problem 192 (The Cosine Rule) Given 4ABC, let the perpendicular from A to BC meet BC at P. If P " C, then we know (by Pythagoras' Theorem) that c <sup>2</sup> " a <sup>2</sup> ` b 2 . Suppose P ‰ C.

(i) Suppose first that P lies on the line segment CB, or on CB extended beyond B. Express the lengths of P C and AP in terms of b and =C. Then apply Pythagoras' Theorem to 4AP B to conclude that

$$c^2 = a^2 + b^2 - 2ab\cos C.$$

(ii) Suppose next that P lies on the line segment BC extended beyond C. Prove once again that

$$c^2 = a^2 + b^2 - 2ab\cos C. \tag{7}$$

Problem 193 Go back to the configuration in Problem 191(b). The required angles are unaffected by scaling, so we may choose AB " BC " 1. Devise a strategy using the Sine Rule and the Cosine Rule to calculate =BDC and =ACD exactly. 4

It is worth reflecting on what the Cosine Rule really tells us:


Hence if we know three sides, or two sides and the angle between them, we can work out all of the angles. The Sine Rule then complements this by ensuring that:


The upshot is that once a triangle is uniquely determined by the given data, we can "solve" to find all three sides and all three angles.

Trigonometry has a long and very interesting history (which is not at all easy to unravel). Euclid (flourished c. 300 BC) understood that corresponding sides in similar figures were "proportional". And he stated and proved the generalization of Pythagoras' Theorem, which we now call the Cosine Rule; but he did this in a theoretical form, without introducing cosines. Euclid's versions for acute-angled and obtuse-angled triangles involved correction terms with opposite signs, so he proved them separately (Elements, Book II, Propositions 12 and 13).

However, the development of trigonometry as an effective theoretical and practical tool seems to have been due to Hipparchus (died c. 125 BC), to Menelaus (c. 70–130 AD), and to Ptolemy (died 168 AD). Once trigonometry moved beyond the purely theoretical, the combination of


liberated astronomers, and later engineers, to calculate lengths and angles efficiently, and as accurately as they required.

In mathematics we either work with exact values, or we have to control errors precisely. But trigonometry can still be a valuable exact tool, provided we remember the lessons of working with fractions such as <sup>2</sup> 3 , or with surds such as ? 2, or with constants such as π, and resist the temptation to replace them by some unenlightening approximate decimal. We can replace cos´<sup>1</sup> ` 1 2 ˘ ` " π 3 ˘ and cos´<sup>1</sup> ` ´ 1 2 ˘ `" 2π 3 ˘ by their exact values; but in general we need to be willing to work with, and to think about, exact forms such as "cos´<sup>1</sup> ` 1 3 ˘ " and "cos´<sup>1</sup> ` ´ 1 3 ˘ ", without switching to some approximate evaluation.

### Problem 194


### Problem 195


Problem 196 Go back to the scenario of Problem 188, with a regular tetrahedron and a regular octahedron both having edges of length 2, and both having one face flat on the table. Suppose we slide the tetrahedron across the table towards the octahedron. What unexpected phenomenon is guaranteed by Problems 194(a) and 195(a)? 4

Problem 197 Consider the cube with edges of length 2 running parallel to the coordinate axes, with its centre at the origin p0, 0, 0q, and with opposite corners at p1, 1, 1q and p´1, ´1, ´1q. The x-, y-, and z-axes, and the xy-, yz-, and zx-planes cut this cube into eight unit cubes – one sitting in each octant.


Problem 198 Consider a single face ABCDE of the regular dodecahedron, with edges of length 1, together with the five pentagons adjacent to it – so that each of the vertices A, B, C, D, E has vertex figure 5<sup>3</sup> . Each vertex figure is rigid, so the whole arrangement of six regular pentagons is also rigid. Let V , W, X, Y , Z be the five vertices adjacent to A, B, C, D, E respectively. Calculate the dihedral angle between the two pentagonal faces that meet at the edge AB. 4

Problem 199 Suppose a regular icosahedron (Problem 189) has edges of length 2. Position vertex A at the 'North pole', and let BCDEF be the regular pentagon formed by its five neighbours.

	- (ii) How many identical regular icosahedra can one fit together, without overlaps, around a single edge?
	- (i) Prove that the three edge lengths of the right-angled triangle 4BOA are the edge lengths of the regular hexagon inscribed in the circle C, of the regular 10-gon inscribed in the circle C, and of the regular 5-gon inscribed in the circle C.
	- (ii) Calculate the distance separating the plane of the regular pentagon BCDEF, and the plane of the corresponding regular pentagon joined to the 'South pole'. 4

Notice that Problem 199(b) shows that the regular icosahedron can be 'constructed' in the Euclidean spirit: part (b)(i) is essentially Proposition 10 of Book XIII of Euclid's Elements, and part (b)(ii) is implicit in Proposition 16 of the Book XIII. Once we are given the radius OB, we can:


It may be worth commenting on a common confusion concerning the regular icosahedron. Each regular polyhedron has a circumcentre, with all vertices lying on a corresponding sphere. If we join any triangular face of the regular icosahedron to the circumcentre O, we get a tetrahedron. These 20 tetrahedra are all congruent and fit together exactly at the point O "without gaps or overlaps". But they are not regular tetrahedra: the circumradius is less than the edge length of the regular icosahedron.

Problem 200 Prove that the only regular polyhedron that tiles 3D (without gaps or overlaps) is the cube. 4

In one sense the result in Problem 200 is disappointing. However, since we know that there are all sorts of interesting 3-dimensional arrangements related to crystals and the way atoms fit together, the message is really that we need to look beyond regular tilings. For example, the construction in Problem 197 shows how the familiar regular tiling of space with cubes incorporates a semi-regular tiling of space with eight regular tetrahedra and two regular octahedra at each vertex.

### 5.8. Circular arcs and circular sectors

Length is defined for straight line segments, and area is defined in terms of rectangles; neither measure is defined for shapes with curved boundaries – unless, that is, they can be cunningly dissected and the pieces rearranged to make a straight line, or a rectangle.

Figure 5: Dumbbell.

Problem 201 Four identical semicircles of radius 1 fit together to make the dumbbell shape shown in Figure 5. Find the exact area enclosed without using the formula for the area of a circle. 4

In general, making sense of length and area for shapes with curved boundaries requires us to combine a little imagination with what we know about straight line segments and polygons. Our goal here is to lead up to the familiar results for the length of circular arcs and the area of circular sectors. But first we need to explore the perimeter and area of regular polygons, and the surface area of prisms and pyramids.

As so often in mathematics, to make sense of the perimeter and area of regular polygons we need to look beyond their actual values (which will vary according to the size of the polygon), and instead interpret these values as a function of some normalizing parameter – such as the radius. The calculations will be simpler if you first prove a general result.

### Problem 202

(a) A regular n-gon and a regular 2n-gon are inscribed in a circle of radius 1. The regular n-gon has edges of length s<sup>n</sup> " s, while the regular 2n-gon has edges of length s2<sup>n</sup> " t. Prove that

$$t^2 = 2 - \sqrt{4 - s^2}.$$


$$s\_5 = \frac{\sqrt{10 - 2\sqrt{5}}}{2}.$$

Use the result in part (a) to calculate the edge length s<sup>10</sup> of a regular 10-gon inscribed in the same circle. 4

### Problem 203

	- (i) Find the exact perimeter p<sup>n</sup> (in surd form): when n " 3; when n " 4; when n " 5; when n " 6; when n " 8; when n " 10; when n " 12.
	- (ii) Check that, for each n:

$$p\_n = c\_n \times r$$

for some constant cn, where

$$c\_3 < c\_4 < c\_5 < c\_6 < c\_8 < c\_{10} < c\_{12} \cdots$$

	- (i) Find the exact perimeter P<sup>n</sup> (in surd form): when n " 3; when n " 4; when n " 5; when n " 6; when n " 8; when n " 10; when n " 12.
	- (ii) Check that, for each n:

$$P\_n = C\_n \times r$$

for some constant Cn, where

$$C\_3 > C\_4 > C\_5 > C\_6 > C\_8 > C\_{10} > C\_{12} \cdots$$

(c) Explain why c<sup>12</sup> ă C12. 4

It follows from Problem 203 that

• the perimeters p<sup>n</sup> and P<sup>n</sup> of regular n-gons inscribed in, or circumscribed about, a circle of radius r all have the same form:

(inscribed) p<sup>n</sup> " c<sup>n</sup> ˆ r; (circumscribed) P<sup>n</sup> " C<sup>n</sup> ˆ r.


### Hence

• the perimeter P of the circle appears to have the form P " K ˆ r, where the ratio

$$K = \frac{\text{perimeter}}{\text{radius}}$$

satisfies

$$c\_3 < c\_4 < c\_5 < c\_6 < c\_8 < c\_{10} < \cdots < K < \cdots < C\_{10} < C\_8 < C\_6 < C\_5 < C\_4 < C\_3.$$

In particular, the value of the constant K lies somewhere between c<sup>12</sup> " 6.21 ¨ ¨ ¨ and C<sup>12</sup> " 6.43 ¨ ¨ ¨ . If we now define the quotient K to be equal to "2π", we see that

$$\text{(perimeter of circle of radius } r\text{)} = 2\pi r,$$

where π denotes some constant lying between 3.1 and 3.22

In this spirit one might reinterpret the first two bullet points as defining two sequences of constants "πn" and "Πn" for n ě 3, such that

• (perimeter of a regular n-gon with circumradius r) " 2πnr, where

$$\begin{aligned} \pi\_3 &= \frac{3\sqrt{3}}{2} = 2.59\dots^1, \pi\_4 = 2\sqrt{2} = 2.82\dots^7, \pi\_5 = \frac{5\sqrt{10-2\sqrt{5}}}{4} = 2.93\dots^7, \pi\_6 = 3, \\ \text{etc.} \end{aligned}$$

etc.,

and

• (perimeter of a regular n-gon with inradius r)" 2Πnr, where

$$
\Pi\_3 = 3\sqrt{3} = 5.19\dots, \Pi\_4 = 4,\\
\Pi\_5 = 5\sqrt{5 - 2\sqrt{5}} = 3.63\dots,\\
\Pi\_6 = 2\sqrt{3} = 3.46\dots,\\
\Pi\_7 = 3\sqrt{3} = 3.46\dots,\\
\Pi\_8 = 3\sqrt{3} = 3.46\dots
$$

etc..

Moreover

• π<sup>3</sup> ă π<sup>4</sup> ă π<sup>5</sup> ă π<sup>6</sup> ă π<sup>8</sup> ă ¨ ¨ ¨ ă π ă ¨ ¨ ¨ ă Π<sup>8</sup> ă Π<sup>6</sup> ă Π<sup>5</sup> ă Π<sup>4</sup> ă Π3.

Problem 204 Find the exact length (in terms of π)


In the next problem we follow a similar sequence of steps to conclude that the quotient

$$L = \frac{\text{area of circle of radius } r}{r^2}$$

is also constant. The surprise lies in the fact that this different constant is so closely related to the previous constant K.

### Problem 205

	- (i) Find the exact area a<sup>n</sup> (in surd form): when n " 3; when n " 4; when n " 5; when n " 6; when n " 8; when n " 10; when n " 12.
	- (ii) Check that, for each n:

$$a\_n = d\_n \times r^2$$

for some constant dn, where

$$d\_3 < d\_4 < d\_5 < d\_6 < d\_8 < d\_{10} < d\_{12} \cdots$$

	- (i) Find the exact area A<sup>n</sup> (in surd form): when n " 3; when n " 4; when n " 5; when n " 6; when n " 8; when n " 10; when n " 12.
	- (ii) Check that, for each n:

$$A\_n = D\_n \times r^2$$

for some constant Dn, where

$$D\_3 > D\_4 > D\_5 > D\_6 > D\_8 > D\_{10} > D\_{12} \cdots$$

(c) Explain why d<sup>12</sup> ă D12. 4

It follows from Problem 205 that

• the areas a<sup>n</sup> and A<sup>n</sup> of regular n-gons inscribed in, or circumscribed about, a circle of radius r all have the same form:

(inscribed) a<sup>n</sup> " d<sup>n</sup> ˆ r 2 ; (circumscribed) A<sup>n</sup> " D<sup>n</sup> ˆ r 2 .


$$L = \frac{\text{area of circle of radius } r}{\text{radius squared}}$$

satisfies

d<sup>3</sup> ă d<sup>4</sup> ă d<sup>5</sup> ă d<sup>6</sup> ă d<sup>8</sup> ă d<sup>10</sup> ¨ ¨ ¨ ă L ă ¨ ¨ ¨ ă D<sup>10</sup> ă D<sup>8</sup> ă D<sup>6</sup> ă D<sup>5</sup> ă D<sup>4</sup> ă D3.

In particular, the value of L lies somewhere between d<sup>12</sup> " 3 and D<sup>12</sup> " 12p2 ´ ? 3q " 3.21 ¨ ¨ ¨ . The surprise lies in the fact that the constant L is exactly half of the constant K – that is, L " π, so

> (area of circle of radius r) " πr<sup>2</sup> .

The next problem offers a heuristic explanation for this surprise.

Figure 6: Circle cut into 8 slices.

Problem 206 A regular 2n-gon ABCDE ¨ ¨ ¨ is inscribed in a circle of radius r. The 2n radii OA, OB, . . . joining the centre O to the 2n vertices cut the circle into 2n sectors, each with angle <sup>π</sup> n (Figure 6).

These 2n sectors can be re-arranged to form an "almost rectangle", by orienting them alternately to point "up" and "down". In what sense does this "almost rectangle" have "height " r" and "width = πr"? 4

### Problem 207

(a) Find a formula for the surface area of a right cylinder with height h and with circular base of radius r.

(b) Find a similar formula for the surface area of a right prism with height h, whose base is a regular n-gon with inradius r. 4

### Problem 208

	- (i) of a semicircle of radius r;
	- (ii) of a quarter circle of radius r;
	- (iii) of a sector of a circle of radius r that subtends an angle θ radians at the centre.

### Problem 209


### Problem 210

(a) Find an expression involving "sin <sup>π</sup> n " for the ratio

> perimeter of inscribed regular n-gon perimeter of circumscribed circle .

(b) Find an expression involving "tan <sup>π</sup> n " for the ratio

$$\frac{\text{perimeter of circumscribed regular } n \text{-gon}}{\text{perimeter of inscribed circle}}. \qquad \qquad \triangle$$

### Problem 211

(a) Find an expression involving "sin <sup>2</sup><sup>π</sup> n " for the ratio

> area of inscribed regular n-gon area of circumscribed circle .

(b) Find an expression involving "tan <sup>π</sup> n " for the ratio

$$\frac{\text{area of circumscribed regular } n \text{-gon}}{\text{area of inscribed circle}}. \qquad \qquad \qquad \triangle$$

### 5.9. Convexity

This short section presents a simple result which to some extent justifies the assumptions made in the previous section – namely that the perimeter (or area) of a regular n-gon inscribed in a circle is less than the perimeter (or area) of the circle, and of the circumscribed regular n-gon.

Problem 212 A convex polygon P<sup>1</sup> is drawn in the interior of another convex polygon P2.


### 5.10. Pythagoras' Theorem in three dimensions

Pythagoras' Theorem belongs in 2-dimensions. But does it generalise to 3-dimensions? The usual answer is to interpret the result in terms of coordinates.

### Problem 213


This extension of Pythagoras' Theorem to 3-dimensions is extremely useful, but not very profound. In contrast, the next result is more intriguing, but seems to be a complete fluke of limited relevance. In 2D, a right angled triangle is obtained by


This suggests that a corresponding figure in 3D might be obtained by


The obvious candidate for the "3D-hypotenuse" is then the sloping face BCD, and the three right angled triangles 4ABC, 4ACD, 4ADB presumably correspond to the 'legs' (the shorter sides) of the right triangle in 2D.

Problem 214 You are given a pyramid ABCD with all three faces meeting at A being right angled triangles with right angles at A. Suppose AB " b, AC " c, AD " d.


More significant (e.g. for navigation on the surface of the Earth) and more interesting than Problem 214 is to ask what form Pythagoras' Theorem takes for "lines on a sphere".

For simplicity we work on a unit sphere. We discovered in the run-up to Problem 34 that lines, or shortest paths, on a sphere are arcs of great circles. So, if the triangle 4ABC on the unit sphere is right angled at A, we may rotate the sphere so that the arc AB lies along the equator and the arc AC runs up a circle of longitude. It is then clear that, once the lengths c, b of AB and AC are known, the locations of B and C are essentially determined, and hence the length of the arc BC on the sphere is determined. So we would like to have a simple formula that would allow us to calculate the length of the arc BC directly in terms of c and b.

Problem 215 Given a spherical triangle 4ABC on the unit sphere with centre O, such that =BAC is a right angle, and such that AB has length c, and AC has length b.

	- (ii) Suppose =B " π 2 , but =C (and hence c) is unconstrained. The output a is then determined – but the formula must give this fixed output for different values of c. What does this suggest as the "simplest possible" formula for a? 4

The answers to Problem 215 give a pretty good idea what form Pythagoras' Theorem must take on the unit sphere. The next problem proves this result as a simple application of the familiar 2D Cosine Rule.

Problem 216 Given any triangle 4ABC on the unit sphere with a right angle at the point A, we may position the sphere so that A lies on the equator, with AB along the equator and AC up a circle of longitude. Let O be the centre of the sphere and let T be the tangent plane to the sphere at the point A. Extend the radii OB and OC to meet the plane T at B<sup>1</sup> and C 1 respectively.


When "solving triangles" on the sphere the same principles apply as in the plane: right angled triangles hold the key – but Pythagoras' Theorem and trig in right angled triangles must be extended to obtain variations of the Sine Rule and the Cosine Rule for spherical triangles. The corresponding results on the sphere are both similar to, and intriguingly different from, those we are used to in the plane. For example, there are two forms of the Cosine Rule extending the result in Problem 216.

Problem 217 Given a (not necessarily right angled) triangle 4ABC on the unit sphere, apply the same proof as in Problem 216 to show (with the usual labelling) that:

$$
\cos a = \cos b \cdot \cos c + \sin b \cdot \sin c \cdot \cos A \qquad \qquad \qquad \triangle
$$

The other form of the Cosine Rule is "dual" to that in Problem 217 (with arcs and angles interchanged, and with an unexpected change of sign) – namely:

cos A " ´ cos B ¨ cos C ` sin B ¨ sin C ¨ cos a.

The next two problems derive a version of the Sine Rule for spherical triangles.

Problem 218 Let 4ABC be a triangle on the unit sphere with a right angle at A. Let A<sup>1</sup> lie on the arc BA produced, and C 1 lie on the arc BC produced so that 4A1BC<sup>1</sup> is right angled at A<sup>1</sup> . With the usual labelling (so that x denotes the length of the side of a triangle opposite vertex X, with arc AC " b, arc BC " a, arc BC<sup>1</sup> " a 1 , and arc A1C <sup>1</sup> " b 1 , prove that:

$$\frac{\sin b}{\sin a} = \frac{\sin b'}{\sin a'}.\tag{7}$$

Problem 219 Let 4ABC be a general triangle on the unit sphere with the usual labelling (so that x denotes the length of the side of a triangle opposite vertex X, and X is used both to label the vertex and to denote the size of the angle at X). Prove that:

$$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B} = \frac{\sin c}{\sin C}. \tag{7}$$

It is natural to ask (cf Problem 32):

"If the three ratios in Problem 219 are all equal, what is it that they are all equal to?"

The answer may not at first seem quite as nice as in the Euclidean 2-dimensional case: one answer is that they are all equal to

> sin a ¨ sin b ¨ sin c volume of the tetrahedron OABC .

Notice that this echoes the result in the Euclidean plane, where the three ratios in the Sine Rule are all equal to 2R, and

$$2R = \frac{abc}{2(\text{area of } \triangle ABC)}.$$

### 5.11. Loci and conic sections

This section offers a brief introduction to certain classically important loci in the plane. The word locus here refers to the set of all points satisfying some simple geometrical condition; and all the examples in this section are based on the notion of distance from a point and from a line.

Given a point O and a positive real r, the locus of points at distance r from O is precisely the circle of radius r with centre O. If r ă 0, then the locus is empty; while if r " 0, the locus consists of the point O alone.

Given a line m and a positive real r, the locus of all points at distance r from the line m consists of a pair of parallel lines – one either side of the line m. Given a circle of radius r, and a positive real number d ă r; the locus of points at distance d from the circle consists of two circles, each concentric with the given circle (one inside the given circle and one outside). If d ą r, the locus consists of a single circle outside the given circle.

Given two points A and B, the locus of points which are equidistant from A and from B is precisely the perpendicular bisector of the line segment AB. And given two lines m, n the locus of points which are equidistant from m and from n takes different forms according as m and n are, or are not, parallel.


Problem 220 Given a point F and a line m, choose m as the x-axis and the line through F perpendicular to m as the y-axis. Let F have coordinates p0, 2aq.


The locus, or curve, in Problem 220 is called a parabola; the point F is called the focus of the parabola, and the line m is called the directrix. In general, the ratio

"the distance from X to F" : "the distance from X to m"

is called the eccentricity of the curve. Hence the parabola has eccentricity e " 1.

The parabola has many wonderful properties: for example, it is the path followed by a projectile under the force of gravity; if viewed as the surface of a mirror, a parabola reflects the sun's rays (or any parallel beam) to a single point – the focus F. Since the only variable in the construction of the parabola is the distance "2a" between the focus and the directrix, we can scale distances to see that any two different-looking parabolas must in fact be similar to one another – just as with any two circles. (It is hard not to infer from the graphs that y " 10x 2 is a "thin" parabola, and that y " ` 1 10 ˘ x 2 is a "fat" parabola. But the first can be rewritten in the form 10y " p10xq 2 , and the second can be rewritten in the form ` y 10 ˘ " ` x 10 ˘2 , so each is a re-scaled version of Y " X<sup>2</sup> .)

So far we have considered loci defined by some pair of distances being equal, or in the ratio 1 : 1. More interesting things begin to happen when we consider conditions in which two distances are in a fixed ratio other than 1 : 1.

### Problem 221


### Problem 222


distance from X to the point F : distance from X to the line m

is a positive constant e ă 1.

(c) Prove that parts (a) and (b) give different ways of specifying the same curve, or locus. 4

### Problem 223


distance from X to the point F : distance from X to the line m

is a constant e ą 1.

(c) Prove that parts (a) and (b) give different ways of specifying the same curve, or locus. 4

Problem 221 is sometimes presented in the form of a mild joke.

Two dragons are sleeping, one at A and one at B. Dragon A can run twice as fast as dragon B. A specimen of homo sapiens is positioned on the line segment AB, twice as far from A as from B, and cunningly decides to crawl quietly away, while maintaining the ratio of his distances from A and from B (so as to make it equally difficult for either dragon to catch him should they wake).

The locus that emerges generally comes as a surprise: if the man sticks to his imposed restriction, by moving so that his position X satisfies XA " 2¨XB, then he follows a circle and lands back where he started! The circle is called the circle of Apollonius, and the points A and B are sometimes referred to as its foci.

The locus in Problem 222 is an ellipse – with foci A (or F " p´ae, 0q) and B (" pae, 0q), and with directrix m (the line y " ´ a e ; the line y " a e is the second directrix of the ellipse). The "focus-focus" description in part (a) is symmetrical under reflection in both the line AB and the perpendicular bisector of AB. The "focus-directrix" description in (b) is clearly symmetrical in the line through F perpendicular to m; but it is a surprise to find that the equation

$$\frac{x^2}{a^2} + \frac{y^2}{a^2(1-e^2)} = 1$$

is also symmetrical under reflection in the y-axis. If we set b <sup>2</sup> " a 2 p1 ´ e 2 q, the equation takes the form

$$
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,
$$

which crosses the x-axis when x " ˘a, and crosses the y-axis when y " ˘b. In its standard form, we usually choose coordinates so that b ă a: the line segment from p´a, 0q to pa, 0q is then called the major axis, and half of it (say from p0, 0q to pa, 0q) – of length a – is called the semi-major axis; the line segment from p0, ´bq to p0, bq is called the minor axis, and half of it (say from p0, 0q to p0, bq) – of length b – is called the semi-minor axis.

The form of the equation shows that an ellipse is obtained from a unit circle by stretching by a factor "a" in the x-direction, and by a factor "b" in the y-direction. This implies that the area of an ellipse is equal to πab (since each small s by s square that arises in the definition of the "area" of the unit circle gets stretched into an "as by bs rectangle"). However, the equation tells us nothing about the perimeter of an ellipse. Attempts to pin down the perimeter of an ellipse gave rise in the 18th century to the subject of "elliptic integrals".

Like parabolas, ellipses arise naturally in many important settings. For example, Kepler (1571–1630) discovered that the planetary orbits are not circular (as had previously been believed), but are ellipses – with the Sun at one focus (a conjecture which was later explained by Isaac Newton (1642–1727)). Moreover, the tangent to an ellipse at any point X is equally inclined to the two lines XA and XB, so that a beam emerging from one focus is reflected at every point of the ellipse so that all the reflected rays pass through the other focus.

The curve in Problem 223 is a hyperbola – with foci A (or F " p´ae, 0q) and B (" pae, 0q), and with directrix m (the line y " ´ a e ; the line y " a e is the second directrix of the hyperbola). The "focus-focus" description in part (a) is symmetrical under reflection in both the line AB and the perpendicular bisector of AB. The "focus-directrix" description in (b) is clearly symmetrical in the line through F perpendicular to m; but it is a surprise to find that the equation

$$\frac{x^2}{a^2} - \frac{y^2}{a^2(e^2 - 1)} = 1$$

is also symmetrical under reflection in the y-axis. Like parabolas and ellipses, hyperbolas arise naturally in many important settings – in mathematics and in the natural sciences.

All these loci were introduced and studied by the ancient Greeks without the benefit of coordinate geometry and equations. They were introduced as planar cross-sections of a cone – that is, as natural extensions of straight lines and circles (since the doubly infinite cone is the surface traced out when one rotates a line about an axis through a point on that line). The equivalence of the focus-directrix definition in Problems 220, 222, and 223 and cross sections of a cone follows from the next problem. All five constructions in Problem 224 work with the doubly-infinite cone, which we may represent as x <sup>2</sup>`y <sup>2</sup> " przq <sup>2</sup> – although this representation is not strictly needed for the derivations. The surface of the double cone extends to infinity in both directions, and is obtained by taking the line y " rz in the yz-plane (where r ą 0 is constant), and rotating it about the z-axis. Images of this rotated line are called generators of the cone; and the point they all pass through (i.e. the origin) is called the apex of the cone.

### Problem 224 (Dandelin's spheres: Dandelin (1794–1847))

(a) Describe the cross-sections obtained by cutting such a double cone by a horizontal plane (i.e. a plane perpendicular to the z-axis). What if the cutting plane is the xy-plane?

Figure 7: Conic sections.

	- (ii) What cross-section is obtained if the cutting plane passes through the apex, but is not vertical?
	- (i) What happens if c " 0?
	- (ii) Now assume the cutting plane is parallel to a generator, but does not pass through the apex of the cone – so we may assume that the plane cuts only the bottom half of the cone. Insert a small sphere inside the bottom half of the cone and above the cutting plane, and inflate the sphere as much as possible – until it touches the cone around a horizontal circle (the "contact circle with the cone"), and touches the plane at a single point F. Let the horizontal plane of the "contact circle with the cone" meet the cutting plane in the line m. Prove that each point of the cross-sectional curve is equidistant from the point F and from the line m – and so is a parabola.
	- (ii) We may assume that the plane cuts only the bottom half of the cone. Insert a small sphere inside the bottom half of the cone and above the cutting plane (i.e. on the same side of the cutting plane as the apex of the cone), and inflate the sphere as much as possible – until it touches the cone around a horizontal circle, and touches the plane at a single

point F. Let the horizontal plane of the contact circle meet the cutting plane in the line m. Prove that, for each point X on the cross-sectional curve, the ratio

"distance from X to F" : "distance from X to m" " e : 1

is constant, with e ă 1, and so is an ellipse.

Figure 8: The conic section arising in Problem 224(d).

	- (ii) We can be sure that the plane cuts the bottom half of the cone (as well as the top half). Insert a small sphere inside the bottom half of the cone and on the same side of the cutting plane as the apex, and inflate the sphere as much as possible – until it touches the cone around a horizontal circle, and touches the plane at a single point F. Let the horizontal plane of the contact circle meet the cutting plane in the line m. Prove that, for each point X on the cross-sectional curve, the ratio

"distance from X to F" : "distance from X to m" " e : 1 is constant, with e ą 1, and so is a hyperbola. 4

Problem 224 reveals a remarkable correspondence. It is not hard to show algebraically that any quadratic equation in two variables x, y represents either a point, or a pair of crossing (possibly identical) straight lines, or a parabola, or an ellipse, or a hyperbola: that is, by changing coordinates, the quadratic equation can be transformed to one of the standard forms obtained in this section. Hence, the possible quadratic curves are precisely the same as the possible cross-sections of a cone. This remarkable equivalence is further reinforced by the many natural contexts in which these conic sections arise.

### 5.12. Cubes in higher dimensions

This final section on elementary geometry seeks to explore fresh territory by going beyond three dimensions. Whenever we try to jump up to a new level, it can help to first take a step back and 'take a longer run up'. So please be patient if we initially take a step or two backwards.

We all know what a unit 3D-cube is. And – going backwards – it is not hard to guess what is meant by a unit "2D-cube": a unit 2D-cube is just another name for a unit square. It is then not hard to notice that a unit 3D-cube can be constructed from two unit 2D-cubes as follows:


Perhaps a unit 2D-cube can be constructed in a similar way from "unit 1D-cubes"! This idea suggests that a unit "1D-cube" is just another name for a unit line segment.

Take the unit 1D-cube to be the line segment from 0 to 1:


Having taken a step back, we repeat (and reformulate) the previous construction of a 3D-cube:

• position two such unit 2D-cubes in 3D: with one 2D-cube joining p0, 0, 0q to p1, 0, 0q, then to p1, 1, 0q, then to p0, 1, 0q and back to p0, 0, 0q, and the other 2D-cube joining p0, 0, 1q to p1, 0, 1q, then to p1, 1, 1q, then to p0, 1, 1q, and back to p1, 0, 0q;


To sum up: a unit cube in 1D, or in 2D, or in 3D:


A 3D-cube is surrounded by six 2D-cubes (or faces), and a 2D-cube is surrounded by four 1D-cubes (or faces). So it is natural to interpret the two end vertices of a 1D-cube as being '0D-cubes'. We can then see that a cube in any dimension is made up from cubes of smaller dimensions. We can also begin to make a reasonable guess as to what we might expect to find in a '4D-cube'.

### Problem 225

	- (ii) How many edges (i.e. 1D-cubes) are there in a 1D-cube?
	- (ii) How many "faces" (i.e. 2D-cubes) are there in a 2D-cube?
	- (iii) How many edges (i.e. 1D-cubes) are there in a 2D-cube?
	- (ii) How many 3D-cubes are there in a 3D-cube?
	- (iii) How many edges (i.e. 1D-cubes) are there in a 3D-cube?
	- (iv) How many "faces" (i.e. 2D-cubes) are there in a 3D-cube?

### Problem 226

	- (ii) Label each vertex of your sketch with coordinates px, yq (x, y " 0 or 1) so that the lower 2D-cube has the equation "y " 0" and the upper 2D-cube has the equation "y " 1".
	- (ii) Label each vertex of your sketch with coordinates px, y, zq (where each x, y, z " 0 or 1) so that the lower 2D-cube has the equation "z " 0" and the upper 2D-cube has the equation "z " 1".

[Hint: In part (b) your sketch was a projection of a 3D-cube onto 2D paper, and this forced you to represent the lower and upper 2D-cubes as rhombuses rather than genuine 2D-cubes (unit squares). In part (c) you face the even more difficult task of representing a 4D-cube on 2D paper; so you must be prepared for other "distortions". In particular, it is almost impossible to see what is going on if you try to physically position one 3D-cube "directly above" the other on 2D paper. So start with the "upper" unit 3D-cube towards the top right of your paper, and then position the "lower" unit 3D-cube not directly below it on the paper, but below and slightly to the left, before pairing off and joining up each vertex of the upper 3D-cube with the corresponding vertex in the lower 3D-cube.]

(ii) Label each vertex of your sketch with coordinates pw, x, y, zq (where each w, x, y, z " 0 or 1) so that the lower 3D-cube has the equation "z " 0" and the upper 3D-cube has the equation "z " 1". 4

Problem 227 The only possible path along the edges of a 2D-cube uses each vertex once and returns to the start after visiting all four vertices.

	- (ii) Look at the sequence of coordinate triples as you follow your path. What do you notice?
	- (ii) Look at the sequence of coordinate 4-tuples as you follow your path. What do you notice? 4

### 5.13. Chapter 5: Comments and solutions

137. Note: The spirit of constructions restricts us to:


The whole thrust of this first problem is to find some way to "jump" from A (or B) to C. So the problem leaves us with very little choice; AB is given, and A and B are more-or-less indistinguishable, so there are only two possible 'first moves' – both of which work with the line segment AC (or BC).

Join AC.

Then construct the point X such that 4ACX is equilateral (i.e. use Euclid's Elements, Book I, Proposition 1). Construct the circle with centre A which passes through B; let this circle meet the line AX at the point Y , where either


In each case, AY " AB. Finally construct the circle with centre X which passes through Y . In case (i), let the circle meet the line segment XC at D; in case (ii), let the circle meet CX produced (beyond X) at D.

In case (i), CX " CD ` DX; therefore

$$\begin{array}{rcl} \underline{CD} &=& \underline{CX} - \underline{DX} \\ &=& \underline{AX} - \underline{YX} \\ &=& \underline{AY} = \underline{AB}. \end{array}$$

In case (ii),

$$\begin{array}{rcl} \underline{CD} &=& \underline{CX} + \underline{XD} \\ &=& \underline{AX} + \underline{XY} \\ &=& \underline{AY} = \underline{AB}. \end{array}$$

QED

QED

138.

$$\begin{aligned} \angle AXC &=& \angle AXB - \angle CXB\\ &=& \angle CXD - \angle CXB \quad \text{(since the two straight angles)}\\ &=& \angle AXB \text{ and } \angle CXD \text{ are equal)}\\ &=& \angle BXD. \end{aligned}$$

139.

AM " AM MB " MC (by construction of the midpoint M) BA " CA (given). 6 4AMB " 4AMC (by SSS-congruence) 6 =AMB " =AMC, so each angle is exactly half the straight angle =BMC. Hence AM is perpendicular to BC. QED

140. Let ABCDEF be a regular hexagon with sides of length 1. Then 4ABC (formed by the first three vertices) satisfies the given constraints, with =ABC " 120˝ .

Let 4B 1C <sup>1</sup>D be an equilateral triangle with sides of length 2, and with A 1 the midpoint of B <sup>1</sup>D. Then 4A 1B 1C 1 satisfies the given constraints with angle =A 1B 1C <sup>1</sup> " 60˝ .

#### 141.


#### 142.

(i) Draw the circle with centre A and passing through B, and the circle with centre B passing through A. Let these two circles meet at C and D.

Wherever the midpoint M of AB may be, we know from Euclid Book I, Proposition 1 and Problem 139:

that 4ABC is equilateral, and that CM is perpendicular to AB, and that 4ABD is equilateral, and that DM is perpendicular to AB.

Hence CMD is a straight line.

So if we join CD, then this line cuts AB at its midpoint M. QED

(ii) We may suppose that BA ď BC.

Then the circle with centre B, passing through A meets BC internally at A 1 (say).

Let the circle with centre A and passing through B meet the circle with centre A 1 and passing through B at the point D.

Claim BD bisects =ABC. Proof

> BA " BA<sup>1</sup> (radii of the same circle with centre B) AD " AB (radii of the same circle with centre A) " A <sup>1</sup>B " A <sup>1</sup>D (radii of same circle with centre A 1 ) BD " BD

Hence 4BAD " 4BA<sup>1</sup>D (by SSS-congruence). 6 =ABD " =A <sup>1</sup>BD. QED

(iii) Suppose first that P A " P B. Then the circle with centre P and passing through A meets the line AB again at B. If we construct the midpoint M of AB as in part (i), then PM will be perpendicular to AB.

Now suppose that one of P A and P B is longer than the other. We may suppose that P A ą P B, so B lies inside the circle with centre P and passing through A. Hence this circle meets the line AB again at A <sup>1</sup> where B lies between A and A 1 . If we now construct the midpoint M of AA<sup>1</sup> as in part (i), then PM will be perpendicular to AA<sup>1</sup> , and hence to AB. QED

143. Let M be the midpoint of AB.


6 =XMA " =XMB, so each must be exactly half a straight angle.

6 X lies on the perpendicular bisector of AB. QED

144. Let X lie on the plane perpendicular to NS, through the midpoint M. 6 4XMN " 4XMS (by SAS congruence, since XM " XM, =XMN " =XMS, MN " MS) 6 XN " XS.

Let X be equidistant from N and from S, then 4XMN " 4XMS (by SSS-congruence, since XM " XM, MN " MS, NX " SX).


145. Let M be the midpoint of AC. Join BM and extend the line beyond M to the point D such that MB " MD. Join CD. Then 4AMB " 4CMD (by SAS-congruence, since

AM " CM (by construction of the midpoint M) =AMB " =CMD (vertically opposite angles) MB " MD (by construction)).

6 =DCM " =BAM. Now

$$
\begin{aligned}
\angle ACX &=& \angle DCM + \angle DCX \\
&>& \angle DCM \\
&=& \angle BAM \\
&=& \angle A.
\end{aligned}
$$

Hence =ACX ą =A.

Similarly, we can extend AC beyond C to a point Y . Let N be the midpoint of BC. Join AN and extend the line beyond N to the point E such that NA " NE. Join CE.

Then 4BNA " 4CNE (again by SAS-congruence). 6 =BCY ą =BCE " =CBA " =B. QED

#### 146.

	- (i) AB " AC, or
	- (ii) AC ą AB.
	- (i) If AB " AC, then 4ABC is isosceles, so =ACB " =ABC contrary to assumption.
	- (ii) If AC ą AB, then =ABC ą =ACB (by part (a)) again contrary to assumption.

Hence, if =ACB ą =ABC, it follows that AB ą AC. QED

(c) Extend AB beyond B to the point D, such that BD " BC. Then 4BDC is isosceles with apex B, so =BDC " =BCD. Now

$$
\angle ACD = \angle ACB + \angle BCD > \angle BCD = \angle BDC.
$$

Hence, by part (b), AD ą AC. By construction, AD " AB ` BD " AB ` BC, so AB ` BC ą AC. QED

147. Suppose =C " =A`=B, but that C does not lie on the circle with diameter AB. Then C lies either inside, or outside the circle. Let O be the midpoint of AB.


148. Suppose, to the contrary, that OP is not perpendicular to the tangent at P. Drop a perpendicular from O to the tangent at P to meet the tangent at Q. Extend P Q beyond Q to some point X. Then =OQP and =OQX are both right angles. Since Q (‰ P) lies on the tangent, Q lies outside the circle, so OQ ą OP. Hence (by Problem 146(a)) =OP Q ą =OQP " =OQX – contrary to the fact that =OQX ą =OP Q (by Problem 145). QED

149. Let Q lie on the line m such that P Q is perpendicular to m. Let X be any other point on the line m, and let Y be a point on m such that Q lies between X and Y . Then =P QX and =P QY are both right angles. Suppose that P X ă P Q. Then =P QY " =P QX ă =P XQ (by Problem 146(a)), which contradicts Problem 145 (since =P QY is an exterior angle of 4P QX). Hence P X ě P Q as required. QED

150. =A ` =B ` =C, and =XCA ` =C are both equal to a straight angle. So =A ` =B " =XCA.

151. Join OA, OB, OC. Since these radii are all equal, this produces three isosceles triangles. There are five cases to consider.

(i) Suppose first that O lies on AB. Then AB is a diameter, so =ACB is a right angle (by Problem 141), =AOB is a straight angle, and the result holds.

(ii) Suppose O lies on AC, or on BC. These are similar, so we may assume that O lies on AC. Then 4OBC is isosceles, so =OBC " =OCB. =AOB is the exterior angle of 4OBC, so

$$
\angle AOB = \angle OBC + \angle OCB = 2 \cdot \angle ACB
$$

(by Problem 150).

(iii) Suppose O lies inside 4ABC. 4OAB, 4OBC, 4OCA are isosceles, so let =OAB " =OBA " x, =OBC " =OCB " y, =OCA " =OAC " z. Then =ACB " y ` z, =ABC " x ` y, =BAC " x ` z. The three angles of 4ABC add to a straight angle, so 2px ` y ` zq equals a straight angle. Hence, in 4OBA,

$$
\angle AOB = 2(x+y+z) - (\angle OAB + \angle OBA) = 2(y+z) = 2 \cdot \angle ACB.
$$

(iv) Suppose O lies outside 4ABC with O and B on opposite sides of AC. 4OAB, 4OBC, 4OCA are isosceles, so let =OAB " =OBA " x, =OBC " =OCB " y, =OCA " =OAC " z. Then =ACB " y ´ z, =ABC " x ` y, =BAC " x ´ z. The three angles of 4ABC add to a straight angle, so 2x ` 2y ´ 2z equals a straight angle. Hence

$$2x + 2y - 2z = \angle AOB + \angle OAB + \angle OBA = \angle AOB + 2x,$$

so =AOB " 2y ´ 2z " 2 ¨ =ACB.

(v) Suppose O lies outside 4ABC with O and B on the same side of AC. 4OAB, 4OBC, 4OCA are isosceles, so let =OAB " =OBA " x, =OBC " =OCB " y, =OCA " =OAC " z. Then =ACB " y ` z, =ABC " y ´ x, =BAC " z ´ x.

The three angles of ABC add to a straight angle, so 2y`2z´2x equals a straight angle.

Since C lies on the minor arc relative to the chord AB, we need to interpret"the angle subtended by the chord AB at the centre O" as the reflex angle outside the triangle 4AOB – which is equal to "2x more than a straight angle", so =AOB " 2y ` 2z " 2 ¨ =ACB. QED

152. The chord AB subtends angles at C and at D on the same arc. Similarly BC subtends angles at A and at D on the same arc. Hence (by Problem 151) =ACB " =ADB, and =BAC " =BDC.

Hence

$$
\angle ADC = \angle ADB + \angle BDC = \angle ACB + \angle BAC,
$$

so =ADC ` =ABC equals the sum of the three angles in 4ABC. QED

153. Let O be the circumcentre of 4ABC, and let =XAB " x. Then =XAO is a right angle, and 4OAB is isosceles. There are two cases.


$$
\angle XAB + \angle BAC + \angle YAC = \angle XAB + \angle BAC + \angle ABC
$$

are both straight angles, so =XAB " =ACB (since the three angles of 4ABC also add to a straight angle). QED

#### 154.

	- (ii) We are told that the points C, Dlie "on the same side of the line AB". This"side" of the line AB (or "half-plane") is split into three parts by the half-lines"ACproduced beyond C" and "BCproduced beyond C".There are two very different possibilities.

Suppose first that Dlies in one of the two overlapping wedge-shaped regions"between ABproduced andACproduced" or "between BAproduced andBCproduced". Then either DAorDBcuts the arc ABat a point X(say), and=ACB " =AXB ą =ADB (by Problem 145applied to 4AXDor to 4BXD).

The only alternative is that Dlies in the wedge shaped region outside the circleat the point C,lying "betweenACproduced and BCproduced". Then C liesinside 4ADB, so C lies inside the circumcircle of 4ADB. Hencepart (i) implies that =ACB ă =ADBas required. QED


155. 4OP Q " 4OP<sup>1</sup>Q (by RHS-congruence: =OP Q " =OP<sup>1</sup>Q are both right angles, OP " OP<sup>1</sup> , OQ " OQ) 6 QP " QP<sup>1</sup> , and =QOP " =QOP<sup>1</sup> . QED 156.


=BY X " =BZX are both right angles XY " XZ, since we are assuming X is equidistant from m and from n BX " BX).

Hence =XBY " =XBZ, so X lies on the bisector of =Y BZ. QED

#### 157.

(i) Join AC. Then 4ABC " 4CDA by ASA-congruence:

$$
\begin{array}{l}
\angle BAC = \angle DCA \text{ (alternate angles, since } AB \parallel DC) \\
\underline{AC} = \underline{CA} \\
\angle ACB = \angle CAD \text{ (alternate angles, since } CB \parallel DA).
\end{array}
$$

In particular, 4ABC and 4CDA must have equal area, and so each is exactly half of ABCD. QED

Note: Once we prove (Problem 161 below) that a parallelogram has the same area as the rectangle on the same base and lying between the same pair of parallels (whose area is equal to "base ˆ height"), the result in part (i) will immediately translate into the familiar formula for the area of the triangle

$$
\frac{1}{2}(\text{base} \times \text{height}).
$$


=XAB " =XCD (alternate angles, since AB k DC) AB " CD (by part (ii)) =XBA " =XDC (alternate angles, since AB k DC).

$$\begin{array}{l} \text{Hence } \underline{XA} \text{ (in } \triangle AXB) = \underline{XC} \text{ (in } \triangle CXD), \text{ and } \underline{XB} \text{ (in } \triangle AXB) = \underline{XD} \text{ (in } \triangle XB), \\ \text{(\triangle CXD)}. \end{array}$$

158. We may assume that m cuts the opposite sides AB at Y and DC at Z. 4XY B " 4XZD by ASA-congruence:

=Y XB " =ZXD (vertically opposite angles) XB " XD (by Problem 157(iii)) =XBY " =XDZ (alternate angles).

Therefore

$$\begin{aligned} \text{area}(YZCB) &= \text{area}(\triangle BCD) - \text{area}(\triangle XZD) + \text{area}(\triangle XYB) \\ &= \text{area}(\triangle BCD) \\ &= \frac{1}{2}\text{area}(ABCD). \end{aligned}$$

QED

#### 159.

(a) Join AC. Then 4ABC " 4CDA by SAS-congruence:

$$\frac{BA}{\angle BAC} = \underline{DC} \text{ (given)}\\\angle BAC = \angle DCA \text{ (alternate angles, since } AB \parallel DC)\\\underline{AC} = \underline{CA}.$$

Hence =BCA " =DAC, so AD k BC as required. QED

(b) Join AC. Then 4ABC " 4CDA by SSS-congruence:

$$\begin{array}{l} \underline{AB} = \underline{CD} \text{ (given)}\\ \underline{BC} = \underline{DA} \text{ (given)}\\ \underline{CA} = \underline{AC} .\end{array}$$

Hence =BAC " =DCA, so AB k DC; and =BCA " =DAC, so BC k AD.

QED

(c) =A ` =B ` =C ` =D " 2=A ` 2=B " 2=A ` 2=D are each equal to two straight angles.

6 =A ` =B is equal to a straight angle, so AD k BC; and =A ` =D is equal to a straight angle, so AB k DC. QED

Note: The fact that the angles in a quadrilateral add to two straight angles is proved in the next chapter. However, if preferred, it can be proved here directly. If we imagine pins located at A, B, C, D, then a string tied around the four points defines their "convex hull" – which is either a 4-gon (if the string touches all four pins), or a 3-gon (if one vertex is inside the triangle formed by the other three). In the first case, either diagonal (AC or BD) will split the quadrilateral internally into two triangles; in the second case, one of the three 'edges' joining vertices of the convex hull to the internal vertex cannot be an edge of the quadrilateral, and so must be a diagonal, which splits the quadrilateral internally into two triangles. 160. AM " MD (by construction of M as the midpoint), and BN " NC. 6 AM " BN (since AD " BC by Problem 157(ii)). 6 ABNM is a parallelogram (by Problem 159(a)), so MN k AB. Let AC cross MN at Y . Then 4AY M " 4CY N (by ASA-congruence, since

$$\begin{array}{l} \angle YAM = \angle YCN \text{ (alternate angles, since } AD \parallel BC) \\ \hline AM = \underline{CN} \\ \angle AMY = \angle CNY \text{ (alternate angles, since } AD \parallel BC). \end{array}$$

Hence AY " CY , so Y is the midpoint of AC – the centre of the parallelogram (where the two diagonals meet (by Problem 157(iii))). QED

### 161.

Note: In the easy case, where the perpendicular from A to the line DC meets the side DC internally at X, it is natural to see the parallelogram ABCD as the "sum" of a trapezium ABCX and a right angled triangle 4AXD. If the perpendicular from B to DC meets DC at Y , then 4AXD " 4BY C. Hence we can rearrange the two parts of the parallelogram DCBA to form a rectangle XY BA.

However, a general proof cannot assume that the perpendicular from A (or from B) to DC meets DC internally. Hence we are obliged to think of the parallelogram in terms of differences. This is a strategy that is often useful, but which can be surprisingly elusive.

Draw the perpendiculars from A and B to the line CD, and from C and D to the line AB. Choose the two perpendiculars which, together with the lines AB and CD define a rectangle that completely contains the parallelogram ABCD (that is, if AB runs from left to right, take the left-most, and the right-most perpendiculars). These will be either the perpendiculars from B and from D, or the perpendiculars from A and from C (depending on which way the sides AC and BD slope).

Suppose the chosen perpendiculars are the one from B – meeting the line DC at P, and the one from D – meeting the line AB at Q.

Then BP k QD (by Problem 159(c)), so BQDP is a parallelogram with a right angle – and hence a rectangle. Hence BQ " P D, and BP " QD (by Problem 157(ii)).

4QAD " 4P CB (by RHS-congruence), so each is equal to half the rectangle on base P C and height P B. Hence

$$\begin{aligned} \text{area}(ABCD) &= \text{area(rectangle } BQDP) \\ &= \text{area(rectangle on base } \underline{PC} \text{ with height } \underline{PB}) \\ &= \text{area(rectangle on base } \underline{CD} \text{ with height } \underline{DQ}). \end{aligned}$$

QED

162.


163. Consider any path from H to V . Suppose this reaches the river at X. The shortest route from H to X is a straight line segment; and the shortest route from X to V is a straight line segment.

If we reflect the point H in the line of the river, we get a point H<sup>1</sup> , where HH<sup>1</sup> is perpendicular to the river and meets the river at Y (say).

Then 4HXY " 4H<sup>1</sup>XY (by SAS-congruence, since HY " H<sup>1</sup>Y , =HY X " =H<sup>1</sup>Y X, and Y X " Y X). Hence HX " H<sup>1</sup>X, so the distance from H to V via X is equal to HX ` XV " H<sup>1</sup>X ` XV , and this is shortest when H<sup>1</sup> , X, and V are collinear. (So to find the shortest route, reflect H in the line of the river to H<sup>1</sup> , then draw H<sup>1</sup>V to cross the line of the river at X, and travel from H to V via X.)

### 164.

(a) Let 4P QR be any triangle inscribed in 4ABC, with P on BC, Q on CA, R on AB (not necessarily the orthic triangle). Let P <sup>1</sup> be the reflection of P in the side AC, and let P <sup>2</sup> be the reflection of P in the side AB. Then P Q " P <sup>1</sup>Q, and P R " P <sup>2</sup>R (as in Problem 163).

6 P Q ` QR ` RP " P <sup>1</sup>Q ` QR ` RP<sup>2</sup> .

Each choice of the point P on AB determines the positions of P 1 and P 2 . Hence the shortest possible perimeter of 4P QR arises when P <sup>1</sup>QRP<sup>2</sup> is a straight line. That is, given a choice of the point P, choose Q and R by:


It remains to decide how to choose P on BC so that P 1P 2 is as short as possible. The key here is to notice that A lies on both AC and on AB. 6 AP " AP<sup>1</sup> , and AP " AP<sup>2</sup> , so 4AP<sup>1</sup>P 2 is isosceles. Also =P AC " =P <sup>1</sup>AC, and =P AB " =P <sup>2</sup>AB. 6 =P <sup>1</sup>AP<sup>2</sup> " 2 ¨ =A.

Hence, for each position of the point P, 4AP<sup>1</sup>P 2 is isosceles with apex angle equal to 2 ¨ =A. Any two such triangles are similar (by SAS-similarity).

Hence the triangle 4AP<sup>1</sup>P <sup>2</sup> with the shortest "base" P 1P 2 occurs when the legs AP<sup>1</sup> and AP<sup>2</sup> are as short as possible. But AP " AP<sup>1</sup> " AP<sup>2</sup> , so this occurs when AP is as short as possible – namely when AP is perpendicular to BC.

Since the same reasoning applies to Q and to R, it follows that the required triangle 4P QR must be the orthic triangle of 4ABC. QED

(b) Let 4P QR be the orthic triangle of 4ABC, with P on BC, Q on CA, R on AB. Let H be the orthocenter of 4ABC.

=BP H and =BRH are both right angles, so add to a straight angle. Hence (by Problem 154(c)), BP HR is a cyclic quadrilateral. Similarly CP HQ and AQHR are cyclic quadrilaterals.

In the circumcircle of CP HQ, we see that the initial "angle of incidence" =CQP " =CHP. Also =CHP " =AHR (vertically opposite angles); and in the circumcircle of AQHR, =AHR " =AQR.

Hence =CQP " =AQR, so a ray of light which traverses P Q will reflect at Q along the line QR. Similarly one can show that =ARQ " =BRP, so that the ray will then reflect at R along RP; and =BP R " =CP Q, so the ray will then reflect at P along P Q. QED

#### 165.

(a)(i) Triangles 4ABL and 4ACL have equal bases BL " CL, and the same apex A – so lie between the same parallels. Hence they have equal area (by Problems 157 and 161).

Similarly, 4GBL and 4GCL have equal bases BL " CL, and the same apex G – so have equal area.

Hence the differences 4ABG " 4ABL ´ 4GBL and 4ACG " 4ACL ´ 4GCL have equal area.


In the same way 4BCM and 4BAM have equal area; and 4GCM and 4GAM have the same area – say y (since CM " AM). Hence 4BCG and 4BAG have equal area.

But then 4ABG " 4ACG " 4BCG and 4ACG " 4AMG ` 4CMG " 2y, 4BCG " 4BLG ` 4CLG " 2x. Hence x " y, so 4AMG, 4CMG, 4CLG, 4BLG all have the same area x, and 4ABG has area 2x.

The segment GN divides 4ABG into two equal parts (4ANG and 4BNG), so each part has area x.

Hence 4CAG ` 4ANG has the same area (3x) as 4CAN. Hence =CGN is a straight angle, and the three medians AL, BM, CN all pass through the point G.

Note: At first sight, the 'proof' of the result in (b) using vectors seems considerably easier. (If A, B, C have position vectors a, b, c respectively, then L has the position vector <sup>1</sup> 2 pb ` cq, and M has position vector <sup>1</sup> 2 pc ` aq, and it is easy to see that AL and BM meet at G with position vector <sup>1</sup> 3 pa`b`cq. One can then check directly that G lies on CN, or notice that the symmetry of the expression <sup>1</sup> 3 pa ` b ` cq guarantees that G is also the point where BM and CN meet.

The inscrutable aspect of this 'proof' lies in the fact that all the geometry has been silently hidden in the algebraic assumptions which underpin the unstated axioms of the 2-dimensional vector space, and the underlying field of real numbers. Hence, although the vector 'proof' may seem simpler, the two different approaches cannot really be compared.

### 166.


167. Since A and B are interchangeable in the result to be proved, we may assume that A is the point on the secant that lies between P and B.

In order to make deductions, we have to create triangles – so join AT and BT. This creates two triangles: 4P AT and 4P T B, in which we see that: =T P A " =BP T, =P T A " =P BT (by Problem 153), 6 =P AT " =P T B (since the three angles in each triangle add to a straight angle). Hence 4P AT " 4P T B (by AAA-similarity). 6 P T : P B " P A : P T, or P A ˆ P B " P T <sup>2</sup> . QED

168. Since A, B are interchangeable in the result to be proved, and C, D are interchangeable, we may assume that A lies between P and B, and that C lies between P and D.


Notice that AC is a chord which links the two secants P AB and P CD. So join AD and CB.

Then 4P AD " 4P CB (by AAA-similarity: since =AP D " =CP B, and =P DA " =P BCq. 6 P A : P C " P D : P B, or P A ˆ P B " P C ˆ P D. QED

169. Join AD and CB. Then 4P AD " 4P CB (by AAA-similarity: since =AP D " =CP B (vertically opposite angles) =P DA " =P BC (angles subtended by chord AC on the same arc) =P AD " =P CB (angles subtended by a chord BD on the same arc)). 6 P A : P C " P D : P B, or P A ˆ P B " P C ˆ P D. QED

#### 170.

(a) If a " b, then ABCD is a parallelogram (by Problem 159(a)). AD " BC (by Problem 157(ii)). AM " BN, so ABNM is a parallelogram (by Problem 159(a)). MN " AB has length a, and MN k AB (by Problem 160).

If a ‰ b, then a ă b, or b ă a. We may assume that a ă b. Construct the line through B parallel to AD, and let this line meet DC at Q. Then ABQD is a parallelogram, so DQ " AB, and AD " BQ. Hence QC has length b ´ a.

Construct the line through M parallel to QC (and hence parallel to BA), and let this meet BQ at P, and BC at N 1 .

Then ABPM and MP QD are both parallelograms.

6 MP " AB has length a, and

$$
\underline{BP} = \underline{AM} = \underline{MD} = \underline{PQ}.
$$

Now 4BP N<sup>1</sup> " 4BQC (by AAA-similarity, since P N<sup>1</sup> k QC); so

$$\underline{BP} : \underline{BQ} = \underline{BN'} : \underline{BC} = 1 : 2.$$

Hence N <sup>1</sup> " N is the midpoint of BC, MN k BC, and MN has length

$$a + \frac{b - a}{2} = \frac{a + b}{2}.$$

(b) Suppose first that a " b. Then ABCD is a parallelogram (by Problem 159(a)), so the area of ABCD is given by a ˆ d ("(length of base) ˆ height"). (by Problem 161). Hence we may suppose that a ă b.

Solution 1: Extend AB beyond B to a point X such that BX " DC (so AX has length a ` b).

Extend DC beyond C to a point Y such that CY " AB (so DY has length a ` b).

Clearly ABCD and Y CBX are congruent, so each has area one half of area(AXY D).

Now AX k DY , and AX " DY , so AXY D is a parallelogram (by Problem 159(a)).

Hence AXY D has area "(length of base) ˆ height" (by Problem 161), so ABCD has area <sup>a</sup>`<sup>b</sup> <sup>2</sup> ˆ d.

Solution 2: [We give a second solution as preparation for Problem 171.] Now a ă b implies that =BAD ` =ABC is greater than a straight angle.

[Proof. The line through B parallel to AD meets DC at Q, and ABQD is a parallelogram.

Hence DQ " AB, so Q lies between D and C, and

$$
\angle ABC = \angle ABQ + \angle QBC > \angle ABQ.
$$

6 =BAD ` =ABC ą =BAD ` =ABQ, which is equal to a straight angle.]

So if we extend DA beyond A, and CB beyond B, the lines meet at X, where X, D are on opposite sides of AB. Then 4XAB " 4XDC (by AAA-similarity), whence corresponding lengths in the two triangles are in the ratio AB : DC " a : b. In particular, if the perpendicular from X to AB has length h, then h <sup>h</sup>`<sup>d</sup> " a b , so hpb ´ aq " ad.

Now

$$\begin{aligned} \text{area}(ABCD) &= \text{area}(\triangle XDC) - \text{area}(\triangle XAB) \\ &= \frac{1}{2}b(h+d) - \frac{1}{2}ah \\ &= \frac{1}{2}bd + \frac{1}{2}h(b-a) \\ &= \frac{1}{2}bd + \frac{1}{2}ad \\ &= \left(\frac{a+b}{2}\right)d. \end{aligned}$$

171. Note: The volume of a pyramid or cone is equal to

$$
\frac{1}{3} \times \text{ (area of base)} \times \text{height.}
$$

There is no elementary general proof of this fact. The initial coefficient of <sup>1</sup> 3 arises because we are "adding up", or integrating, cross-sections parallel to the base, whose area involves a square x 2 , where x is the distance from the apex (just as the coefficient <sup>1</sup> 2 in the formula for the area of a triangle arises because we are integrating linear cross-sections whose size is a multiple of x – the distance from the apex). Special cases of this formula can be checked – for example, by noticing that a cube ABCDEF GH of side s, with base ABCD, and upper surface EF GH, with E above D, F above A, and so on, can be dissected into three identical pyramids – all with apex E: one with base ABCD, one with base BCHG, and one with base ABGF. Hence each pyramid has volume <sup>1</sup> 3 s 3 , which may be interpreted as

$$
\frac{1}{3} \times \text{(area of base = s}^2\text{)} \times \text{(height = s)}.
$$

To obtain the frustum of height d, a pyramid with height h (say) is cut off a pyramid with height h ` d.

$$\begin{aligned} \text{(i.: volume(frustum)} &= \left[\frac{1}{3} \times b^2 \times (h+d)\right] - \left[\frac{1}{3} \times a^2 \times h\right] \\ &= \left[\frac{1}{3} \times b^2 \times d\right] + \left[\frac{1}{3} \times (b^2 - a^2) \times h\right]. \end{aligned}$$

Let N be the midpoint of BC, and let the line AN meet the upper square face of the frustum at M.

Let the perpendicular from the apex A to the base BCDE meet the upper face of the frustum at Y and the base BCDE at Z.

Then 4AY M " 4AZN (by AAA-similarity), so AY : AZ " Y M : ZN. 6 h <sup>h</sup>`<sup>d</sup> " a b , so hpb ´ aq " ad.

$$\begin{aligned} \text{(i.' volume(frustum))} &= \quad \frac{1}{3}b^2d + \frac{1}{3}\left(b^2 - a^2\right)h \\ &= \quad \frac{1}{3}b^2d + \frac{1}{3}(b+a)ad \\ &= \quad \frac{1}{3}\left(b^2 + ab + a^2\right)d. \end{aligned}$$

172. Construct the line through A which is parallel to A 1B 1 , and let it meet the line BB<sup>1</sup> at P.

Similarly, construct the line through B which is parallel to B 1C 1 and let it meet the line CC<sup>1</sup> at Q.

Then 4ABP " 4BCQ (by AAA-similarity), so AB : BC " AP : BQ. Now AA<sup>1</sup>B <sup>1</sup>P is a parallelogram, so AP " A 1B 1 , and BB<sup>1</sup>C <sup>1</sup>Q is a parallelogram, so BQ " B 1C 1 .

$$
\Box : \underline{\check{AB}} : \underline{BC} = \underline{A'B'} : \underline{B'C'}. \tag{20}
$$

#### 173.

(a) Suppose AB has length a and CD has length b. Problem 137 allows us to construct a point X such that AX " CD, so AX has length b. Now construct the point Y where the circle with centre A and radius AX meets BA produced (beyond A). Then Y B has length a ` b.

If a ą b, let Z be the point where the circle with centre A and passing through X meets the segment AB internally Then ZB has length a ´ b.


Construct the midpoint M of HB; draw the circle with centre M and passing through H and B.

b

Construct the perpendicular to HB at the point A to meet the circle at K. Then 4HAK " 4KAB, so AB : AK " AK : AH. Hence AK " ? a.

174. The key to this problem is to use Problem 158: a parallelogram (and hence any rectangle) is divided into two congruent pieces by any straight cut through the centre. If A is the centre of the rectangular piece of fruitcake, and B is the centre of the combined rectangle consisting of "fruitcake plus icing", then the line AB gives a straight cut that divides both the fruitcake and the icing exactly in two.

### 175.

(a) The minute hand is pointing exactly at "7", but the hour hand has moved <sup>7</sup> <sup>12</sup> of the way from "1" to "2": that is, <sup>7</sup> <sup>12</sup> of 30˝ , or 17 <sup>1</sup> 2 ˝ . Hence the angle between the hands is 162 <sup>1</sup> 2 ˝ .

The same angle arises whenever the hands are trying to point in opposite directions, but are off-set by <sup>7</sup> <sup>12</sup> of 30˝ , or 17 <sup>1</sup> 2 ˝ . This suggests that instead of "35 minutes after 1" we should consider "35 minutes before 11", or 10: 25.

(b) The two hands coincide at midnight. The minute hand then races ahead, and the hands do not coincide again until shortly after 1:00 – and indeed after 1:05. More precisely, in 60 minutes, the minute hand turns through 360˝ , so in x minutes, it turns through 6x ˝ . In 60 minutes the hour hand turns through 30˝ , so in x minutes, the hour hand turns through <sup>1</sup> 2 x ˝ .

The hands overlap when

$$
\frac{1}{2}x \equiv 6x \pmod{360},
$$

or when <sup>11</sup> 2 x is a multiple of 360. This occurs when x " 0 (i.e. at midnight); then not until

$$x = \frac{720}{11} = 65\frac{5}{11},$$

and the time is 5 <sup>5</sup> <sup>11</sup> minutes past 1: that is, after 1 <sup>1</sup> <sup>11</sup> hours. It occurs again 1 <sup>1</sup> <sup>11</sup> hours, or 65 <sup>5</sup> <sup>11</sup> minutes, later – namely at 10 <sup>10</sup> <sup>11</sup> minutes past 2, and so on.

Hence the phenomenon occurs at midnight, and then 11 more times until noon (with noon as the 12th time; and then 11 more times until midnight – and hence 23 times in all (including both midnight occurrences).

(c) If we add a third hand (the 'second hand'), all three hands coincide at midnight. In x minutes, the second hand turns through 360x ˝ .

We now know exactly when the hour hand and minute hand coincide, so we can check where the second hand is at these times. For example, at 5 <sup>5</sup> <sup>11</sup> minutes past 1, the second hand has turned through p360 ˆ 65 <sup>5</sup> <sup>11</sup> q ˝ , and

$$360 \times 65 \frac{5}{11} = 360 \times \frac{5}{11} \pmod{360},$$

so the second hand is nowhere near the other two hands.

The k th occasion when the hour hand and minute hands coincide occurs at k ˆ 1 1 <sup>11</sup> hours after midnight, when the two hands point in a direction ` 360k 11 ˘˝ clockwise beyond "12" . At the same time, the second hand has turned through p360 ˆ 65 <sup>5</sup> <sup>11</sup> ˆ kq ˝ , and

$$360 \times 65 \frac{5}{11} \times k = 360 \times \frac{5}{11} \times k \pmod{360},$$

or five times as far round, and these two are never equal pmod 360q.

#### 176.

Note: One of the things that makes it possible to calculate distances exactly here is that the angles are all known exactly, and give rise to lots of right angled triangles.

The rotational symmetry of the clockface means we only have to consider segments with one endpoint at 12 o'clock (say A). The reflectional symmetry in the line AG means that we only have to find AB, AC, AD, AE, AF, and AG.

Clearly AG " 2. If O denotes the centre of the clockface, then AD is the hypotenuse of an isosceles triangle 4OAD with legs of length 1, so AD " ? 2.

4ACO is isosceles (OA " OC " 1) with apex angle =AOC " 60˝ , so the triangle is equilateral. Hence AC " OA " 1.

It follows that ACEGIK is a regular hexagon, so AE " ? 3 (if OC meets AE at X, then 4ACX is a 30-60-90 triangle, and so is half of an equilateral triangle, whence AX " ? 3 2 ).

It remains to find AB and AF.

Let OB meet AC at Y . Then OY " ? 3 2 (since 4OY A is a 30-60-90 triangle). 6 BY " 1 ´ ? 3 2 , AY " 1 2 , so AB " a 2 ´ ? 3. Finally, AB subtends =AOB " 30˝ at the centre, whence =OAB " =OBA " 75˝ and =AGB " 15˝ . 6 =ABG " 90˝ so we may apply Pythagoras' Theorem to 4ABG to find BG " a

AF " 2 ` ? 3.

177.


? 7 cannot be realized as a distance between integer lattice points in 3D.


? 15 cannot be realized as a distance between integer lattice points in 3D.


Note: The underlying question extends Problem 32:

Which integers can be represented as a sum of three squares?

This question was answered by Legendre (1752–1833):

Theorem. A positive integer can be represented as a sum of three squares if and only if it is not of the form 4<sup>a</sup> p8b ` 7q.

### 178.

	- (i) Consider first the 7-gonal star ADGCF BE. Let GD and BE cross at X and let GC and BF cross at Y . Join DE, BG. As in part (a),

$$
\angle BGC + \angle GBF = \angle BFC + \angle GCF,
$$

and

$$
\angle BGD + \angle GBE = \angle BED + \angle GDE,
$$

so the angles in the 7-gonal star have the same sum as the angles in 4ADE. Hence the seven angles have sum π radians.

(ii) Similar considerations with the 7-gonal star ACEGBDF show that its seven angles have sum 3π radians.

Note: Notice that the three possible "stars" (including the polygon ABCDEF G) have angle sums π radians, 3π radians, and 5π radians.

(c) if n " 2k ` 1 is prime, we may join A to its immediate neighbour B (1-step), or to its second neighbor C (2-step), . . . , or to its k th neighbour (k-step), so there are k different stars, with angle sums

$$(n-2)\pi, (n-4)\pi, (n-6)\pi, \dots, 3\pi, \pi$$

respectively.

If n is not prime, the situation is slightly more complicated, since, for each divisor m of n, the km-step stars break up into separate components.

#### 179.

	- (ii) In the same way it follows that MP passes through L and Q, that P S passes through O and T, etc. so that the figure fits snugly inside the pentagon MP SV Y , whose angles are all equal to 108˝ . Moreover, 4ANX " 4BQL (by SAS-congruence), so XN " LQ, whence MP " Y M. Similarly MP " P S " SV " V Y , so MP SV Y is a regular pentagon.
	- (iii) In the regular pentagon EAXY Z the diagonal EY k AX. Moreover XAC is a straight line, and =ACB `=NBC " 180˝ , so AC k NB. Hence Y E k NB, and Y E " NB " τ (the Golden Ratio <sup>1</sup>` ? 5 2 ), so Y EBN is a parallelogram. 2 .

Hence Y N " EB " τ , so Y M " 1 ` τ " τ

$$\text{(b)(i)}$$

$$
\triangle MPY = \triangle PSM = \triangle SVP = \triangle VYS = \triangle YMV
$$

(by SAS-congruence), so

$$\underline{Y}\underline{P} = \underline{M}\underline{S} = \underline{PV} = \underline{S}\underline{Y} = \underline{VM}$$

Also =PMS " =MP Y " 36˝ ; 6 4BMP has equal base angles, and so is isosceles. Hence BM " BP, and =MBP " 108˝ " =ABC. Similarly =AMY " =AY M " 36˝ . 6 4AMY " 4BPM (by ASA-congruence), so

$$\underline{AY} = \underline{AM} = \underline{BM} = \underline{BP}.$$

4MAB " 4P BC – by ASA-congruence:

$$
\begin{array}{rcl}
\angle BPC &=& 108^\circ - \angle BPM - \angle CPS = 36^\circ = \angle AMB, \\
\underline{MA} &=& \underline{PB}, \text{ and} \\
\angle PBC &=& \angle MBA = \angle MAB.
\end{array}
$$

Hence AB " BC.

Continuing round the figure we see that

$$\underline{AB} = \underline{BC} = \underline{CD} = \underline{DE} = \underline{EA}$$

and that

$$
\angle A = \angle B = \angle C = \angle D = \angle E.
$$

(ii) Extend DB to meet MP at L, and extend DA to meet MY at N. Then 36˝ " =DBC " =LBM (vertically opposite angles). Hence 4LBM has equal base angles and so is isosceles: LM " LB. Similarly NM " NA. Now 4LBM " 4NAM (by ASA-congruence, since MA " MB), so LM " LB " NA " NM. In the regular pentagon ABCDE we know that =DBC " 36˝ ; and in 4LBM, =LBM " =DBC (vertically opposite angles), so =MLB " 108˝ . Hence =BLP " 72˝ " =LBP, so 4P LB is isosceles: P L " P B. In the regular pentagon MP SV Y , 4PMA is isosceles, so PM " P A. 6 LM " PM ´ P L " P A ´ P B " BA. Hence ABLMN is a pentagon with five equal sides. It is easy to check that the five angles are all equal. (iii) We saw in (i) that the five diagonals of MP SV Y are equal. We showed

in Problem 3 that each has length τ , and that MP DY is a rhombus, so DY " PM " 1. Similarly SE " 1. Hence SD " τ ´ 1, and DE " SE ´ SD " 2 ´ τ " pτ 2 q ´1 .

#### 180.

(a) Since the tiles fit together "edge-to-edge", all tiles have the same edge length. The number k of tiles meeting at each vertex must be at least 3 (since the angle at each vertex of a regular n-gon " ` 1 ´ 2 n ˘ π ă π), and can be at most 6 (since the smallest possible angle in a regular n-gon occurs when n " 3, and is then π 3 ).

We consider each possible value of k in turn.


Hence n " 3, or 4, or 6.

	- (ii) If k " 6, n " 3, let the vertices correspond to the complex numbers p ` qω, where p, q are integers, and where ω is a complex cube root of 1 (that is, a solution of the equation ω <sup>3</sup> " 1 ‰ ω, or ω <sup>2</sup> ` ω ` 1 " 0), with the edges being the line segments of length 1 joining nearest neighbours (at distance 1).
	- (iii) If k " 3, n " 6, take the same vertices as in (ii), but eliminate all those for which p ` q " 0 pmod 3q, then let the edges be the line segments of length 1 joining nearest neighbours.

#### 181.

	- ∗ If t " 6, then the vertex figure must be 3 6 .
	- ∗ If t " 5, the remaining gap of 60˝ could only take a sixth triangle, so this case cannot occur.
	- ∗ If t " 4, we are left with angle of 120˝ , so the only possible vertex figure is 3 4 .6.
	- ∗ If t " 3, then we are left with an angle of 180˝ , so the only possible configurations are 3 3 .4 2 (with the two squares together), or 3 2 .4.3.4 (with the two squares separated by a triangle).
	- ∗ If t " 2, we are left with an angle of 240˝ , which cannot be filled with 3 or more tiles (since the average angle size would then be at most 80˝ , and no more triangles are allowed), so the only possible vertex figures are 3 2 .4.12, 3.4.3.12, 3 2 .6 2 , 3.6.3.6 (since 3 2 .5.n or 3.5.3.n would require a regular n-gon with an angle of 132˝ , which is impossible).

Note: The compactness of the argument based on the parameter t is about to end. We continue the same approach, with the focus shifting from the parameter t to a new parameter s – namely the number of squares in each vertex figure.

	- If s " 0, the 300˝ cannot be filled with 3 or more tiles (since then the average angle size would be at most 100˝ , and no squares can be used), so there are exactly two additional tiles. Since each tile has angle ď 180˝ , we cannot use a hexagon, so the smallest tile has at least 7 sides; and since the average of the two remaining angle sizes is 150˝ , the smallest tile has at most 12 sides. It is now easy to check that the only possible vertex figures are 3.7.42, 3.8.24, 3.9.20, 3.10.15, 3.12<sup>2</sup> .

If s " 1, we would be left with an angle of 210˝ , which would require two larger tiles with average angle size 105˝ , which is impossible.

If s " 2, the only possible vertex figures are 3.4 2 .6, or 3.4.6.4.

Clearly we cannot have s " 3 (or we would be left with a gap of 30˝ ); and t " 1, s ą 3 is also impossible.

∗ Hence we may assume that t " 0, in which case s is at most 4. If s " 4, then the vertex figure is 4 4 . If s " 3, then the remaining gap could only take a fourth square, so this case does not occur. If s " 2, we are left with an angle of 180˝ , which cannot be filled.

If s " 1, we are left with an angle of 270˝ , so there must be exactly two additional tiles and the only possible vertex figures are 4.5.20, 4.6.12, or 4.8 2 (since a regular 7-gon would leave an angle of 141 <sup>3</sup> 7 ˝ ).

∗ Hence we may assume that t " s " 0, and proceed using the parameter f – namely the number of regular pentagons. Clearly f is at most 3, and cannot equal 3 (or we would leave an angle of 36˝ ).

If f " 2, we are left with an angle of 144˝ , so the only vertex figure is 5 2 .10.

If f " 1, we are left with an angle of 252˝ , which requires exactly two further tiles, whose average angle is 126˝ ; but this forces us to use a hexagon – leaving an angle of 132˝ , which is impossible.

∗ Hence we may assume that t " s " f " 0. So the smallest possible tile is a hexagon, and since we need at least 3 tiles at each vertex, the only possible vertex figure is 6 3 .

Hence, the simple minded necessary condition (namely that the vertex figure should have no gaps) gives rise to a list of twenty-one possible vertex figures:

3 6 , 3 4 .6, 3 3 .4 2 , 3 2 .4.3.4, 3 2 .4.12, 3.4.3.12, 3 2 .6 2 , 3.6.3.6, 3.7.42, 3.8.24, 3.9.20, 3.10.15, 3.12<sup>2</sup> , 3.4 2 .6, 3.4.6.4, 4 4 , 4.5.20, 4.6.12, 4.8 2 , 5 2 .10, 6 3 .

(ii) Lemma. The vertex figures 3 2 .4.12, 3.4.3.12, 3 2 .6 2 , 3.7.42, 3.8.24, 3.9.20, 3.10.15, 3.4 2 .6, 4.5.20, 5 2 .10 do not extend to semi-regular tilings of the plane.

Proof. Suppose to the contrary that any of these vertex figures could be realized by a semi-regular tiling of the plane. Choose a vertex B and consider the tiles around vertex B.

In the first eight of the listed vertex figures we may choose a triangle T " ABC, which is adjacent to polygons of different sizes on the edges BA (say) and BC.

In the two remaining vertex figures, there is a face T " ABC ¨ ¨ ¨ with an odd number of edges, which has the same property – namely that of being adjacent to an a-gon on the edge BA (say) and a b-gon on the edge BC with a ‰ b. (For example, in the vertex figure 3 2 .4.12, T " ABC is a triangle, and the faces on BA and on BC are – in some order – a 3-gon and a 4-gon, or a 3-gon and a 12-gon.)

In each case let the face T be a p-gon.

If the face adjacent to T on the edge BA is an a-gon, and that on edge BC is a b-gon, then the vertex figure symbol must include the sequence ". . . a.p.b . . . ".

If we now switch attention from vertex B to the vertex A, then we know that A has the same vertex figure, so must include the sequence ". . . a.p.b . . . ", so the face adjacent to the other edge of T at A must be a b-gon. As one traces round the edges of the face T , the faces adjacent to T are alternately a-gons and b-gons – contradicting the fact that T has an odd number of edges.

Hence none of the listed vertex figures extends to a semi-regular tiling of the plane. QED

(b) It transpires that the remaining eleven vertex figures

3 6 , 3 4 .6, 3 3 .4 2 , 3 2 .4.3.4, 3.6.3.6, 3.12<sup>2</sup> , 3.4.6.4, 4 4 , 4.6.12, 4.8 2 , 6 3

can all be realized as semi-regular tilings (and one – namely 3 4 .6 – can be realized in two different ways, one being a reflection of the other).

In the spirit of Problem 180(b), one should want to do better than to produce plausible pictures of such tilings, by specifying each one in some canonical way. We leave this challenge to the reader.

182. [We construct a regular hexagon, and take alternate vertices.]

Draw the circle with centre O passing through A. The circle with centre A passing through O meets the circle again at X and Y .

The circle with centre X and passing through A and O meets the circle again at B; and the circle with centre Y and passing through A and O, meets the circle again at C.

Then 4AOX, 4AOY , 4XOB, 4Y OC are equilateral, so =AXB " 120˝ " =AY C, and =XAB " 30˝ " =Y AC.

Hence 4AXB " 4AY C, so AB " AC. 4ABC is isosceles so =B " =C, with apex angle =BAC " =XAY ´ =XAB ´ =Y AC " 60˝ ; hence 4ABC is equiangular and so equilateral.

### 183.

(a) Draw the circle with centre O and passing through A.

Extend AO beyond O to meet the circle again at C.

Construct the perpendicular bisector of AC, and let this meet the circle at B and at D.

Then BA " BC (since the perpendicular bisector of AC is the locus of points equidistant from A and from C); similarly DA " DC.

4OAB and 4OCB are both isosceles right angled triangles, so =ABC is a right angle (or appeal to "the angle subtended on the circle by the diameter AC"). Similarly =A, =C, =D are right angles, so ABCD is a rectangle with BA " BC, and hence a square.

Note: This construction starts with the regular 2-gon AC inscribed in its circumcircle, and doubles it to get a regular 4-gon, by constructing the perpendicular bisectors of the "two sides" to meet the circumcircle at B and at D.

(b) Erect the perpendiculars to AB at A and at B.

Then draw the circles with centre A and passing through B, and with centre B and passing through A.

Let these circles meet the perpendiculars to AB (on the same side of AB) at D and at C.

Then AD " BC, and AD k BC, so ABCD is a parallelogram (by Problem 159(a)), and hence a (being a parallelogram with a right angle), and so a square (since AB " AD).

### 184.

(a)(i) First construct the regular 3-gon ACE with circumcentre O. Then construct the perpendicular bisectors of the three sides AC, CE, EA, and let these meet the circumcircle at B, D, F.

Note: Here we emphasise the general step from inscribed regular n-gon to inscribed regular 2n-gon – even though this may seem perverse in the case of a regular 3-gon (since we constructed the inscribed regular 3-gon in Problem 182 by first constructing the regular 6-gon and then taking alternate vertices).


(ii) Extend AB beyond B, and let this line meet the circle with centre B and passing through A at X.

Now construct a square BXY Z on the side BX as in 183(a), and let the diagonal BY meet the circle at C.

Construct the circumcentre O of 4ABC (the point where the perpendicular bisectors of AB, BC meet).

Construct the next vertex D as the point where the circle with centre O and passing through A meets the circle with centre C and passing through B. The remaining points E, F, G, H can be found in a similar way.

#### 185.

(a)(i) There are various ways of doing this – none of a kind that most of us might stumble upon. Draw the circumcircle with centre O and passing through A. Extend the line AO beyond O to meet the circle again at X.

Construct the perpendicular bisector of AX, and let this meet the circle at Y and at Z. Construct the midpoint M of OZ, and join MA.

Let the circle with centre M and passing through A meet the line segment OY at the point F.

Finally let the circle with centre A and passing through F meet the circumcircle at B. Then AB is a side of the required regular 5-gon. (The vertex C on the circumcircle is then obtained as the second meeting point of the circumcircle with the circle having centre B and passing through A. The points D, E can be found in a similar way.)

The proof that this construction does what is claimed is most easily accomplished by calculating lengths. a

Let OA " 2. Then OF " ? 5 ´ 1, so AF " 10 ´ 2 ? 5 " AB. It remains to prove that this is the correct length for the side of a regular pentagon inscribed in a circle of radius 2. Fortunately the work has already been done, since 4OAB is isosceles with apex angle equal to 72˝ . If we drop a perpendicular from O to AB, then we need to check whether it is true that

$$
\sin 36^\circ = \frac{\sqrt{10 - 2\sqrt{5}}}{4}
$$

But this was already shown in Problem 3(c).


Then =ABP " 72˝ ; so if we extend the line P B beyond B to X, then =ABX " 108˝ . Let BX meet the circle with centre B through A at the point C. Then BA " BC and =ABC " 108˝ , so we are up and running. If we let the perpendicular bisectors of AB and BC meet at O, then the circle with centre O and passing through A also passes through B and C (and the yet to be located points D and E). The circle with centre C and passing through B meets this circle again at D; and the circle with centre A and passing through B meets the circle again at E.

(ii) To construct the regular 10-gon ABCDEF GHIJ, treat B as the point O in (a)(i) and construct a regular 5-gon AXCY Z inscribed in the circle with centre B and passing through A.

Then =ABC " 144˝ , and BA " BC, so C is the next vertex of the required regular 10-gon. We may now proceed as in (a)(ii) to first construct the circumcentre O of the required regular 10-gon as the point where the perpendicular bisectors of AB and BC meet, then draw the circumcircle, and finally step off successive vertices D, E, . . . of the 10-gon around the circumcircle.

186. The number k of faces meeting at each vertex can be at most five (since more would produce an angle sum that is too large). And k ě 3 (in order to create a genuine corner.


187. The respective midpoints have coordinates: ` ˘ ` ˘ `

of AB: 1 2 , 1 2 , 0 ; of AC: 1 2 , 0, 1 2 ; of AD: 1, 1 2 , 1 2 ˘ ; of BC: ` 0, 1 2 , 1 2 ˘ ; of BD: ` 1 2 , 1, 1 2 ˘ ; of CD: ` 1 2 , 1 2 , 1 ˘ . 6 P Q " ?<sup>1</sup> 2 " P R " P S " P T " QR " RS " ST " T Q.

### 188.

(a) There are infinitely many planes through the apex A and the base vertex B. Among these planes, the one perpendicular to the base BCD is the one that passes through the midpoint M of CD. Let the perpendicular from A to the base, meet the base BCD at the point X, which must lie on BM. Let AX have length h. To find h we calculate the area of 4ABM in two ways. First, BM has length ? 3, so areap4ABMq " <sup>1</sup> 2 `? 3 ˆ h ˘

Second, <sup>4</sup>ABM is isosceles with base AB and apex <sup>M</sup>, so has height ? 2. 6 areap4ABMq " <sup>1</sup> 2 p2 ˆ ? <sup>2</sup>q " ? 2. b

If we now equate the two expressions for areap4ABMq, we see that h " 8 3 .

	- (ii) Let M be the midpoint of BC and N the midpoint of DE. Then NM and AF cross at X and so define a single plane. In this plane, 4ANM " 4FMN (by SSS-congruence, since AN " FM " ? 3, NM " MN, MA " NF " ? 3); hence =ANM " =FMN, so AN k MF. Similarly, if P is the midpoint of AE and Q is the midpoint of F C, then 4DP Q " 4BQP, so DP k QB. Hence the top face DEA is parallel to the bottom face BCF, so the height of the octahedron sitting on the table is equal to the height of <sup>4</sup>FMN. But this triangle has sides of lengths 2, ? ? 3, 3, so this height is exactly the same as the height h in part (a).

#### 189.


Note: The rectangle NP RQ is a "1 by τ ´1" rectangle, and 1 : τ ´1 " τ : 1. Hence the regular icosahedron can be constructed from three congruent, and pairwise perpendicular, copies of a "Golden rectangle".

190. We mimic the classification of possible vertex figures for semi-regular tilings. We are assuming that the angles meeting at each vertex add to ă 360˝ , so the number k of faces at each vertex lies between 3 and 5. Because faces are regular, but not necessarily congruent, k does not determine the shape of the faces. Hence we let t denote the number of triangles at each vertex, which can range from 0 up to 5.


If the next smallest face is a 4-gon, then we get the possible vertex figures

3 2 .4 2 and 3.4.3.4, 3 2 .4.5 and 3.4.3.5, 3 2 .4.6 and 3.4.3.6, 3 2 .4.7 and 3.4.3.7, 3 2 .4.8 and 3.4.3.8, 3 2 .4.9 and 3.4.3.9, 3 2 .4.10 and 3.4.3.10, 3 2 .4.11 and 3.4.3.11.

If the next smallest face is a 5-gon, then we get the possible vertex figures

3 2 .5 2 and 3.5.3.5, 3 2 .5.6 and 3.5.3.6, 3 2 .5.7 and 3.5.3.7.

Note: Before proceeding further it is worth deciding which among the putative vertex figures identified so far seem to correspond to semi-regular polyhedra – and then to prove that these observations are correct.


To avoid further proliferation of spurious 'putative vertex figures' we inject a version of the Lemma used for tilings somewhat earlier than we did for tilings, and then apply the underlying idea to eliminate other spurious possibilities as they arise.

Lemma. The vertex figures

3 2 .4 2 , 3 2 .n (n ą 3), 3 2 .4.5, 3.4.3.5, 3 2 .4.6, 3.4.3.6, 3 2 .4.7, 3.4.3.7, 3 2 .4.8, 3.4.3.8, 3 2 .4.9, 3.4.3.9, 3 2 .4.10, 3.4.3.10, 3 2 .4.11, 3.4.3.11, 3 2 .5 2 , 3 2 .5.6, 3.5.3.6, 3 2 .5.7, 3.5.3.7

do not arise as vertex figures of any semi-regular polyhedron.

Proof outline. Each of these requires that the vertex figure of any vertex B includes a tile T " ABC ¨ ¨ ¨ with an odd number of edges, for which the edge BA is adjacent to an a-gon, the edge BC is adjacent to a b-gon (where a ‰ b), and where the subsequent faces adjacent to T are forced to alternate – a-gon, b-gon, a-gon, . . . – which is impossible. QED

For the rest we introduce the additional parameter s to denote the number of 4-gons in the vertex figure.

Suppose t " 1. Then the remaining polygons have angle sum ă 300˝ , so there are at most 3 additional faces in the vertex figure (since if there were 4 or more extra faces, the average angle size would be ă 75˝ , with no more triangles allowed).

If there are 3 additional faces, the average angle size is ă 100˝ , so s ą 0.

	- 3.5.n (4 ă n), which is impossible as in the Lemma;
	- 3.6 2 , which corresponds to the truncated tetrahedron;
	- 3.6.n (6 ă n), which is impossible as in the Lemma;
	- 3.7.n (6 ă n), which is impossible as in the Lemma;
	- 3.8 2 , which corresponds to the truncated cube;
	- 3.8.n (8 ă n), which is impossible as in the Lemma;
	- 3.9.n (8 ă n), which is impossible as in the Lemma;
	- 3.10<sup>2</sup> , which corresponds to the truncated dodecahedron;
	- 3.10.n (10 ă n), which is impossible as in the Lemma;
	- 3.11.n (10 ă n), which is impossible as in the Lemma.

Thus we may assume that t " 0 – in which case, s ă 4.


#### 191.

	- (ii) 4ABC is equilateral, so AC " AB " AD. Hence 4ADC is isosceles, so =ACD " 70˝ " =ADC, whence =BDC " 70˝ ´ =ADB " 30˝ .
	- (ii) No lengths are specified, so we may choose the length of AC. The point B then lies on the perpendicular bisector of AC, and =CAB " 70˝ determines the location of B exactly. The line AD makes an angle of 40˝ with AC , and BD makes an angle of 80˝ with AC, so the location of D is determined. Hence, despite our failure in part (i), the angles are determined.

192.

(i) If P lies on CB, then P C " b cos C, AP " b sin C, and in the right angled triangle 4AP B we have:

$$\begin{aligned} c^2 &= \quad \left(b\sin C\right)^2 + \left(a - b\cos C\right)^2\\ &= \quad a^2 + b^2(\sin^2 C + \cos^2 C) - 2ab\cos C\\ &= \quad a^2 + b^2 - 2ab\cos C. \end{aligned}$$

If P lies on CB extended beyond B, then P C " b cos C, AP " b sin C as before, and in the right angled triangle 4AP B we have:

$$\begin{aligned} c^2 &=& (b\sin C)^2 + (b\cos C - a)^2 \\ &=& a^2 + b^2(\sin^2 C + \cos^2 C) - 2ab\cos C \\ &=& a^2 + b^2 - 2ab\cos C. \end{aligned}$$

	- c <sup>2</sup> " pb sin Cq <sup>2</sup> ` pa ` P Cq 2 " pb sin Cq <sup>2</sup> ` pa ´ b cos Cq 2 " a <sup>2</sup> ` b 2 psin<sup>2</sup> C ` cos <sup>2</sup> Cq ´ 2ab cos C " a <sup>2</sup> ` b <sup>2</sup> ´ 2ab cos C.

193. There are many ways of doing this – once one knows the Sine Rule and Cosine Rule. If we let =BDC " y and =ACD " z, then one route leads to the identity cospz ´10˝ q " 2 sin 10˝ ¨ sin z, from which it follows that z " 80˝ , y " 20˝ .

#### 194.

(a) The angle between two faces, or two planes, is the angle one sees "end-on" – as one looks along the line of intersection of the two planes. This is equal to the angle between two perpendiculars to the line of intersection – one in each plane. If M is the midpoint of BC, then 4ABC is isosceles with apex A, so the median AM is perpendicular to BC; similarly 4DBC is isosceles with apex D, so the median DM is perpendicular to BC. ?

4MAD is isosceles with apex M, MA " MD " 3, AD " 2, so we can use the Cosine Rule to conclude that 2<sup>2</sup> " 3 ` 3 ´ 2 ¨ 3 ¨ cosp=AMDq, whence cosp=AMDq " <sup>1</sup> 3 .

(b) cosp=AMDq " <sup>1</sup> 3 , and <sup>1</sup> <sup>3</sup> ă 1 2 , so =AMD ą 60˝ ; hence we cannot fit six regular tetrahedra together so as to share an edge. Since =AMD is acute, =AMD ă 90˝ , so we can certainly fit four regular tetrahedra together with lots of room to spare. We can now appeal to "trigonometric tables", or a calculator, to see that ˙

$$
\arccos\left(\frac{1}{3}\right) < 1.24,
$$

that is 1.24 radians, so five tetrahedra use up less than 6.2 radians – which is less than 2π. Hence we can fit five regular tetrahedra together around a common edge with room to spare (but not enough to fit a sixth).

#### 195.

(a) The angle between the faces ABC and F BC is equal to the angle between two perpendiculars to the common edge BC. Since the two triangles are isosceles with the common base BC, it suffices to find the angle between the two medians AM and FM.

In Problem 188 we saw that BCDE is a square, with sides of length 2. If we switch attention from the opposite pair of vertices A, F to the pair C, E, then the same proof shows that ABF D is a square with sides of length 2. Hence the diagonal AF " 2 ? 2.

Now apply the Cosine Rule to 4AMF to conclude that:

$$\left(2\sqrt{2}\right)^2 = 3 + 3 - 2 \cdot 3 \cdot \cos(\angle AMF),$$

so it follows that cosp=AMFq " ´<sup>1</sup> 3 .

(b) cosp=AMFq " ´<sup>1</sup> <sup>3</sup> ă 0, so =AMF ą 90˝ ; hence we cannot fit four regular octahedra together so as to share a common edge. Moreover, ´ 1 <sup>2</sup> ă ´<sup>1</sup> 3 , so =AMD ă 120˝ ; hence we can fit three octahedra together to share an edge with room to spare (but not enough room to fit a fourth).

196. The angle <sup>=</sup>AMD " arccos ` 1 3 ˘ in Problem 194 is acute, and the angle <sup>=</sup>AMF " arccos ` ´ 1 3 ˘ in Problem 195 is obtuse. Hence these angles are supplementary; so the regular tetrahedron fits exactly into the wedge-shaped hole between the face ABC of the regular octahedron and the table.

### 197.


Note: The six vertices p˘1, 0, 0), p0, ˘1, 0), p0, 0, ˘1q span a regular octahedron, with a regular tetrahedron fitting exactly on each face. The resulting compound star-shaped figure is called the stellated octahedron. Johannes Kepler (1571–1630) made an extensive study of polyhedra and this figure is sometimes referred to as Kepler's 'stella octangula'. It is worth making in order to appreciate the way it appears to consist of two interlocking tetrahedra.

198. Let the unlabeled vertex of the pentagonal face ABW ´ V be P. In the pentagon ABW P V the edge AB is parallel to the diagonal V W. Hence ABW V is an isosceles trapezium. The sides V A and W B (produced) meet at S in the plane of the pentagon ABW P V .

4SAB has equal base angles, so SA " SB.

Hence SV " SW.

Similarly BC k W X, and W B and XC meet at some point S 1 on the line W B.

Now 4S <sup>1</sup>BC " 4SAB, so S <sup>1</sup>B " SA " SB. Hence S <sup>1</sup> " S, and the lines V A, W B, XC, Y D, ZE all meet at S.

Since V W k AB, we know that 4SAB " 4SV W, with scale factor

$$\tau: 1 = \underline{V}\underline{W}: \underline{AB} = \underline{SV}: \underline{SA}.$$

If SA " x, then x ` 1 : x " τ : 1, so x " τ .

Let M be the midpoint of AB and let O denote the circumcentre of the regular pentagon ABCDE.

4OAB is isosceles, so OM is perpendicular to AB, and the required dihedral angle between the two pentagonal faces is =OMP.

It turns out to be better to find not the dihedral angle =OMP, but its supplement: namely the angle =SMO.

From 4OAM we see that

$$\underline{OA} = \frac{1}{2\sin 36^\circ},$$

and we know that SA " τ " 2 cos 36˝ . One can then check that OS " cot 36˝ . Similarly, in 4OAM we have

$$
\underline{OM} = \frac{\cot 36^{\circ}}{2}.
$$

Hence in 4OMS we have

$$\tan(\angle SMO) = 2.$$

Hence the required dihedral angle is equal to π ´ arctan 2 « 116.56˝ .

#### 199.

(a)(i) Let M be the midpoint of AC. The angle between the two faces is equal to =BMD.

In 4BMD, we have BM " DM " ? 3, BD " 2τ " 1 ` ? 5. So the Cosine Rule in 4BMD gives:

$$6 + 2\sqrt{5} = 3 + 3 - 2 \cdot 3 \cdot \cos(\angle BMD),$$

so cosp=BMDq " ´ ? 5 3 ,

$$
\angle BMD = \arccos\left(-\frac{\sqrt{5}}{3}\right) = \pi - \arccos\left(\frac{\sqrt{5}}{3}\right) \approx 138.19^{\circ}.
$$


$$
\underline{OB} = \frac{1}{\sin 36^\circ};
$$

and =MBV " 18˝ , so

$$\underline{BV} = \frac{1}{\cos 18^{\circ}}.$$

It is easiest to use the converse of Pythagoras' Theorem, and to write everything first in terms of cos 36˝ , then (since τ " 2 cos 36˝ , so cos 36˝ " 1` ? 5 4 ) write everything in terms of ? 5.

If we use sin<sup>2</sup> 36˝ " 1 ´ cos<sup>2</sup> 36˝ , and 2 cos<sup>2</sup> 18˝ ´ 1 " cos 36˝ " 1` ? 5 4 , then

$$\begin{aligned} \underline{BV}^2 + \underline{OB}^2 &= \quad \left(\frac{1}{\cos 18^\circ}\right)^2 + \left(\frac{1}{\sin 36^\circ}\right)^2 \\ &= \quad \frac{8}{5 + \sqrt{5}} + \frac{8}{5 - \sqrt{5}} \\ &= \quad \frac{80}{20} = 2^2 \\ &= \quad \underline{AB}^2. \end{aligned}$$

Hence, in the right angled triangle 4AOB, we must have OA " BV as claimed.

(ii) Miraculously no more work is needed. Let the vertex at the 'south pole' be L, and let the pentagon formed by its five neighbours be GHIJK. It helps if we can refer to the circumcircle of BCDEF as the 'tropic of Cancer', and to the circumcircle of GHIJK as the 'tropic of Capricorn'.

The pentagon GHIJK is parallel to BCDEF, but the vertices of the southern pentagon have been rotated through <sup>π</sup> 5 relative to BCDEF, so that G (say) lies on the circumcircle of the pentagon GHIJK, but sits directly below the midpoint of the minor arc BC. Let X denote the point on the 'tropic of Capricorn' which is directly beneath B. Then 4BXG is a right angled triangle with BG " AB " 2, and XG " BV is equal to the edge length of a regular 10-gon inscribed in the circumcircle of GHIJK. Hence, by the calculation in (i), BX is equal to the edge length of the regular hexagon inscribed in the same circle – which is also equal to the circumradius.

200. A necessary condition for copies of a regular polyhedron to "tile 3D (without gaps or overlaps)" is that an integral number of copies should fit together around an edge. That is, the dihedral angle of the polyhedron should be an exact submultiple of 2π. Only the cube satisfies this necessary condition.

Moreover, if we take as vertices the points pp, q, rq with integer coordinates p, q, r, and as our regular polyhedra all possible translations of the standard unit cube having opposite corners at p0, 0, 0q and p1, 1, 1q, then we see that it is possible to tile 3D using just cubes.

201. The four diameters form the four edges of a square ABCD of edge length 2. The protruding semicircular segments on the left and right can be cut off and inserted to exactly fill the semicircular indentations above and below.

Hence the composite shape has area exactly equal to 2<sup>2</sup> " 4 square units.

#### 202.

(a) If the regular n-gon is ACEG ¨ ¨ ¨ and the regular 2n-gon is ABCDEF G ¨ ¨ ¨ , then AC " s<sup>n</sup> " s and AB " s2<sup>n</sup> " t. If M is the midpoint of AC, AM " s 2 and c ´

$$\underline{MB} = 1 - \sqrt{\left(1 - \left(\frac{s}{2}\right)^2\right)}$$

$$\begin{aligned} t.^2 \qquad \qquad t^2 &= \underline{AB}^2 \\ &= \left(\frac{s}{2}\right)^2 + \left[1 - \sqrt{1 - \left(\frac{s}{2}\right)^2}\right]^2 \\ &= 1 + 1 - 2\sqrt{1 - \left(\frac{s}{2}\right)^2} \\ &= 2 - \sqrt{4 - s^2}. \end{aligned} \tag{1}$$

(b) Put s " s<sup>2</sup> " 2 in (1) to get t " s<sup>4</sup> " ? 2. Then put s " s<sup>4</sup> " ? 2 in (1) to get

$$t = ss = \sqrt{2 - \sqrt{2}}$$

(c) Put t " s<sup>6</sup> " 1 in (1) to get s " s<sup>3</sup> " ? 3. Then put s " s<sup>6</sup> " 1 to get

$$t = s\_{12} = \sqrt{2 - \sqrt{3}}.$$

(d) Put  $s = s\_5 = \frac{\sqrt{10 - 2\sqrt{5}}}{2}$  to get

$$t = s\_{10} = \sqrt{\frac{3 - \sqrt{5}}{2}}$$
.

203.

$$\begin{aligned} \text{(a)} \ (\text{i)} \ n &= 3 \colon p\_3 = 3\sqrt{3} \times r \\ n &= 4 \colon p\_4 = 4\sqrt{2} \times r \\ n &= 5 \colon p\_5 = \frac{5\sqrt{10 - 2\sqrt{5}}}{2} \times r \\ n &= 6 \colon p\_6 = 6 \times r \\ n &= 8 \colon ps = 8\sqrt{2 - \sqrt{2}} \times r \\ n &= 10 \colon p\_{10} = 5\sqrt{6 - 2\sqrt{5}} \times r \\ n &= 12 \colon p\_{12} = 12\sqrt{2 - \sqrt{3}} \times r . \\ \text{(ii)} \end{aligned}$$

$$\begin{aligned} c\_3 &= 5.19 \dots \quad < \quad c\_4 = 5.65 \dots \\ &< \quad c\_5 = 5.87 \dots \\ &< \quad c\_6 = 6 \\ &< \quad c\_8 = 6.12 \dots \\ &< \quad c\_{10} = 6.18 \dots \\ &< \quad c\_{12} = 6.21 \dots \end{aligned}$$

$$\begin{aligned} \text{(b)} \text{(i)} \quad n=3: &P\_3=6\sqrt{3} \times r\\ \quad n=4: &P\_4=8 \times r\\ \quad n=5: &P\_5=10\sqrt{5-2\sqrt{5}} \times r\\ \quad n=6: &P\_6=4\sqrt{3} \times r\\ \quad n=8: &P\_8=8\left(2\sqrt{2}-2\right) \times r\\ \quad n=10: &P\_{10}=4\sqrt{25-10\sqrt{5}} \times r\\ \quad n=12: &P\_{12}=12\left(4-2\sqrt{3}\right) \times r. \\ \text{(ii)} \end{aligned}$$

$$\begin{aligned} C\_3 &= 10.39 \dots \quad > \quad C\_4 = 8\\ &> \quad C\_5 = 7.26 \dots \\ &> \quad C\_6 = 6.92 \dots \\ &> \quad C\_8 = 6.62 \dots \\ &> \quad C\_{10} = 6.49 \dots \\ &> \quad C\_{12} = 6.43 \dots \end{aligned}$$

(c) Let O be the centre of the circle of radius r. Let A, B lie on the circle with =AOB " 30˝ . Let M be the midpoint of AB – so that 4OAB is isosceles, with apex O and height h " OM. Let A 1 lie on OA produced, and let B 1 lie on OB produced, such that OA<sup>1</sup> " OB<sup>1</sup> and A 1B 1 is tangent to the circle. Then 4OAB " 4OA<sup>1</sup>B <sup>1</sup> with 4OA<sup>1</sup>B 1 larger than 4OAB, so the scale factor 1 <sup>h</sup> " 2 a 2 ´ ? 3 ą 1. Hence P<sup>12</sup> " 2 a 2 ´ ? 3 ˆ p<sup>12</sup> ą p12, so C<sup>12</sup> ą c12.

204. (i) πr (ii) <sup>π</sup> 2 r (iii) θr

#### 205.

(a)(i) Note: This could be a long slog. However we have done much of the work before: when the radius is 1, the most of the required areas were calculated exactly back in Problem 3 and Problem 19.

Alternatively, the area of each of the n sectors is equal to <sup>1</sup> 2 sin θ, where θ is the angle subtended at the centre of the circle, and the exact values of the required trig functions were also calculated back in Chapter 1.

$$\begin{aligned} a\_3 &=& \frac{3\sqrt{3}}{4} \times r^2\\ a\_4 &=& 2 \times r^2\\ a\_5 &=& \frac{5}{8}\sqrt{10+2\sqrt{5}} \times r^2\\ a\_6 &=& \frac{3\sqrt{3}}{2} \times r^2\\ a\_8 &=& 2\sqrt{2} \times r^2\\ a\_{10} &=& \frac{5}{4}\sqrt{10-2\sqrt{5}} \times r^2\\ a\_{12} &=& 3 \times r^2 \end{aligned}$$

(ii)

$$d\_3 = 1.29\dots \quad <\quad d\_4 = 2$$

$$<\quad d\_5 = 2.37\dots$$

$$<\quad d\_6 = 2.59\dots$$

$$<\quad d\_8 = 2.82\dots$$

$$<\quad d\_{10} = 2.93\dots$$

$$<\quad d\_{12} = 3.$$

(b)(i) Note: This could also be a long slog. However we have done much of the work before. But notice that, when the radius is 1, the area of each of the n sectors is equal to half the edge length times the height (" 1); so if r " 1, then the total area is numerically equal to "half the perimeter Pn".

$$\begin{array}{rcl} A\_3 & = & 3\sqrt{3} \times r^2 \\ A\_4 & = & 4 \times r^2 \\ A\_5 & = & 5\sqrt{5 - 2\sqrt{5}} \times r^2 \\ A\_6 & = & 2\sqrt{3} \times r^2 \\ A\_8 & = & 8\left(\sqrt{2} - 1\right) \times r^2 \\ A\_{10} & = & 2\sqrt{25 - 10\sqrt{5}} \times r^2 \\ A\_{12} & = & 12\left(2 - \sqrt{3}\right) \times r^2 \end{array}$$

(ii)

$$\begin{aligned} D\_3 &= 5.19 \dots \quad > \quad D\_4 = 4\\ &> \quad D\_5 = 3.63 \dots \\ &> \quad D\_6 = 3.46 \dots \\ &> \quad D\_8 = 3.31 \dots \\ &> \quad D\_{10} = 3.24 \dots \\ &> \quad D\_{12} = 3.21 \dots \end{aligned}$$

(c) Let O be the centre of the circle of radius r. Let A, B lie on the circle with =AOB " 30˝ .

Let M be the midpoint of AB – so that 4OAB is isosceles, with apex O and height h " OM.

Let A 1 lie on OA produced, and let B 1 lie on OB produced, such that OA<sup>1</sup> " OB<sup>1</sup> and A 1B 1 is tangent to the circle.

Then 4OAB " 4OA<sup>1</sup>B <sup>1</sup> with 4OAB contained in 4OA<sup>1</sup>B 1 , so the scale factor 1 <sup>h</sup> " 2 a 2 ´ ? 3 ą 1. Hence

$$4(2 - \sqrt{3}) \cdot a\_{12} = A\_{12} > a\_{12},$$

so D<sup>12</sup> ą d12.

206. The rearranged shape (shown in Figure 9) is an "almost rectangle", where OA forms an "almost height" and OA " r. Half of the 2n circular arcs such as AB form the upper "width", and the other half form the "lower width", so each of these "almost widths" is equal to half the perimeter of the circle – namely πr.

Figure 9: Rectification of a circle.

Hence the area of the rearranged shape is very close to

$$r \times \pi r = \pi r^2.$$

#### 207.

(a) Cut along a generator, open up and lay the surface flat to obtain: a 2πr by h rectangle and two circular discs of radius r. Hence the total surface area is

$$2\pi r^2 + 2\pi rh = 2\pi r(r+h).$$

(b) The lateral surface consists of n rectangles, each with dimensions s<sup>n</sup> by h (where s<sup>n</sup> is the edge length of the regular n-gon), and hence has area Pnh " 2Πnrh. Adding in the two end discs (each with area <sup>1</sup> 2 Pnr) then gives total surface area 2Πnrpr ` hq.

#### 208.


#### 209.

(a) Focus first on the sloping surface. If we cut along a "generator" (a straight line segment joining the apex to a point on the circumference of the base), the surface opens up and lays flat to form a sector of a circle of radius l. The outside arc of this sector has length 2πr, so the sector angle at the centre is equal to r l ¨ 2π, and hence its area is <sup>r</sup> l ¨ πl<sup>2</sup> " πrl.

Adding the area of the base gives the total surface area of the cone as

$$
\pi r(r+l) = \frac{1}{2} \cdot 2\pi r(r+l).
$$

(b) Let M be the midpoint of the edge BC and let l denote the 'slant height' AM. Then the area of the n sloping faces is equal to <sup>1</sup> 2 P<sup>n</sup> ¨ l, while the area of the base is equal to <sup>1</sup> 2 P<sup>n</sup> ¨ r.

Hence the surface area is precisely <sup>1</sup> <sup>2</sup>Pnpr ` lq.

### 210.


### 211.


#### 212.


In the end, any attempt to prove it seems to underline the need to use "proof by induction" – for example, on the number of edges of the inner polygon. This method is not formally treated until Chapter 6, but is needed here.

' Suppose the inner polygon P<sup>1</sup> " ABC has just n " 3 edges, and has perimeter p1.

Draw the line through A parallel to BC, and let it meet the (boundary of the) polygon P<sup>2</sup> at the points U and V . The triangle inequality (Problem 146(c)) guarantees that the length UV is less than or equal to the length of the compound path from U to V along the perimeter of the polygon P<sup>2</sup> (keeping on the opposite side of the line UV from B and C). So, if we cut off the part of P<sup>2</sup> on the side of the line UV opposite to B and C, we obtain a new outer convex polygon P3, which contains P1, and whose perimeter is no larger than that of P2.

Now draw the line through B parallel to AC, and let it meet the boundary of P<sup>3</sup> at points W, X. If we cut off the part of P<sup>3</sup> on the side of the line W X opposite to A and C, we obtain a new outer convex polygon P4, which contains P1, and whose perimeter is no larger than that of P3.

If we now do the same by drawing a line through C parallel to AB, and cut off the appropriate part of P4, we obtain a final outer convex polygon P5, which contains the polygon P1, and whose perimeter is no larger than that of P<sup>4</sup> – and hence no larger than that of the original outer polygon P2.

All three vertices A, B, C of P<sup>1</sup> now lie on the boundary of the outer polygon P5, so the triangle inequality guarantees that AB is no larger than the length of the compound path along the boundary of P<sup>5</sup> from A to B (staying on the opposite side of the line AB from C). Similarly BC is no larger than the length of the compound path along the boundary of P<sup>5</sup> from B to C; and CA is no larger than the length of the compound path along the boundary of P<sup>5</sup> from C to A.

Hence the perimeter p<sup>1</sup> of the triangle P<sup>1</sup> is no larger than the perimeter of the outer polygon P5, whose perimeter was no larger than the perimeter of the original outer polygon P2. Hence the result holds when the inner polygon is a triangle.

' Now suppose that the result has been proved when the inner polygon is a k-gon, for some k ě 3, and suppose we are presented with a pair of polygons P1, P<sup>2</sup> where the inner polygon P<sup>1</sup> " ABCD ¨ ¨ ¨ is a convex pk ` 1q-gon.

Draw the line m through C parallel to BD. Let this line meet the outer polygon P<sup>2</sup> at U and V . Cut off the part of P<sup>2</sup> on the opposite side of the the line UCV to B and D, leaving a new outer convex polygon P with perimeter no greater than that of P2. We prove that the perimeter of polygon P<sup>1</sup> is less than that of polygon P – and hence less than that of P2. Equivalently, we may assume that UCV is an edge of P2.

Translate the line m parallel to itself, from m to BD and beyond, until it reaches a position of final contact with the polygon P1, passing through the vertex X (and possibly a whole edge XY ) of the inner polygon P1. Let this final contact line parallel to m be m<sup>1</sup> .

Since P<sup>1</sup> is convex and k ě 3, we know that X is different from B and from D. As before, we may assume that m<sup>1</sup> is an edge of the outer polygon P2. Cut both P<sup>2</sup> and P<sup>1</sup> along the line CX to obtain two smaller configurations, each of which consists of an inner convex polygon inside an outer convex polygon, but in which


Then (by induction on the number of edges of the inner polygon) the perimeter of each inner polygon is no larger than the perimeter of the corresponding outer polygon; so for each inner polygon, the partial perimeter running from C to X (omitting the edge CX) is no larger than the partial perimeter of the corresponding outer polygon running from C to X (omitting the edge CX). So when we put the two parts back together again, we see that the perimeter of P<sup>1</sup> is no larger than the perimeter of P2.

Hence the result holds when P<sup>1</sup> is a triangle; and if the result holds whenever the inner polygon has k ě 3 edges, it also holds whenever the inner polygon has pk ` 1q edges.

It follows that the result holds whatever the number of edges of the inner polygon may be. QED

#### 213.

(a) Join P Q. Then the lines y " b and x " d meet at R to form the right angled triangle P QR.

Pythagoras' Theorem then implies that pd ´ aq <sup>2</sup> ` pe ´ bq <sup>2</sup> " P Q<sup>2</sup> .

(b) Join P Q. The points P " pa, b, cq and R " pd, e, cq lie in the plane z " c. If we work exclusively in this plane, then part (a) shows that

$$
\underline{PR}^2 = \left(d - a\right)^2 + \left(e - b\right)^2.
$$

QR " |f ´ c|, and 4P RQ has a right angle at R. Hence

$$
\underline{PQ}^2 = \underline{PR}^2 + \underline{RQ}^2 = \left(d - a\right)^2 + \left(e - b\right)^2 + \left(f - c\right)^2.
$$

#### 214.

$$\text{(a) } \text{ area}(\triangle ABC) = \frac{bc}{2}, \text{ area}(\triangle ACD) = \frac{cd}{2}, \text{ area}(\triangle ABD) = \frac{bd}{2}.$$


$$\text{area}(\triangle BCD)^2 = \left(\frac{bc}{2}\right)^2 + \left(\frac{cd}{2}\right)^2 + \left(\frac{bd}{2}\right)^2,$$

so that

$$\text{area}(\triangle ABC)^2 + \text{area}(\triangle ACD)^2 + \text{area}(\triangle ABD)^2 = \text{area}(\triangle BCD)^2.$$

#### 215.

(a) b and c are indeed lengths of arcs of great circles on the unit sphere: that is, arcs of circles of radius 1 (centred at the centre of the sphere). However, back

in Chapter 1 we used the 'length' of such circular arcs to define the angle (in radians) subtended by the arc at the centre. So b and c are also angles (in radians).


formula might involve "adding sin b and adding sin c".)

(ii) Suppose that =B " π 2 . Since we can imagine AB along the equator, and since there is a right angle at A, it follows that AC and BC both lie along circles of longitude, and so meet at the North pole. Hence C will be at the North pole, so a " b " π 2 .

The inputs to any spherical version of Pythagoras' Theorem are then b " π 2 , and c. And c is not constrained, so every possible input value of c must lead to the same output a " π 2 . This tends to suggest that the formula involves some multiple of a product combining "cos b" with some function of c. And since the inputs "b" and "c" must appear symmetrically, we might reasonably expect some multiple of "cos b ¨ cos c".

#### 216.


$$\begin{aligned} \tan^2 b + \tan^2 c &= \sec^2 b + \sec^2 c - 2\sec b \cdot \sec c \cdot \cos(\angle B'OC')\\ &= \sec^2 b + \sec^2 c - 2\sec b \cdot \sec c \cdot \cos a. \end{aligned}$$

Hence cos a " cos b ¨ cos c. QED

217. Construct the plane tangent to the sphere at A. Extend OB to meet this plane at B 1 , and extend OC to meet the plane at C 1 .

4OAB<sup>1</sup> has a right angle at A with =AOB<sup>1</sup> " =AOB " c. Hence AB<sup>1</sup> " tan c. Similarly AC<sup>1</sup> " tan b. Hence B 1C <sup>1</sup><sup>2</sup> " tan<sup>2</sup> b ` tan<sup>2</sup> c ´ 2 ¨ tan b ¨ tan c ¨ cos A. In 4OAB<sup>1</sup> we see that OB<sup>1</sup> " sec c. Similarly OC<sup>1</sup> " sec b. We can now apply the Cosine Rule to 4OB<sup>1</sup>C 1 to obtain:

$$\begin{aligned} \tan^2 b + \tan^2 c - 2 \cdot \tan b \cdot \tan c \cdot \cos A \\ &= \sec^2 b + \sec^2 c - 2 \sec b \cdot \sec c \cdot \cos(\angle B'OC') \\ &= \sec^2 b + \sec^2 c - 2 \sec b \cdot \sec c \cdot \cos a . \end{aligned}$$

Hence cos a " cos b ¨ cos c ` sin b ¨ sin c ¨ cos A. QED

218. We show that sin <sup>b</sup> sin <sup>a</sup> " sin B (where B denotes the angle =ABC at the vertex B).

Construct the plane T which is tangent to the sphere at B. Let O be the centre of the sphere; let OA produced meet the plane T at A 2 , and let OC produced meet the plane T at C 2 .

Imagine BA positioned along the equator; then BA<sup>2</sup> is horizontal; AC lies on a circle of longitude, so A <sup>2</sup>C 2 is vertical. Hence =C <sup>2</sup>BA<sup>2</sup> " =B, and =BA<sup>2</sup>C 2 is a right angle; so sin B " A2C<sup>2</sup> BC<sup>2</sup> .

4OA<sup>2</sup>C 2 is right angled at A 2 ; and =A <sup>2</sup>OC<sup>2</sup> " b; so sin b " A2C<sup>2</sup> OC<sup>2</sup> . 4OBC<sup>2</sup> is right angled at B; and =BOC<sup>2</sup> " a; so sin a " BC<sup>2</sup> OC<sup>2</sup> .

Hence sin <sup>b</sup> sin <sup>a</sup> " sin B depends only on the angle at B, so sin <sup>b</sup> sin <sup>a</sup> " sin b 1 sin <sup>a</sup><sup>1</sup> " sin B.

219. We show that

$$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B}.$$

Position the triangle (or rather "rotate the sphere") so that AB runs along the equator, with AC leading into the northern hemisphere.

(i) If =A is a right angle, then

$$\frac{\sin b}{\sin a} = \sin B = \frac{\sin B}{\sin A}$$

by Problem 218. Hence

$$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B}.$$

The same is true if =B is a right angle.

(ii) If =A and =B are both less than a right angle, one can draw the circle of longitude from C to some point X on AB. One can then apply Problem 218 to the two triangles 4CXA and 4CXB (each with a right angle at X). Let x denote the length of the CX. Then sin <sup>x</sup> sin <sup>b</sup> " sin A, and sin <sup>x</sup> sin <sup>a</sup> " sin B.

Hence sin b ¨ sin A " sin x " sin a ¨ sin B, so

$$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B}.$$

(iii) If =A (say) is greater than a right angle, the circle of longitude from C meets the line BA extended beyond A at a point X (say). If we let CX " x, then one can argue similarly using the triangles 4CXA and 4CXB to get sin <sup>x</sup> sin <sup>a</sup> " sin B, and sin <sup>x</sup> sin <sup>b</sup> " sin A, whence

$$\frac{\sin a}{\sin A} = \frac{\sin b}{\sin B}.$$

220. Let X " px, yq be an arbitrary point of the locus.

(i) Then the distance from X to m is equal to y, so setting this equal to XF gives the equation:

$$y^2 = x^2 + \left(y - 2a\right)^2,$$

or

$$x^2 = 4a(y - a).$$

(ii) If we change coordinates and choose the line y " a as a new x-axis, then the equation becomes x <sup>2</sup> " 4aY . The curve is then tangent to the new x-axis at the (new) origin, and is symmetrical about the y-axis.

#### 221.

(a) Choose the line AB as x-axis, and the perpendicular bisector of AB as the y-axis. Then A " p´3, 0q and B " p3, 0q. The point X " px, yq is a point of the unknown locus precisely when

$$\left(\left(x+3\right)^2+y^2\right)-\underline{X}\underline{A}^2=\left(2\cdot\underline{X}\underline{B}\right)^2=2^2\left(\left(x-3\right)^2+y^2\right)^2$$

that is, when

$$\left(x-5\right)^{2}+y^{2}=4^{2}\dots$$

This is the equation of a circle with centre p5, 0q and radius r " 4.

(b) Choose the line AB as x-axis, and the perpendicular bisector of AB as the y-axis.

If f " 1, the locus is the perpendicular bisector of AB.

We may assume that f ą 1 (since if f ă 1, then BX : AX " f ´1 : 1 and f ´<sup>1</sup> ą 1, so we may simply swap the labelling of A and B).

Now A " p´b, 0) and B " pb, 0q, and the point X " px, yq is a point of the unknown locus precisely when

$$\left(\left(x+b\right)^2+y^2\right)-\underline{X}\underline{A}^2=\left(f\cdot\underline{X}\underline{B}\right)^2=f^2\left[\left(x-b\right)^2+y^2\right]^{\frac{1}{2}}$$

that is, when

$$x^2(f^2 - 1) - 2bx(f^2 + 1) + y^2(f^2 - 1) + b^2(f^2 - 1) = 0$$

$$\left(x - \frac{b(f^2 + 1)}{f^2 - 1}\right)^2 + y^2 = \left(\frac{2fb}{f^2 - 1}\right)^2 \dots$$

This is the equation of a circle with centre ´ bpf <sup>2</sup>`1q f2´1 , 0 and radius r " 2f b f2´1 .

### 222.

(a) Choose the line AB as x-axis, and the perpendicular bisector of AB as the y-axis.

Then A " p´c, 0q and B " pc, 0q. The point X " px, yq is a point of the unknown locus precisely when

$$2a = \underline{AX} + \underline{BX} = \sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2},$$

that is, when

$$\begin{array}{rcl} 2a - \sqrt{(x+c)^2 + y^2} &=& \sqrt{(x-c)^2 + y^2} \\ \vdots \text{.} \ 4a^2 - 4a\sqrt{(x+c)^2 + y^2} + \left[(x+c)^2 + y^2\right] &=& \left[(x-c)^2 + y^2\right] \\ \vdots \text{.} & \quad \begin{array}{rcl} a^2 + cx &=& a\sqrt{(x+c)^2 + y^2} \\ (a^2 - c^2)x^2 + a^2y^2 &=& a^2(a^2 - c^2) \end{array} \\ \end{array}$$

Setting <sup>c</sup> <sup>a</sup> " e then yields the equation for the locus in the form:

$$\frac{x^2}{a^2} + \frac{y^2}{a^2(1-e^2)} = 1.$$

Note: In the derivation of the equation we squared both sides (twice). This may introduce spurious solutions. So we should check that every solution px, yq of the final equation satisfies the original condition.

(b) The real number e ă 1 is given, so we may set the distance from F to m be a e ` 1 ´ e 2 ˘ . Choose the line through F and perpendicular to m as x-axis. To start with, we choose the line m as y-axis and adjust later if necessary. ` ˘ ˘

Hence <sup>F</sup> has coordinates ` a e 1 ´ e 2 , 0 , and the point X " px, yq is a point of the unknown locus precisely when

$$\left(x - \frac{a}{e}\left(1 - e^2\right)\right)^2 + y^2 = (ex)^2,$$

which can be rearranged as

$$
\left(1 - e^2\right)x^2 - 2\frac{a}{e}(1 - e^2)x + \left(\frac{a}{e}\right)^2(1 - e^2)^2 + y^2 = 0,
$$

and further as

$$\begin{aligned} \left(1 - e^2\right) \left[x^2 - 2\frac{a}{e}x + \left(\frac{a}{e}\right)^2\right] + y^2 &=& \left(\frac{a}{e}\right)^2 \left(1 - e^2\right) - \left(\frac{a}{e}\right)^2 \left(1 - e^2\right)^2\\ &=& \left(\frac{a}{e}\right)^2 \left(e^2 - e^4\right) \\ &=& a^2 \left(1 - e^2\right) \\ \therefore \left(x - \frac{a}{e}\right)^2 + \frac{y^2}{1 - e^2} &=& a^2. \end{aligned}$$

If we now move the y-axis to the line x " a e the equation takes the simpler form:

$$\frac{x^2}{a^2} + \frac{y^2}{a^2(1-e^2)} = 1.$$

(c) This was done in the derivations in the solutions to parts (a) and (b).

#### 223.

(a) The triangle inequality shows that, if AX ą BX, then AB ` BX ě AX; hence the locus is non-empty only when a ď c. If a " c, then X must lie on the line AB, but not between A and B, so the locus consists of the two half-lines on AB outside AB. Hence we may assume that a ă c.

Choose the line AB as x-axis, and the perpendicular bisector of AB as y-axis. Then A " p´c, 0q and B " pc, 0q. The point X " px, yq is a point of the unknown locus precisely when

$$2a = \left| \underline{AX} - \underline{BX} \right| = \left| \sqrt{(x+c)^2 + y^2} - \sqrt{(x-c)^2 + y^2} \right|.$$

If AX ą BX, we can drop the modulus signs and calculate as in Problem 222.

$$\begin{array}{rcl}2a+\sqrt{(x-c)^2+y^2}&=&\sqrt{(x+c)^2+y^2}.\\3.&\left(a^2+4a\sqrt{(x-c)^2+y^2}+\left(x-c\right)^2+y^2\right)&=&\left(x+c\right)^2+y^2\\\therefore&a^2-cx&=&-a\sqrt{(x-c)^2+y^2}\\\therefore&\left(c^2-a^2\right)x^2-a^2y^2&=&a^2\left(c^2-a^2\right)\end{array}$$

Setting <sup>c</sup> <sup>a</sup> " e (ą 1), then yields the equation for the locus in the form:

$$\frac{x^2}{a^2} - \frac{y^2}{a^2(e^2 - 1)} = 1.$$

Note: In the derivation of the equation we squared both sides (twice). This may introduce spurious solutions. So we should check that every solution px, yq of the final equation satisfies the original condition.In fact, the squaring process introducesadditional solutions in the form of a second branch of the locus, corresponding precisely to points X where AX ă BX.

(b) The real number e ą 1 is given, so we may set the distance from F to m be a e pe <sup>2</sup> ´ 1q. Choose the line through F and perpendicular to m as x-axis. To start with, we choose the line m as y-axis and adjust later if necessary. ˘

Hence <sup>F</sup> has coordinates ` a e pe <sup>2</sup> ´ 1q, 0 , and the point X " px, yq is a point of the unknown locus precisely when

$$\begin{aligned} \left(x - \frac{a}{e}(e^2 - 1)\right)^2 + y^2 &=& \left(ex\right)^2\\ \therefore \quad \left(e^2 - 1\right)x^2 + \frac{2a}{e}(e^2 - 1)x - y^2 &=& \left(\frac{a}{e}\right)^2(e^2 - 1)^2\\ \therefore \quad \left(e^2 - 1\right)\left[x^2 + \frac{2a}{e}x + \left(\frac{a}{e}\right)^2\right] - y^2 &=& \left(\frac{a}{e}\right)^2(e^2 - 1)^2 + \left(\frac{a}{e}\right)^2(e^2 - 1)^2\\ &=& \left(\frac{a}{e}\right)^2(e^4 - e^2)\\ \therefore \quad \left(e^2 - 1\right)\left[x^2 + \frac{2a}{e}x + \left(\frac{a}{e}\right)^2\right] - y^2 &=& a^2(e^2 - 1)\\ \therefore \quad \left(x + \frac{a}{e}\right)^2 - \frac{y^2}{e^2 - 1} &=& a^2 \end{aligned}$$

If we now move the y-axis to the line x " ´ a e the equation takes the simpler form:

$$\frac{x^2}{a^2} - \frac{y^2}{a^2(e^2 - 1)} = 1.$$

(c) This was done in derivations in the solutions to parts (a) and (b).

#### 224.

(a) When z " k is a constant, the equation reduces to that of a circle

$$x^2 + y^2 = (rk)^2$$

in the plane z " k. When the cutting plane is the xy-plane "z " 0", the circle has radius 0, so is a single point.

	- (ii) If the cutting plane through the apex is less steep than a generator, then it cuts the cone only at the apex.

If the cutting plane through the apex is parallel to a generator, then it cuts the cone in a generator – a single line (the next paragraph indicates that this line may be better thought of as a pair of "coincident" lines).

What happens when the cutting plane through the apex is steeper than a generator may not be intuitively clear. One way to make sense of this is to treat the cross-section as the set of solutions of two simultaneous equations – one for the cone, and the other for the plane (say y " nz, with n ă r). This leads to the equation

$$x^2 = (r^2 - n^2)z^2, \quad y = nz,$$

with solution set

$$x = \pm z\sqrt{r^2 - n^2}, \quad xy = nz$$

which specifies a pair of lines crossing at the apex.

A slightly easier way to visualize the cross-section in this case is to let the apex of the double cone be A, and to let X be any other point of the cross-section. Then the line AX is a generator of the cone, so lies on the cone's surface. But A and X also lie in the cutting plane – so the whole line AX must lie in the cutting plane. Hence the cross-section contains the whole line AX.

	- (ii) Thus we assume that the cutting plane does not pass through the apex, and may assume that it cuts the bottom half of the cone. If V is the point nearest the apex where the cutting plane meets the cone, then the cross-section curve starts at V and becomes wider as we go down the cone. Because the plane is parallel to a generator, the plane never cuts the "other side" of the bottom half of the cone, so the cross section never "closes up" – but continues to open up wider and wider as we go further and further down the bottom half of the cone.

Let S be the sphere, which is inscribed in the cone above the cutting plane, and which is tangent to the cutting plane at F. Let C be the circle of contact between S and the cone. Let A be the apex of the cone, and let the apex angle of the cone be equal to 2θ. Let X be an arbitrary point of the cross-section.

To illustrate the general method, consider first the special case where X " V is the "highest" point of the cross-section. The line segment V A is tangent to the sphere S, so crosses the circle C at some point M. Now V F lies in the cutting plane, so is also tangent to the sphere S at the point F. Any two tangents to a sphere from the same exterior point are equal, so it follows that V M " V F. Moreover, V M is exactly equal to the distance from V to the line m (since


Now let X be an arbitrary point of the cross-sectional curve, and use a similar argument. First the line XA is always a generator of the cone, so is tangent to the sphere S, and crosses the circle C at some point Y . Moreover, XF is also tangent to the sphere. Hence XY " XF. It remains to see that XY is equal to the distance XY ˚ from X to the closest point Y ˚ on the line m – since


Hence the cross-sectional curve is a parabola with focus F and directrix m.

	- (ii) The derivation is very similar to that in part (c), and we leave the reader to reconstruct it.

An alternative approach is to insert a second sphere S <sup>1</sup> below the cutting plane, and inflate it until it makes contact with the cone along a circle C 1 while at the same time touching the cutting plane at a point F 1 . If X is an arbitrary point of the cross-sectional curve, then XA is tangent to both spheres, so meets the circle C at some point Y and meets the circle C 1 at some point Y 1 . Then Y , X, Y 1 are collinear. Moreover, XY " XF (since both are tangents to the sphere S from the point X), and XY <sup>1</sup> " XF<sup>1</sup> (since both are tangents to the sphere S 1 from the point X), so

$$\underline{XF} + \underline{XF'} = \underline{XY} + \underline{XY'} = \underline{YY'}$$

But Y Y <sup>1</sup> is equal to the slant height of the cone between the two fixed circles C and C 1 , and so is equal to a constant k. Hence, the focus-focus specification in Problem 222 shows that the cross-section is an ellipse.

(e)(i) If the cutting plane is steeper than a generator, the cross-section cuts both the bottom half and the top half of the cone to give two separate parts of the cross-section. Neither part "closes up", so each part opens up more and more widely.

If V is the highest point of the cross-section on the lower half of the cone, and W is the lowest point of the cross-section on the upper half of the cone, then it seems clear that the cross-section is symmetrical under reflection in the line V W. But it is quite unclear that the two halves of the cross-section are exactly congruent (though again it was known to the ancient Greeks).

(ii) The formal derivation is very similar to that in part (c), and we leave the reader to reconstruct it.

An alternative approach is to copy the alternative in (d), and to insert a second sphere S 1 in the upper part of the cone, on the same side of the cutting plane as the apex, inflate it until it makes contact with the cone along a circle C <sup>1</sup> while at the same time touching the cutting plane at a point F 1 . If X is an arbitrary point of the cross-sectional curve, then XA is tangent to both spheres, so meets the circle C at some point Y and meets the circle C 1 at some point Y 1 . Then Y , X, Y 1 are collinear. Moreover, X, F, and F 1 all lie on the cutting plane. Now XY " XF (since both are tangents to the sphere S from the point X), and XY <sup>1</sup> " XF<sup>1</sup> (since both are tangents to the sphere S 1 from the point X). If X is on the upper half of the cone, then

$$
\underline{XF} - \underline{XF'} = \underline{XY} - \underline{XY'} = \underline{YY'}.
$$

But Y Y <sup>1</sup> is equal to the slant height of the cone between the two fixed circles C and C 1 , and so is equal to a constant k. Hence, the focus-focus specification in Problem 223 shows that the cross-section is a hyperbola.

#### 225.


#### 226.

(c) (i) If you look carefully at the diagram shown here you should be able to see not only the upper and lower 3D-cubes, but also the four other 3D-cubes formed by joining each 2D-cube in the upper 3D-cube to the corresponding 2D-cube in the lower 3D-cube.

Note: Once we have the 3D-cube expressed in coordinates, we can specify precisely which planes produce which cross-sections in Problem 38. The plane x`y `z " 1 passes through the three neighbours of p0, 0, 0q and creates an equilateral triangular cross-section. Any plane of the form z " c (where c is a constant between 0 and 1) produces a square cross-section. And the plane x ` y ` z " 3 2 is the perpendicular bisector of the line joining (0, 0, 0q to p1, 1, 1q, and creates a regular hexagon as cross-section.

#### 227.

(a) View the coordinates as px, y, zq. Start at the origin p0, 0, 0q and travel round 3 edges of the lower 2D-cube "z " 0" to the point p0, 1, 0q. Copy this path of length 3 on the upper 2D-cube "z " 1" (from p0, 0, 1q to p0, 1, 1q. Then join p0, 0, 0q to p0, 0, 1q and join p0, 1, 0q to p0, 1, 1). The result

p0, 0, 0q, p1, 0, 0q, (1, 1, 0q, p0, 1, 0q, p0, 1, 1q, p1, 1, 1q, p1, 0, 1q, p0, 0, 1q (and back to p0, 0, 0q)

has the property that exactly one coordinate changes when we move from each vertex to the next. This is an example of a Gray code of length 3.

Note: How many such paths/circuits are there in the 3D-cube? We can certainly count those of the kind described here. Each such circuit has a "direction": the 12 edges of the 3D-cube lie in one of 3 "directions", and each such circuit contains all four edges in one of these 3 directions. Moreover this set of four edges can be completed to a circuit in exactly 2 ways. So there are 3 ˆ 2 such circuits. In 3D this accounts for all such circuits. But in higher dimensions the numbers begin to explode (in the 4D-cube there are 1344 such circuits).

(b) View the coordinates as pw, x, y, zq. Start at the origin p0, 0, 0, 0q and travel round the 8 vertices of the lower 3D-cube "z " 0" to the point p0, 0, 1, 0q. Then copy this path on the upper 3D-cube "z " 1" from p0, 0, 0, 1q to p0, 0, 1, 1q. Finally join p0, 0, 0, 0q to p0, 0, 0, 1q and join p0, 0, 1, 0q to p0, 0, 1, 1q. The result

p0, 0, 0.0q, p1, 0, 0, 0q, p1, 1, 0, 0q, p0, 1, 0, 0q, p0, 1, 1, 0q, p1, 1, 1, 0q, p1, 0, 1, 0q, p0, 0, 1, 0q p0, 0, 1, 1q, p1, 0, 1, 1q, p1, 1, 1, 1q, p0, 1, 1, 1q, p0, 1, 0, 1q, p1, 1, 0, 1q, p1, 0, 0, 1q, p0, 0, 0, 1q (and back to p0, 0, 0, 0q)

has the property that exactly one coordinate changes when we move from each vertex to the next. This is an example of a Gray code of length 4.

Note: The general construction in dimension n ` 1 depends on the previous construction in dimension n, so makes use of mathematical induction (see Problem 262 in Chapter 6).

# VI. Infinity: recursion, induction, infinite descent

Mathematical induction – i.e. proof by recurrence – is . . . imposed on us, because it is . . . the affirmation of a property of the mind itself. Henri Poincar´e (1854–1912)

Allez en avant, et la foi vous viendra. (Press on ahead, and understanding will follow.) Jean le Rond d'Alembert (1717–1783)

Mathematics has been called "the science of infinity". However, infinity is a slippery notion, and many of the techniques which are characteristic of modern mathematics were developed precisely to tame this slipperiness. This chapter introduces some of the relevant ideas and techniques.

There are aspects of the story of infinity in mathematics which we shall not address. For example, astronomers who study the night sky and the movements of the planets and stars soon note its immensity, and its apparently 'fractal' nature – where increasing the detail or magnification reveals more or less the same level of complexity on different scales. And it is hard then to avoid the question of whether the stellar universe is finite or infinite.

In the mental universe of mathematics, once counting, and the process of halving, become routinely iterative processes, questions about infinity and infinitesimals are almost inevitable. However, mathematics recognises the conceptual gulf between the finite and the infinite (or infinitesimal), and rejects the lazy use of "infinity" as a metaphor for what is simply "very large". Large finite numbers are still numbers; and long finite sums are conceptually quite different from sums that "go on for ever". Indeed, in the third century BC, Archimedes (c. 287–212 BC) wrote a small booklet called The Sand Reckoner, dedicated to King Gelon, in which he introduced the arithmetic of powers (even though the ancient Greeks had no convenient notation for writing such numbers), in order to demonstrate that – contrary to what some people had claimed – the number of grains of sand in the known universe must be finite (he derived an upper bound of approximately 8 ˆ 1063).

The influence wielded by ideas of infinity on mathematics has been profound, even if we now view some of these ideas as flights of fancy –


In contrast, we focus here on the delights of the mathematics, and in particular on how an initial doorway into "ideas of infinity" can be forged from careful reasoning with finite entities. Readers who would like to explore what we pass over in silence could do worse than to start with the essay on "infinity" in the MacTutor History of Mathematics archive:

http://www-history.mcs.st-and.ac.uk/HistTopics/Infinity.html.

The simplest infinite processes begin with recursion – a process where we repeat exactly the same operation over and over again (in principle, continuing for ever). For example, we may start with 0, and repeat the operation "add 1", to generate the sequence:

$$\left|0, 1, 2, 3, 4, 5, 6, 7, \dots\right|$$

Or we may start with 2<sup>0</sup> " 1 and repeat the operation "multiply by 2", to generate:

$$1, 2, 4, 8, 16, 32, 64, 128, \dots$$

Or we may start with 1.000000 ¨ ¨ ¨ , and repeat the steps involved in "dividing by 7" to generate the infinite decimal for <sup>1</sup> 7 :

$$\frac{1}{7} = 0.1428571428571428571\dots \dots$$

We can then vary this idea of "recursion" by allowing each operation to be "essentially" (rather than exactly) the same, as when we define triangular numbers by "adding n" at the n th stage to generate the sequence:

$$0, 1, 3, 6, 10, 15, 21, 28, \dots$$

In other words, the sequence of triangular numbers is defined by a recurrence relation:

T<sup>0</sup> " 0; and when n ě 1, T<sup>n</sup> " Tn´<sup>1</sup> ` n.

We can vary this idea further by allowing more complicated recurrence relations – such as that which defines the Fibonacci numbers:

F<sup>0</sup> " 0, F<sup>1</sup> " 1; and when n ě 1, Fn`<sup>1</sup> " F<sup>n</sup> ` Fn´1.

All of these "images of infinity" hark back to the familiar counting numbers.


The intuition that this process is, in principle, never-ending (so is never actually completed), yet somehow manages to count all positive integers, is what Poincar´e called a "property of the mind itself": that is, the idea that we can define an infinite sequence, or process, or chain of deductions (involving digits, or numbers, or objects, or statements, or truths) by


This idea is what lies behind "proof by mathematical induction", where we prove that some assertion Ppnq holds for all n ě 1 – so demonstrating infinitely many separate statements at a single blow. The validity of this method of proof depends on a fundamental property of the positive integers, or of the counting sequence

$${"{1, 2, 3, 4, 5, \dots}},$$

namely:

The Principle of Mathematical Induction: If a subset S of the positive integers

• contains the integer "1",

and has the property that

• whenever an integer k is in the set S, then the next integer k `1 is always in S too,

then S contains all positive integers.

### 6.1. Proof by mathematical induction I

When students first meet "proof by induction", it is often explained in a way that leaves them feeling distinctly uneasy, because it appears to break the fundamental taboo:

never assume what you are trying to prove.

This tends to leave the beginner in the position described by d'Alembert's quote at the start of the chapter: they may "press on" in the hope that "understanding will follow", but a doubt often remains. So we encourage readers who have already met proof by induction to take a step back, and to try to understand afresh how it really works. This may require you to study the solutions (Section 6.10), and to be prepared to learn to write out proofs more carefully than, and rather differently from, what you are used to.

When we wish to prove a general result which involves a parameter n, where n can be any positive integer, we are really trying to prove infinitely many results all at once. If we tried to prove such a collection of results in turn, "one at a time", not only would we never finish, we would scarcely get started (since completing the first million, or billion, cases leaves just as much undone as at the start). So our only hope is:


Once the result to be proved has been formulated in this way, we can

	- as soon as we know that "Ppkq is true, for some (particular, but unspecified) k ě 1",
	- we can prove in a uniform way that the next result Ppk ` 1q is then automatically true.

Having implemented the first of the two induction steps, we know that Pp1q is true.

The second bullet point above then comes into play and assures us that (since we know that Pp1q is true), Pp2q must be true.

And once we know that Pp2q is true, the second bullet point assures us that Pp3q is also true.

And once we know that Pp3q is true, the second bullet point assures us that Pp4q is also true.

And so on for ever.

We can then conclude that the whole sequence of infinitely many statements are all true – namely that:

"every statement Ppnq is true",

or that

"Ppn) is true, for all n ě 1."

In other words, if we define S to be the set of positive integers n for which the statement Ppnq is true, then S contains the element "1", and whenever k is in S, so is k ` 1; hence, by the Principle of Mathematical Induction we know that S contains all positive integers.

At this stage we should acknowledge an important didactical (rather than mathematical) ploy in our recommended layout here. It is important to underline the distinction between

(i) the individual statements Ppnq which are the separate ingredients in the overall statement to be proved, namely:

"Ppnq is true, for all n ě 1",

where infinitely many individual statements have been compressed into a single compound statement, and

(ii) the induction step, where we


To underline this distinction we consistently use a different "dummy variable" (namely "k") in the latter case. This distinction is a psychological ploy rather than a logical necessity. However, we recommend that readers should imitate this distinction (at least initially).

### 6.2. 'Mathematical induction' and 'scientific induction'

The idea of a "list that goes on for ever" arose in the sequence of powers of 4 back in Problem 16, where we asked

Do the two sequences arising from successive powers of 4:

• the leading digits:

$$4, 2, 6, 2, 1, 4, 2, 6, 2, 1, 4, \dots, 1$$

and

• the units digits:

4, 6, 4, 6, 4, 6, 4, 6, . . . ,

really "repeat for ever" as they seem to?

This example illustrates the most basic misconception that sometimes arises concerning mathematical induction – namely to confuse it with the kind of pattern spotting that is often called 'scientific induction'.

In science (as in everyday life), we routinely infer that something that is observed to occur repeatedly, apparently without exception (such as the sun rising every morning; or the Pole star seeming to be fixed in the night sky) may be taken as a "fact". This kind of "scientific induction" makes perfect sense when trying to understand the world around us – even though the inference is not warranted in a strictly logical sense.

Proof by mathematical induction is quite different. Admittedly, it often requires intelligent guesswork at a preliminary stage to make a guess that allows us to formulate precisely what it is that we should be trying to prove. But this initial guess is separate from the proof, which remains a strictly deductive construction. For example,

the fact that "1", "1 ` 3", "1 ` 3 ` 5", "1 ` 3 ` 5 ` 7", etc. all appear to be successive squares gives us an idea that perhaps the identity

Ppnq: 1 ` 3 ` 5 ` ¨ ¨ ¨ ` p2n ´ 1q " n 2

is true, for all n ě 1.

This guess is needed before we can start the proof by mathematical induction. But the process of guessing is not part of the proof. And until we construct the "proof by induction" (Problem 231), we cannot be sure that our guess is correct.

The danger of confusing 'mathematical induction' and 'scientific induction' may be highlighted to some extent if we consider the proof in Problem 76 above that "we can always construct ever larger prime numbers", and contrast it with an observation (see Problem 228 below) that is often used in its place – even by authors who should know better.

In Problem 76 we gave a strict construction by mathematical induction:


This construction was very carefully worded, so as to be logically correct.

In contrast, one often finds lessons, books and websites that present the essential idea in the above proof, but "simplify" it into a form that encourages anti-mathematical "pattern-spotting" which is all-too-easily misconstrued. For example, some books present the sequence

$$(2;) \ 2 + 1 = \mathbf{3}; \ 2 \times 3 + 1 = \mathbf{7}; \ 2 \times 3 \times 5 + 1 = \mathbf{31}; \ 2 \times 3 \times 5 \times 7 + 1 = \mathbf{211}; \dots$$

as a way of generating more and more primes.

### Problem 228


We have already met two excellent historical examples of the dangers of plausible pattern-spotting in connection with Problem 118. There you proved that:

"if 2<sup>n</sup> ´ 1 is prime, then n must be prime."

You then showed that 2<sup>2</sup> ´ 1, 2<sup>3</sup> ´ 1, 2<sup>5</sup> ´ 1, 2<sup>7</sup> ´ 1 are all prime, but that 2 <sup>11</sup> ´ 1 " 2047 " 23 ˆ 89 is not. This underlines the need to avoid jumping to (possibly false) conclusions, and never to confuse a statement with its converse.

In the same problem you showed that:

"if a <sup>b</sup> ` 1 is to be prime and a ‰ 1, then a must be even, and b must be a power of 2."

You then looked at the simplest family of such candidate primes namely the sequence of Fermat numbers fn:

$$f\_0 = 2^1 + 1 = 3,\\ f\_1 = 2^2 + 1 = 5,\\ f\_2 = 2^4 + 1 = 17,\\ f\_3 = 2^8 + 1 = 257,\\ f\_4 = 2^{16} + 1.$$

It turned out that, although f0, f1, f2, f3, f<sup>4</sup> are all prime, and although Fermat (1601–1665) claimed (in a letter to Marin Mersenne (1588–1648))

that all Fermat numbers f<sup>n</sup> are prime, we have yet to discover a sixth Fermat prime!

There are times when a mathematician may appear to guess a general result on the basis of what looks like very modest evidence (such as noticing that it appears to be true in a few small cases). Such "informed guesses" are almost always rooted in other experience, or in some unnoticed feature of the particular situation, or in some striking analogy: that is, an apparent pattern strikes a chord for reasons that go way beyond the mere numbers. However those with less experience need to realise that apparent patterns or trends are often no more than numerical accidents.

Pell's equation (John Pell (1611–1685)) provides some dramatic examples.

• If we evaluate the expression "n <sup>2</sup> ` 1" for n " 1, 2, 3, . . . , we may notice that the outputs 2, 5, 10, 17, 26, . . . never give a perfect square. And this is to be expected, since the next square after n 2 is

$$(n+1)^2 = n^2 + 2n + 1,$$

and this is always greater than n <sup>2</sup> ` 1.

• However, if we evaluate "991n <sup>2</sup>`1" for n " 1, 2, 3, . . . , we may observe that the outputs never seem to include a perfect square. But this time there is no obvious reason why this should be so – so we may anticipate that this is simply an accident of "small" numbers. And we should hesitate to change our view, even though this accident goes on happening for a very, very, very long time: the smallest value of n for which 991n <sup>2</sup> `1 gives rise to a perfect square is apparently

$$n = 12\,055\,735\,790\,331\,359\,447\,442\,538\,767\,.$$

### 6.3. Proof by mathematical induction II

Even where there is no confusion between mathematical induction and 'scientific induction', students often fail to appreciate the essence of "proof by induction". Before illustrating this, we repeat the basic structure of such a proof.

A mathematical result, or formula, often involves a parameter n, where n can be any positive integer. In such cases, what is presented as a single result, or formula, is a short way of writing an infinite family of results. The proof that such a result is correct therefore requires us to prove infinitely many results at once. We repeat that our only hope of achieving such a mind-boggling feat is

• to formulate the stated result for each value of n separately: that is, as a statement Ppnq which depends on n;

	- as soon as we know that Ppkq is true, for some (unknown) k ě 1,
	- we can prove that the next result Ppk ` 1q is then automatically true.

We can then conclude that

```
"every statement Ppnq is true",
```
or that

```
"Ppnq is true, for all n ě 1".
```
Problem 229 Prove the statement:

"52n`<sup>2</sup> ´ 24n ´ 25 is divisible by 576, for all n ě 1". 4

When trying to construct proofs in private, one is free to write anything that helps as 'rough work'. But the intended thrust of Problem 229 is two-fold:


The central lesson in completing the "induction step" is to recognize that:

to prove that Ppk ` 1q is true, one has to start by looking at what Ppk ` 1q says.

In Problem 229 Ppk ` 1q says that

"5<sup>2</sup>pk`1q`<sup>2</sup> ´ 24pk ` 1q ´ 25 is divisible by 576".

Hence one has to start the induction step with the relevant expression

$$5^{2(k+1)+2} - 24(k+1) - 25,$$

and look for some way to rearrange this into a form where one can use Ppkq (which we assume is already known to be true, and so are free to use).

It is in general a false strategy to work the other way round – by "starting with Ppkq, and then fiddling with it to try to get Ppk ` 1q". (This strategy can be made to work in the simplest cases; but it does not work in general, and so is a bad habit to get into.) So the induction step should always start with the hypothesis of Ppk ` 1q.

The next problem invites you to prove the formula for the sum of the angles in any polygon. The result is well-known; yet we are fairly sure that the reader will never have seen a correct proof. So the intention here is for you to recognise the basic character of the result, to identify the flaws in what you may until now have accepted as a proof, and to try to find some way of producing a general proof.

Problem 230 Prove by induction the statement:

"for each n ě 3, the angles of any n-gon in the plane have sum equal to pn ´ 2qπ radians." 4

The formulation certainly involves a parameter n ě 3; so you clearly need to begin by formulating the statement Ppnq. For the proof to have a chance of working, finding the right formulation involves a modest twist! So if you get stuck, it may be worth checking the first couple of lines of the solution. No matter how Ppn) is formulated, you should certainly know how to prove the statement Pp3q (essentially the formula for the sum of the angles in a triangle). But it is far from obvious how to prove the "induction step":

"if we know that Ppkq is true for some particular k ě 1, then Ppk ` 1q must also be true".

When tackling the induction step, we certainly cannot start with Ppkq! The statement Ppk`1q says something about polygons with k`1 sides: and there is no way to obtain a typical pk ` 1q-gon by fiddling with some statement about polygons with k sides. (If you start with a k-gon, you can of course draw a triangle on one side to get a pk ` 1q-gon; but this is a very special construction, and there is no easy way of knowing whether all pk`1q-gons can be obtained in this way from some k-gon.) The whole thrust of mathematical induction is that we must always start the induction step by thinking about the hypothesis of Ppk ` 1q – that is in this case, by considering an arbitrary pk ` 1q-gon and then finding some guaranteed way of "reducing" it in order to make use of Ppk).

The next two problems invite you to prove some classical algebraic identities. Most of these may be familiar. The challenge here is to think carefully about the way you lay out your induction proof, to learn from the examples above, and (later) to learn from the detailed solutions provided.

Problem 231 Prove by induction the statement:

$$n^n 1 + 3 + 5 + \dots + (2n - 1) = n^2 \text{ holds, for all } n \gg 1^\text{"{}.} \tag{7}$$

The summation in Problem 231 was known to the ancient Greeks. The mystical Pythagorean tradition (which flourished in the centuries after Pythagoras) explored the character of integers through the 'spatial figures' which they formed. For example, if we arrange each successive integer as a new line of dots in the plane, then the sum "1 ` 2 ` 3 ` ¨ ¨ ¨ ` n" can be seen to represent a triangular number. Similarly, if we arrange each odd number 2k ´ 1 in the sum "1 ` 3 ` 5 ` ¨ ¨ ¨ ` p2n ´ 1q" as a "k-by-k reverse L-shape", or gnomon (a word which we still use to refer to the L-shaped piece that casts the shadow on a sundial), then the accumulated L-shapes build up an n by n square of dots – the "1" being the dot in the top left hand corner, the "3" being the reverse L-shape of 3 dots which make this initial "1" into a 2 by 2 square, the "5" being the reverse L-shape of 5 dots which makes this 2 by 2 square into a 3 by 3 square, and so on. Hence the sum "1 ` 3 ` 5 ` ¨ ¨ ¨ ` p2n ´ 1q" can be seen to represent a square number. There is much to be said for such geometrical illustrations; but there is no escape from the fact that they hide behind an ellipsis (the three dots which we inserted in the sum between "5" and "2n ´ 1", which were then summarised when arranging the reverse L-shapes by ending with the words "and so on"). Proof by mathematical induction, and its application in Problem 231, constitute a formal way of avoiding both the appeal to pictures, and the hidden ellipsis.

Problem 232 The sequence

$$2, 5, 13, 35, \dots$$

is defined by its first two terms u<sup>0</sup> " 2, u<sup>1</sup> " 5, and by the recurrence relation:

$$u\_{n+2} = 5u\_{n+1} - 6u\_n.$$


Problem 233 The sequence of Fibonacci numbers

$$0, 1, 1, 2, 3, 5, 8, 13, \dots$$

is defined by its first two terms F<sup>0</sup> " 0, F<sup>1</sup> " 1, and by the recurrence relation:

$$F\_{n+2} = F\_{n+1} + F\_n \text{ when } n \gg 0.$$

Prove by induction that, for all n ě 0,

$$F\_n = \frac{\alpha^n - \beta^n}{\sqrt{5}}, \quad \text{where} \quad \alpha = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad \beta = \frac{1 - \sqrt{5}}{2}. \tag{7}$$

Problem 234 Prove by induction that

$$5^{2n+1} \cdot 2^{n+2} + 3^{n+2} \cdot 2^{2n+1}$$

is divisible by 19, for all n ě 0. 4

Problem 235 Use mathematical induction to prove that each of these identities holds, for all n ě 1:

(a) 1 ` 2 ` 3 ` ¨ ¨ ¨ ` n " npn`1q 2 (b) <sup>1</sup> <sup>1</sup>¨<sup>2</sup> ` 1 <sup>2</sup>¨<sup>3</sup> ` 1 <sup>3</sup>¨<sup>4</sup> ` ¨ ¨ ¨ ` <sup>1</sup> <sup>n</sup>pn`1<sup>q</sup> " 1 ´ 1 n`1 (c) 1 ` q ` q <sup>2</sup> ` q <sup>3</sup> ` ¨ ¨ ¨ ` q <sup>n</sup>´<sup>1</sup> " 1 <sup>1</sup>´<sup>q</sup> ´ q n 1´q (d) 0 ¨ 0! ` 1 ¨ 1! ` 2 ¨ 2! ` ¨ ¨ ¨ ` pn ´ 1q ¨ pn ´ 1q! " n! ´ 1 (e) pcos θ ` isin θq <sup>n</sup> " cos nθ ` isin nθ. 4

Problem 236 Prove by induction the statement:

"p1 ` 2 ` 3 ` ¨ ¨ ¨ ` nq <sup>2</sup> " 1 <sup>3</sup> ` 2 <sup>3</sup> ` 3 <sup>3</sup> ` ¨ ¨ ¨ ` n 3 , for all n ě 1". 4

We now know that, for all n ě 1:

$$1 + 1 + 1 + \dots + 1 \text{ ( $n$  terms)} = n.$$

And if we sum these "outputs" (that is, the first n natural numbers), we get the n th triangular number:

$$1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} = T\_n. \dots$$

$$\Delta$$

The next problem invites you to find the sum of these "outputs": that is, to find the sum of the first n triangular numbers.

### Problem 237

(a) Experiment and guess a formula for the sum of the first n triangular numbers:

$$T\_1 + T\_2 + T\_3 + \dots + T\_n = 1 + 3 + 6 + \dots + \frac{n(n+1)}{2}.$$

(b) Prove by induction that your guessed formula is correct for all n ě 1. 4

We now know closed formulae for

$${"{1} + 2 + 3 + \dots + n"}$$

and for

$$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + (n-1)n\text{''} \text{ :} $$

The next problem hints firstly that these identities are part of something more general, and secondly that these results allow us to find identities for the sum of the first n squares:

$$1^2 + 2^2 + 3^2 + \dots + n^2$$

for the first n cubes:

$$1^3 + 2^3 + 3^3 + \dots + n^3$$

and so on.

#### Problem 238

(a) Note that

$$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n(n+1) = 1 \cdot (1+1) + 2 \cdot (2+1) + 3 \cdot (3+1) + \dots + n \cdot (n+1).$$

Use this and the result of Problem 237 to derive a formula for the sum:

$$1^2 + 2^2 + 3^2 + \dots + n^2.$$

(b) Guess and prove a formula for the sum

$$1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + 3 \cdot 4 \cdot 5 + \dots + (n-2)(n-1)n.$$

Use this to derive a closed formula for the sum:

$$1^3 + 2^3 + 3^3 + \dots + n^3. \tag{7}$$

It may take a bit of effort to digest the statement in the next problem. It extends the idea behind the "method of undetermined coefficients" that is discussed in Note 2 to the solution of Problem 237(a).

### Problem 239

(a) Given n ` 1 distinct real numbers

a0, a1, a2, . . . , an,

find all possible polynomials of degree n which satisfy

fpa0q " fpa1q " fpa2q " ¨ ¨ ¨ " fpan´1q " 0, fpanq " b

for some specified number b.

(b) For each n ě 1, prove the following statement: Given two labelled sets of n ` 1 real numbers

$$a\_0, a\_1, a\_2, \dots, a\_n,$$

and

$${b\_0, b\_1, b\_2, \dots, b\_n, \}$$

where the a<sup>i</sup> are all distinct (but the b<sup>i</sup> need not be), there exists a polynomial f<sup>n</sup> of degree n, such that

$$f\_n(a\_0) = b\_0, \ f\_n(a\_1) = b\_1, \ f\_n(a\_2) = b\_2, \ \dots, \ f\_n(a\_n) = b\_n. \qquad \triangle$$

We end this subsection with a mixed bag of three rather different induction problems. In the first problem the induction step involves a simple construction of a kind we will meet later.

Problem 240 A country has only 3 cent and 4 cent coins.


4

### Problem 241

(a) Solve the equation z ` 1 <sup>z</sup> " 1. Calculate z 2 , and check that z <sup>2</sup> ` 1 z <sup>2</sup> is also an integer.


Problem 242 Let p be any prime number. Use induction to prove:

$$n^n n^p - n \text{ is divisible by } p \text{ for all } n \gg 1\text{"{}.} \tag{7}$$

### 6.4. Infinite geometric series

Elementary mathematics is predominantly about equations and identities. But it is often impossible to capture important mathematical relations in the form of exact equations. This is one reason why inequalities become more central as we progress; another reason is because inequalities allow us to make precise statements about certain infinite processes.

One of the simplest infinite process arises in the formula for the "sum" of an infinite geometric series:

$$1 + r + r^2 + r^3 + \cdots + r^n + \cdots \quad \text{(for every)}.$$

Despite the use of the familiar-looking "+" signs, this can be no ordinary addition. Ordinary addition is defined for two summands; and by repeating the process, we can add three summands (thanks in part to the associative law of addition). We can then add four, or any finite number of summands. But this does not allow us to "add" infinitely many terms as in the above sum. To get round this we combine ordinary addition (of finitely many terms) and simple inequalities to find a new way of giving a meaning to the above "endless sum". In Problem 116 you used the factorisation

$$r^{n+1} - 1 = (r - 1)(1 + r + r^2 + r^3 + \dots + r^n)$$

to derive the closed formula:

$$1 + r + r^2 + r^3 + \dots + r^n = \frac{1 - r^{n+1}}{1 - r}.$$

This formula for the sum of a finite geometric series can be rewritten in the form

$$1 + r + r^2 + r^3 + \dots + r^n = \frac{1}{1 - r} - \frac{r^{n+1}}{1 - r}.$$

At first sight, this may not look like a clever move! However, it separates the part that is independent of n from the part on the RHS that depends on n; and it allows us to see how the second part behaves as n gets large:

when |r| ă 1, successive powers of r get smaller and smaller and converge rapidly towards 0,

so the above form of the identity may be interpreted as having the form:

1 ` r ` r <sup>2</sup> ` r <sup>3</sup> ` ¨ ¨ ¨ ` r <sup>n</sup> " 1 1 ´ r ´ pan "error term"q.

Moreover if |r| ă 1, then the "error term" converges towards 0 as n Ñ 8. In particular, if 1 ą r ą 0, the error term is always positive, so we have proved, for all n ě 1:

$$1 + r + r^2 + r^3 + \dots + r^n < \frac{1}{1 - r}$$

and

the difference between the two sides tends rapidly to 0 as n Ñ 8.

We then make the natural (but bold) step to interpret this, when |r| ă 1, as offering a new definition which explains precisely what is meant by the endless sum

$$1 + r + r^2 + r^3 + \cdots \text{ (for } \mathbf{even}),$$

declaring that, when |r| ă 1,

$$1 + r + r^2 + r^3 + \cdots \text{ (for } \text{vec} \text{)} = \frac{1}{1 - r} \text{.} $$

More generally, if we multiply every term by a, we see that

$$(a+ar+ar^2+ar^3+\cdots \text{ (for ever)} = \frac{a}{1-r}.)$$

Problem 243 Interpret the recurring decimal 0.037037037 ¨ ¨ ¨ (for ever) as an infinite geometric series, and hence find its value as a fraction. 4 Problem 244 Interpret the following endless processes as infinite geometric series.


$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dotsb \text{ (for every) } = 1, 2$$

so the whole process is completed in exactly 1 minute. How much of the cake do I have at the end, and how much do you have? 4

Problem 245 When John von Neumann (1903–1957) was seriously ill in hospital, a visitor tried (rather insensitively) to distract him with the following elementary mathematics problem.

Have you heard the one about the two trains and the fly? Two trains are on a collision course on the same track, each travelling at 30 km/h. A super-fly starts on Train A when the trains are 120 km apart, and flies at a constant speed of 50 km/h – from Train A to Train B, then back to Train A, and so on. Eventually the two trains collide and the fly is squashed. How far did the fly travel before this sad outcome? 4

### 6.5. Some classical inequalities

The fact that our formula for the sum of a geometric series gives us an exact sum is very unusual – and hence very precious. For almost all other infinite series – no matter how natural, or beautiful, they may seem – you can be fairly sure that there is no obvious exact formula for the value of the sum. Hence in those cases where we happen to know the exact value, you may infer that it took the best efforts of some of the finest mathematical minds to discover what we know.

One way in which we can make a little progress in estimating the value of an infinite series is to obtain an inequality by comparing the given sum with a geometric series.

#### Problem 246

(a)(i) Explain why

$$
\frac{1}{3^2} < \frac{1}{2^2},
$$

so

$$\frac{1}{2^2} + \frac{1}{3^2} < \frac{2}{2^2} = \frac{1}{2}.$$

(ii) Explain why <sup>1</sup> , , are all ă , so

$$
\frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2} < \frac{4}{4^2} = \frac{1}{4}.
$$

(b) Use part (a) to prove that

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} < 2, \quad \text{for all} \ n \gg 1.$$

(c) Conclude that the endless sum

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} + \dots \text{ (for every)}$$

has a definite value, and that this value lies somewhere between <sup>17</sup> and 2. 

The next problem presents a rather different way of deriving a similar equality. Once the relevant inequality has been guessed, or given (see Problem 247(a) and (b)), the proof by mathematical induction is often relatively straightforward. And after a little thought about Problem 246, it should be clear that much of the inaccuracy in the general inequality arises from the rather poor approximations made for the first few terms (when n " 1, when n " 2, when n " 3, etc.); hence by keeping the first few terms as they are, and only approximating for n ě 2, or n ě 3, or n ě 4, we can often prove a sharper result.

#### Problem 247

(a) Prove by induction that

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} \le 2 - \frac{1}{n}, \text{ for all } n \gg 1.$$

(b) Prove by induction that

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} < 1.68 - \frac{1}{n}, \text{ for all } \ n \gg 4. \tag{7}$$

The infinite sum

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} + \dots \text{ (for every)}$$

is a historical classic, and has many instructive stories to tell. Recall that, in Problems 54, 62, 63, 236, 237, 238 you found closed formulae for the sums

$$1 + 2 + 3 + \dots + n$$

$$1^2 + 2^2 + 3^2 + \dots + n^2$$

$$1^3 + 2^3 + 3^3 + \dots + n^3$$

and for the sums

$$1 \times 2 + 2 \times 3 + 3 \times 4 + \dots + (n-1)n$$

$$1 \times 2 \times 3 + 2 \times 3 \times 4 + 3 \times 4 \times 5 + \dots + (n-2)(n-1)n.$$

Each of these expressions has a "natural" feel to it, and invites us to believe that there must be an equally natural compact answer representing the sum. In Problem 235 you took this idea one step further by finding a beautiful closed expression for the sum

$$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \dots + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}$$

When we began to consider infinite series, we found the elegant closed formula

$$1 + r + r^2 + r^3 + \dots + r^n = \frac{1}{1 - r} - \frac{r^{n+1}}{1 - r}.$$

We then observed that the final term on the RHS could be viewed as an "error term", indicating the amount by which the LHS differs from <sup>1</sup> 1´r , and noticed that, for any given value of r between ´1 and `1, this error term "tends towards 0 as the power n increases". We interpreted this as indicating that one could assign a value to the endless sum

$$1 + r + r^2 + r^3 + \cdots \text{ (for every)} = \frac{1}{1 - r}.$$

In the same way, in the elegant closed formula

$$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \dots + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}$$

the final term on the RHS indicates the amount by which the finite sum on the LHS differs from 1; and since this "error term" tends towards 0 as n increases, we may assign a value to the endless sum

$$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \dots \text{ (for every)} = 1.$$

It is therefore natural to ask whether other infinite series, such as

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} + \dots \text{ (for every)}$$

may also be assigned some natural finite value. And since the series is purely numerical (without any variable parameters, such as the "r" in the geometric series formula), this answer should be a strictly numerical answer. And it should be exact – though all we have managed to prove so far (in Problems 246 and 247) is that this numerical answer lies somewhere between <sup>17</sup> <sup>12</sup> and 1.68.

This question arose naturally in the middle of the seventeenth century, when mathematicians were beginning to explore all sorts of infinite series (or "sums that go on for ever"). With a little more work in the spirit of Problems 246 and 247 one could find a much more accurate approximate value. But what is wanted is an exact expression, not an unenlightening decimal approximation. This aspiration has a serious mathematical basis, and is not just some purist preference for elegance. The actual decimal value is very close to

1.649934 ¨ ¨ ¨ .

But this conveys no structural information. One is left with no hint as to why the sum has this value. In contrast, the eventual form of the exact expression suggests connections whose significance remains of interest to this day.

The greatest minds of the seventeenth and early eighteenth century tried to find an exact value for the infinite sum – and failed. The problem became known as the Basel problem (after Jakob Bernoulli (1654–1705) who popularised the problem in 1689 – one of several members of the Bernoulli family who were all associated with the University in Basel). The problem was finally solved in 1735 – in truly breathtaking style – by the young Leonhard Euler (1707–1783) (who was at the time also in Basel). The answer

$$\frac{\pi^2}{6}$$

illustrates the final sentence of the preceding paragraph in unexpected ways, which we are still trying to understand.

In the next problem you are invited to apply similar ideas to an even more important series. Part (a) provides a relatively crude first analysis. Part (b) attacks the same question; but it does so using algebra and induction (rather than the formula for the sum of a geometric series) in a way that is then further refined in part (c).

### Problem 248

(a)(i) Choose a suitable r and prove that

$$\frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} < 1 + r + r^2 + \dots + r^{n-1} < 2.1$$

(ii) Conclude that

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} < 3, \text{ for every } n \gg 0,$$

and hence that the endless sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots \text{ (for every)}$$

can be assigned a value "e" satisfying 2 ă e ď 3.

(b)(i) Prove by induction that

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} \leqslant 3 - \frac{1}{n.n!}, \text{ for all } n \gg 1.$$

(ii) Use part (i) to conclude that the endless sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots \text{ (for every)}$$

can be assigned a definite value "e", and that this value lies somewhere between 2.5 and 3.

	- (i) Prove by induction that

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} \lessapprox 2.75 - \frac{1}{n.n!}, \text{ for all } n \gg 2.$$

(ii) Use part (i) to conclude that the endless sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots \text{ (for every)}$$

can be assigned a definite value "e", and that this value lies somewhere between 2.6 and 2.75.

(d)(i) Prove by induction that

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} \leqslant 2.722\dots \text{(for every)} - \frac{1}{n.n!}, \text{ for all } n \gg 3.$$

(ii) Use part (i) to conclude that the endless sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots \text{ (for every)}$$

can be assigned a definite value "e", and that this value lies somewhere between 2.708 and 2.7222 ¨ ¨ ¨ (for ever). 4

We end this section with one more inequality in the spirit of this section, and two rather different inequalities whose significance will become clear later.

Problem 249 Prove by induction that

$$
\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} \gg \sqrt{n}, \text{ for all } n \gg 1. \tag{7}
$$

Problem 250 Let a, b be real numbers such that a ‰ b, and a ` b ą 0. Prove by induction that

$$2^{n-1}(a^n + b^n) \gg (a+b)^n, \text{ for all } n \gg 1. \tag{7}$$

Problem 251 Let x be any real number ě ´1. Prove by induction that

$$(1+x)^n \geqslant 1+nx,\text{ for all } n \geqslant 1. \tag{7}$$

### 6.6. The harmonic series

The great foundation of mathematics is the principle of contradiction, or of identity, that is to say that a statement cannot be true and false at the same time, and that thus A is A, and cannot be not A. And this single principle is enough to prove the whole of arithmetic and the whole of geometry, that is to say all mathematical principles. Gottfried W. Leibniz (1646–1716) We have seen how some infinite series, or sums that go on for ever, can be assigned a finite value for their sum:

$$1 + r + r^2 + r^3 + \cdots \text{ (for every)} = \frac{1}{1 - r}$$

$$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \cdots + \frac{1}{n(n+1)} + \cdots \text{ (for every)} = 1$$

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2} + \cdots \text{ (for every)} = \frac{\pi^2}{6}$$

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} + \cdots \text{ (for every)} = e.$$

We say that these series converge (meaning that they can be assigned a finite value).

This section is concerned with another very natural series, the so-called harmonic series

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} + \dots \text{ (for every)}.$$

It is not entirely clear why this is called the harmonic series. The natural overtones that arise in connection with plucking a stretched string (as with a guitar or a harp) have wavelengths that are <sup>1</sup> 2 the basic wavelength, or <sup>1</sup> 3 of the basic wavelength, and so on. It is also true that, just as each term of an arithmetic series is the arithmetic mean of its two neighbours, and each term of a geometric series is the geometric mean of its two neighbours, so each term of the harmonic series after the first is equal to the harmonic mean (see Problems 85, 89) of its two neighbours:

$$\frac{1}{k} = \frac{2}{\left(\frac{1}{k-1}\right)^{-1} + \left(\frac{1}{k+1}\right)^{-1}}.$$

Unlike the first two series above, there is no obvious closed formula for the finite sum

$$s\_n = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}.$$

Certainly the sequence of successive sums

$$s\_1 = 1, \ s\_2 = \frac{3}{2}, \ s\_3 = \frac{11}{6}, \ s\_4 = \frac{25}{12}, \ s\_5 = \frac{137}{60}, \ \dots$$

does not suggest any general pattern.

Problem 252 Suppose we denote by S the "value" of the endless sum

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} + \dots \text{ (for every)}$$


The Leibniz quotation above emphasizes that the reliability of mathematics stems from a single principle – namely the refusal to tolerate a contradiction. We have already made explicit use of this principle from time to time (see, for example, the solution to Problem 125). The message is simple: whenever we hit a contradiction, we know that we have "gone wrong" – either by making an error in calculation or logic, or by beginning with a false assumption. In Problem 252 the observations you were expected to make are paradoxical: you obtained two different series, which both correspond to " <sup>1</sup> 2 S", but every term in one series is larger than the corresponding term in the other! What one can conclude may not be entirely clear. But it is certainly clear that something is wrong: we have somehow created a contradiction. The three steps ((i), (ii), (iii)) appear to be relatively sensible. But the final observation " 1 2 S ă 1 2 S" (since <sup>1</sup> <sup>2</sup> <sup>ă</sup> 1, <sup>1</sup> <sup>4</sup> ă 1 3 , etc.) makes no sense. And the only obvious assumption we have made is to assume that the endless sum

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} + \dots \text{ (for every)}$$

can be assigned a value "S", which can then be manipulated as though it were a number.

The conclusion would seem to be that, whether or not the endless sum has a meaning, it cannot be assigned a value in this way. We say that the series diverges. Each finite sum

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$$

has a value, and these values "grow more and more slowly" as n increases:


• it takes 12 367 terms before the series reaches a sum ą 10.

However, this slow growth is not enough to guarantee that the corresponding endless sum corresponds to a finite numerical value.

The danger signals should already have been apparent in Problem 249, where you proved that

$$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} \gg \sqrt{n}$$

The n th term ? 1 n tends to 0 as n increases; so the finite sums grow ever more slowly as n increases. However, the LHS can be made larger than any integer K simply by taking K<sup>2</sup> terms. Hence there is no way to assign a finite value to the endless sum

$$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} + \dots \text{ (for every)}.$$

#### Problem 253

(a)(i) Explain why

$$
\frac{1}{2} + \frac{1}{3} < 1.
$$

(ii) Explain why

$$
\frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} < 1.
$$

(iii) Extend parts (i) and (ii) to prove that

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2^n - 1} < n, \quad \text{for all} \ n \gg 2.$$

(iv) Finally use the fact that, when n ě 3,

$$\frac{1}{2^n} < \frac{1}{2} - \frac{1}{3}$$

to modify the proof in (iii) slightly, and hence show that

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2^n} < n, \text{ for all } n \gg 3.$$

(b)(i) Explain why

$$
\frac{1}{3} + \frac{1}{4} > \frac{1}{2}.
$$

(ii) Explain why

$$
\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} > \frac{1}{2}.
$$

(iii) Extend parts (i) and (ii) to prove that

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2^n} > 1 + \frac{n}{2}, \quad \text{for all } n \gg 2.$$

(c) Combine parts (a) and (b) to show that, for all n ě 2, we have the two inequalities

$$1 + \frac{n}{2} < \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2^n} < n.$$

Conclude that the endless sum

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} + \dotsb \quad \text{(for even)}$$

cannot be assigned a finite value. 4

The result in Problem 253(c) has an unexpected consequence.

Problem 254 Imagine that you have an unlimited supply of identical rectangular strips of length 2. (Identical empty plastic CD cases can serve as a useful illustration, provided one focuses on their rectangular side profile, rather than the almost square frontal cross-section.) The goal is to construct a 'stack' in such a way as to stick out as far as possible beyond a table edge. One strip balances exactly at its midpoint, so can protrude a total distance of 1 without tipping over.


The next problem illustrates, in the context of the harmonic series, what is in fact a completely general phenomenon: an endless sum of steadily decreasing

Figure 10: Overhanging strips, n " 10.

positive terms may converge or diverge; but provided the terms themselves converge to 0, then the the corresponding "alternating sum" – where the same terms are combined but with alternately positive and negative signs – always converges.

### Problem 255

$$\text{(a) Let}$$

$$s\_n = \frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots \pm \frac{1}{n}$$

(where the final operation is "`" if n is odd and "´" if n is even).

(i) Prove that

$$s\_{2n-2} \le s\_{2n} \le s\_{2m+1} \le s\_{2m-1},$$

for all m, n ě 1.

(ii) Conclude that the endless alternating sum

$$\frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dotsb \text{ (for every)}$$

can be assigned a value s that lies somewhere between s<sup>6</sup> " 37 <sup>60</sup> and s<sup>5</sup> " 47 60 .

(b) Let

$$a\_1, a\_2, a\_3, \dots$$

be an endless, decreasing sequence of positive terms (that is, a<sup>n</sup>`<sup>1</sup> ă a<sup>n</sup> for all n ě 1). Suppose that the sequence of terms a<sup>n</sup> converges to 0 as n Ñ 8.

(i) Let

$$s\_n = a\_1 - a\_2 + a\_3 - a\_4 + a\_5 - \dots \pm a\_n$$

(where the final operation is "`" if n is odd and "´" if n is even). Prove that

$$s\_{2n-2} < s\_{2n} < s\_{2m+1} < s\_{2m-1}, \text{ for all } m, n \gg 1.$$

(ii) Conclude that the endless alternating sum

$$(a\_1 - a\_2 + a\_3 - a\_4 + a\_5 - \dotsb \text{ (for ever)})$$

can be assigned a value s that lies somewhere between s<sup>2</sup> " a<sup>1</sup> ´a<sup>2</sup> and s<sup>3</sup> " a<sup>1</sup> ´ a<sup>2</sup> ` a3. 4

Just as with the series

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} + \dots \text{ (for ever)}$$

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots \text{ (for ever)},$$

we can show relatively easily that

$$\frac{1}{1} - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dotsb \text{ (for every)}$$

can be assigned a value s. It is far less clear whether this value has a familiar name! (It is in fact equal to the natural logarithm of 2: "log<sup>e</sup> 2".) A similarly intriguing series is the alternating series of odd terms from the harmonic series:

$$\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dotsb \text{ (for every)}$$

You should be able to show that this endless series can be assigned a value somewhere between s<sup>2</sup> " 2 3 and s<sup>3</sup> " 13 <sup>15</sup> ; but you are most unlikely to guess that its value is equal to <sup>π</sup> 4 . This was first discovered in 1674 by Leibniz (1646–1716). One way to obtain the result is using the integral of p1`x 2 q ´1 from 0 to 1: on the one hand the integral is equal to arctan x evaluated when x " 1 (that is, <sup>π</sup> 4 ); on the other hand, we can expand the integrand as a power series 1 ´ x <sup>2</sup> ` x <sup>4</sup> ´ x <sup>6</sup> ` ¨ ¨ ¨ , integrate term by term, and prove that the resulting series converges when x " 1. (It does indeed converge, though it does so very, very slowly.)

The fact that the alternating harmonic series has the value log<sup>e</sup> 2 seems to have been first shown by Euler (1707–1783), using the power series expansion for logp1 ` xq.

### 6.7. Induction in geometry, combinatorics and number theory

We turn next to a mixed collection of problems designed to highlight a range of applications.

Problem 256 Let f<sup>1</sup> " 2, fk`<sup>1</sup> " fkpf<sup>k</sup> ` 1q. Prove by induction that f<sup>k</sup> has at least k distinct prime factors. 4

### Problem 257

	- (ii) Prove by induction that n straight lines in the plane divide the plane into at most R<sup>n</sup> regions.
	- (ii) Prove by induction that n planes in 3-dimensions divide space into at most S<sup>n</sup> regions. 4

Problem 258 Given a square, prove that, for each n ě 6, the initial square can be cut into n squares (of possibly different sizes). 4

Problem 259 A tree is a connected graph, or network, consisting of vertices and edges, but with no cycles (or circuits). Prove that a tree with n vertices has exactly n ´ 1 edges. 4

The next problem concerns spherical polyhedra. A spherical polyhedron is a polyhedral surface in 3-dimensions, which can be inflated to form a sphere (where we assume that the faces and edges can stretch as required). For example, a cube is a spherical polyhedron; but the surface of a picture frame is not. A spherical polyhedron has

• faces (flat 2-dimensional polygons, which can be stretched to take the form of a disc),


Each face must clearly have at least 3 edges; and there must be at least three edges and three faces meeting at each vertex.

If a spherical polyhedron has V vertices, E edges, and F faces, then the numbers V , E, F satisfy Euler's formula

$$V - E + F = 2.$$

For example, a cube has V " 8 vertices, E " 12 edges, and F " 6 faces, and 8 ´ 12 ` 6 " 2.

### Problem 260

	- (ii) Describe a spherical polyhedron with exactly 8 edges.

Problem 261 A map is a (finite) collection of regions in the plane, each with a boundary, or border, that is 'polygonal' in the sense that it consists of a single sequence of distinct vertices and – possibly curved – edges, that separates the plane into two parts, one of which is the polygonal region itself. A map can be properly coloured if each region can be assigned a colour so that each pair of neighbouring regions (sharing an edge) always receive different colours. Prove that the regions of such a map can be properly coloured with just two colours if and only if an even number of edges meet at each vertex. 4

Problem 262 (Gray codes) There are 2<sup>n</sup> sequences of length n consisting of 0s and 1s. Prove that, for each n ě 2, these sequences can be arranged in a cyclic list such that any two neighbouring sequences (including the last and the first) differ in exactly one coordinate position. 4 Problem 263 (Calkin-Wilf tree) The binary tree in the plane has a distinguished vertex as 'root', and is constructed inductively. The root is joined to two new vertices; and each new vertex is then joined to two further new vertices – with the construction process continuing for ever (Figure 11). Label the vertices of the binary tree with positive fractions as follows:


Problem 264 A collection of n intervals on the x-axis is such that every pair of intervals have a point in common. Prove that all n intervals must then have at least one point in common. 4

### 6.8. Two problems

Problem 265 Several identical tanks of water sit on a horizontal base. Each pair of tanks is connected with a pipe at ground level controlled by a valve, or tap. When a valve is opened, the water level in the two connected tanks becomes equal (to the average, or mean, of the initial levels). Suppose we start with tank T which contains the least amount of water. The aim is to open and close valves in a sequence that will lead to the final water level in tank T being as high as possible. In what order should we make these connections? 4

Figure 11: A (rooted) binary tree.

Problem 266 I have two flasks. One is 'empty', but still contains a residue of a dangerous chemical; the other contains a fixed amount of solvent that can be used to wash away the remaining traces of the dangerous chemical. What is the best way to use the fixed quantity of solvent? Should I use it all at once to wash out the first flask? Or should I first wash out the flask using just half of the solvent, and then repeat with the other half? Or is there a better way of using the available solvent to remove as much as possible of the dangerous chemical? 4

### 6.9. Infinite descent

In this final section we touch upon an important variation on mathematical induction. This variation is well-illustrated by the next (probably familiar) problem.

Problem 267 Write out for yourself the following standard proof that ? 2 is irrational.


Problem 267 has the classic form of a proof which reaches a contradiction by infinite descent.


$$n > n' > n'' > \dots > 0$$

of positive integers, which is impossible (since such a chain can have length at most n).

4. Hence the initial assumption that the claim was false must itself be false – so the claim must be true (as required).

Proof by "infinite descent" is an invaluable tool. But it is important to realise that the method is essentially a variation on proof by mathematical induction. As a first step in this direction it is worth reinterpreting Problem 267 as an induction proof.

Problem 268 Let Ppnq be the statement:

" ? 2 cannot be written as a fraction with positive denominator ď n".


Problem 268 shows that, in the particular case of Problem 267 one can translate the standard proof that "? 2 is irrational" into a proof by induction. But much more is true. The contradiction arising in step 3. above is an application of an important principle, namely

The Least Element Principle: Every non-empty set S of positive integers has a smallest element.

The Least Element Principle is equivalent to The Principle of Mathematical Induction which we stated at the beginning of the chapter:

The Principle of Mathematical Induction: If a subset S of the positive integers


then S contains all positive integers.

### Problem 269


To round off this final chapter you are invited to devise a rather different proof of the irrationality of ? 2.

Problem 270 This sequence of constructions presumes that we know – for example, by Pythagoras' Theorem – that, in any square OP QR, the ratio

$$\text{\textquotedblleft diagonal : side\textquotedblright} = \underline{OQ} : \underline{OP} = \sqrt{2} : 1.$$

Let OP QR be a square. Let the circle with centre Q and passing through P meet OQ at P 1 . Construct the perpendicular to OQ at P 1 , and let this meet OR at Q<sup>1</sup> .


### 6.10. Chapter 6: Comments and solutions

Note: It is important to separate the underlying idea of "induction" from the formal way we have chosen to present proofs. As ever in mathematics, the ideas are what matter most. But the process of struggling with (and slowly coming to understand why we need) the formal structure behind the written proofs is part of the way the ideas are tamed and organised.

Readers should not be intimidated by the physical extent of the solutions to this chapter. As explained in the main text it is important for all readers to review the way they approach induction proofs: so we have erred in favour of completeness – knowing that as each reader becomes more confident, s/he will increasingly compress, or abbreviate, some of the steps.

#### 228.

(a) Yes.

(b) Yes.

$$2 \times 3 \times 5 \times 7 \times 11 + 1 = 2311,$$

and ? 2311 " 48.07 . . . , so we only need to check prime factors up to 47.

(c) No.

2 ˆ 3 ˆ 5 ˆ 7 ˆ 11 ˆ 13 ` 1 " 30 031,

and ? 30 031 " 173.29 . . . so we might have to check all 40 possible prime factors up to 173. However, we only have to start at 17 [Why?], and checking with a calculator is very quick. In fact 30 031 factorises rather prettily as 59 ˆ 509.

229. Note: The statement in the problem includes the quantifier "for all n ě 1".

What is to be proved is the compound statement

"Ppnq is true for all n ě 1".

In contrast, each individual statement Ppnq refers to a single value of n.

It is essential to be clear when you are dealing with the compound statement, and when you are referring to some particular statement Pp1q, or Ppnq, or Ppkq.

Let Ppnq be the statement:

"5<sup>2</sup>n`<sup>2</sup> ´ 24n ´ 25 is divisible by 576".

• Pp1q is the statement:

"5<sup>4</sup> ´ 24 ˆ 1 ´ 25 is divisible by 576".

That is:

"625 ´ 49 " 576 is divisible by 576",

which is evidently true.

• Now suppose that we know Ppkq is true for some k ě 1. We must show that Ppk ` 1q is then also true.

To prove that Ppk ` 1q is true, we have to consider the statement Ppk ` 1q. It is no use starting with Ppkq. However, since we know that Ppk) is true, we are free to use it at any stage if it turns out to be useful.

To prove that Ppk ` 1q is true, we have to show that

"5<sup>2</sup>pk`1q`<sup>2</sup> ´ 24pk ` 1q ´ 25 is divisible by 576".

So we must start with 5<sup>2</sup>pk`1q`<sup>2</sup>´24pk`1q´25 and try to "simplify" it (knowing that "simplify" in this case means "rewrite it in a way that involves 5<sup>2</sup>k`<sup>2</sup>´24k´ 25").

$$\begin{aligned} 5^{2(k+1)+2} - 24(k+1) - 25 \\ &= [5^{2k+4}] - 24k - 24 - 25 \\ &= [5^{2(k+2)+2} - 24k - 25) + 5^2 \cdot (24k) + 5^2 \cdot 25 \\ &\quad - 24k - (24 + 25) \\ &= 5^2(5^{2k+2} - 24k - 25) + [(5^2 - 1) \times (24k)] \\ &\quad + [5^2 \times 25 - 24 - 25] \\ &= 5^2(5^{2k+2} - 24k - 25) + 24^2k + [5^2 \times 25 - 24 - 25] \\ &= 5^2(5^{2k+2} - 24k - 25) + 24^2k + [5^4 - 24 - 25]. \end{aligned}$$

The first term on the RHS is a multiple of p5 <sup>2</sup>k`<sup>2</sup> ´ 24k ´ 25q, so is divisible by 576 (since we know that Ppkq is true); the second term on the RHS is a multiple of 24<sup>2</sup> " 576; and the third term on the RHS is the expression arising in Pp1q, which we saw is equal to 576.

6 the whole RHS is divisible by 576

6 the LHS is divisible by 576, so Ppk ` 1q is true.

#### Hence


230. Let Ppnq be the statement:

"the angles of any p-gon, for any value of p with 3 ď p ď n, have sum exactly pp ´ 2qπ radians".

1. Pp3q is the statement:

"the angles of any triangle have sum π radians".

This is a known fact: given triangle 4ABC, draw the line XAY through A parallel to BC, with X on the same side of AC as B and Y on the same

side of AB as C. Then =XAB " =CBA and =Y AC " =BCA (alternate angles), so

$$
\angle B + \angle A + \angle C = \angle XAB + \angle A + \angle YAC = \angle XAY = \pi.
$$

2. Now we suppose that Ppkq is known to be true for some k ě 3. We must show that Ppk ` 1q is then necessarily true.

To prove that Ppk ` 1q is true, we have to consider the statement Ppk ` 1q: that is,

"the angles of a p-gon, for any value of p with 3 ď p ď k ` 1, have sum exactly pp ´ 2qπ radians".

This can be reworded by splitting it into two parts:

"the angles of any p-gon for 3 ď p ď k have sum exactly pp ´ 2qπ radians;"

and

"the angles of any pk`1q-gon have sum exactly ppk`1q´2qπ radians".

The first part of this revised version is precisely the statement Ppkq, which we suppose is known to be true. So the crux of the matter is to prove the second part – namely that the angles of any pk ` 1q-gon have sum pk ´ 1qπ.

Let A0A1A<sup>2</sup> ¨ ¨ ¨ A<sup>k</sup> be any pk ` 1q-gon.

[Note: The usual move at this point is to say "draw the chord AkA<sup>1</sup> to cut the polygon into the triangle AkA1A<sup>0</sup> (with angle sum π (by Pp3q), and the k-gon A1A<sup>2</sup> ¨ ¨ ¨ A<sup>k</sup> (with angle sum pk ´ 2qπ (by Ppkq), whence we can add to see that A0A1A<sup>2</sup> ¨ ¨ ¨ A<sup>k</sup> has angle sum ppk ` 1q ´ 2qπ. However, this only works


So what is usually presented as a "proof" does not work in general.

If we want to prove the general result – for polygons of all shapes – we have to get round this unwarranted assumption. Experiment may persuade you that "there is always some vertex that sticks out and which can be safely "cut off"; but it is not at all clear how to prove this fact (we know of no simple proof). So we have to find another way.]

Consider the vertex A1, and its two neighbours A<sup>0</sup> and A2.

Imagine each half-line, which starts at A1, and which sets off into the interior of the polygon. Because the polygon is finite, each such half-line defines a line segment A1X, where X is the next point of the polygon which the half line hits (that is, X is one of the vertices Am, or a point on one of the sides AmA<sup>m</sup>`<sup>1</sup>).

Consider the locus of all such points X as the half line swings from A1A<sup>0</sup> (produced) to A1A<sup>2</sup> (produced). There are two possibilities: either


Because of the way the point X was chosen, the chord A1X " A1A<sup>m</sup> does not meet any other point of the pk ` 1q-gon A0A1A<sup>2</sup> ¨ ¨ ¨ Ak, and so splits the pk ` 1q-gon into an m-gon A1A2A<sup>3</sup> ¨ ¨ ¨ A<sup>m</sup> (with angle sum pm ´ 2qπ by Ppkq) and a pk ´ m ` 3q-gon AmA<sup>m</sup>`<sup>1</sup>A<sup>m</sup>`<sup>2</sup> ¨ ¨ ¨ AkA0A<sup>1</sup> (with angle sum pk ´ m ` 1qπ by Ppkq). So the pk ` 1q-gon A0A1A<sup>2</sup> ¨ ¨ ¨ A<sup>k</sup> has angle sum ppk ` 1q ´ 2qπ as required.

Hence Ppk ` 1q is true.

6 Ppnq is true for all n ě 3. QED

231. Let Ppnq be the statement

$$1 + 3 + 5 + \dots + (2n - 1) = n^2 \dots$$


$$1 + 3 + 5 + \dots + (2k - 1) = k^2 \dots$$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is an equation, so we start with the LHS of Ppk ` 1q and try to simplify it in an appropriate way to get the RHS of Ppk ` 1q:

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{array}{c} 1 + 3 + 5 + \cdots + (2(k+1) - 1) \\ \end{array} \\ &= \begin{array}{c} (1 + 3 + 5 + \cdots + (2k - 1)) + (2k + 1) . \end{array} \end{aligned}$$

If we now use Ppkq, which we are supposing to be true, then the first bracket is equal to k 2 , so this sum is equal to:

$$\begin{aligned} &=\quad k^2 + (2k+1) \\ &=\quad \left(k+1\right)^2 \\ &=\quad \text{RHS of } \mathbf{P}(k+1). \end{aligned}$$

Hence Ppk ` 1q holds.

Combining these two bullet points then shows that "Ppnq holds, for all n ě 1". QED

#### 232.


Let Ppnq be the statement:

"u<sup>m</sup> " 2 <sup>m</sup> ` 3 <sup>m</sup> for all m, 0 ď m ď n".

' LHS of Pp0q " u<sup>0</sup> " 2; RHS of Pp0q " 2 <sup>0</sup> ` 3 <sup>0</sup> " 1 ` 1. Since these two are equal, Pp0q is true.

Pp1q combines Pp0q, and the equality of u<sup>1</sup> " 5 and 2<sup>1</sup> ` 3 1 ; since these two are equal, Pp1q is true.

' Suppose that Ppkq is true for some particular (unspecified) k ě 1; that is, we know that, for this particular k,

"u<sup>m</sup> " 2 <sup>m</sup> ` 3 <sup>m</sup> for all m, 0 ď m ď k."

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q requires us to prove that

"u<sup>m</sup> " 2 <sup>m</sup> ` 3 <sup>m</sup> for all m, 0 ď m ď k ` 1."

Most of this is guaranteed by Ppkq, which we assume to be true. It remains for us to check that the equality holds for u<sup>k</sup>`<sup>1</sup>. We know that

$$u\_{k+1} = 5u\_k - 6u\_{k-1} - 5$$

And we may use Ppk), which we are supposing to be true, to conclude that:

$$\begin{aligned} u\_{k+1} &= -5\left(2^k + 3^k\right) - 6\left(2^{k-1} + 3^{k-1}\right), \\ &= -(10-6)2^{k-1} + (15-6)3^{k-1} \\ &= -2^{k+1} + 3^{k+1}. \end{aligned}$$

Hence Ppk ` 1q holds.

Combining these two bullet points then shows that "Ppnq holds, for all n ě 0". QED 233. Let Ppnq be the statement:

$$\text{``}F\_m = \frac{\alpha^m - \beta^m}{\sqrt{5}} \text{ for all } m, \, 0 \in m \lessapprox n\text{''},$$

where α " 1` ? 5 2 and β " 1´ ? 5 2 .

	- LHS of Pp1q " F<sup>1</sup> " 1; RHS of Pp1q " <sup>α</sup>?´<sup>β</sup> 5 " 1. Since these two are equal, Pp1q is true.

$$^{"\mu}F\_m = \frac{\alpha^m - \beta^m}{\sqrt{5}}\text{ for all }m, \, 0 \lessapprox m \lessapprox k.$$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q requires us to prove that

$$^{\omega}F\_m = \frac{\alpha^m - \beta^m}{\sqrt{5}} \text{ for all } m, \, 0 \lesssim m \lesssim k+1."$$

Most of this is guaranteed by Ppkq, which we assume to be true. It remains to check this for F<sup>k</sup>`<sup>1</sup>. We know that

$$F\_{k+1} = F\_k + F\_{k-1} \cdots$$

And we may use Ppkq, which we are supposing to be true to conclude that:

$$\begin{aligned} F\_{k+1} &= -\frac{\alpha^k - \beta^k}{\sqrt{5}} + \frac{\alpha^{k-1} - \beta^{k-1}}{\sqrt{5}} \\ &= -\frac{\alpha^k + \alpha^{k-1}}{\sqrt{5}} - \frac{\beta^k + \beta^{k-1}}{\sqrt{5}} \\ &= -\frac{\alpha^{k+1} - \beta^{k+1}}{\sqrt{5}} \end{aligned}$$

(since α and β are roots of the equation x <sup>2</sup> ´ x ´ 1 " 0) Hence Ppk ` 1q holds.

Combining these two bullet points then shows that "Ppnq holds, for all n ě 1". QED

Note: You may understand the above solution and yet wonder how such a formula could be discovered. The answer is fairly simple. There is a general theory about linear recurrence relations which guarantees that the set of all solutions of a second order recurrence (that is, a recurrence in which each term depends on the two previous terms) is "two dimensional" (that is, it is just like the familiar 2D plane, where every vector pp, qq is a combination of the two "base vectors" p1, 0q and p0, 1q:

$$p(p,q) = p(1,0) + q(0,1).$$

Once we know this, it remains:


234. Let Ppnq be the statement:

$$3^{4n}5^{2n+1} \cdot 2^{n+2} + 3^{n+2} \cdot 2^{2n+1} \text{ is divisible by } 19^{n}.$$


To prove that Ppk ` 1q is true, we have to show that

"5<sup>2</sup>k`<sup>3</sup> ¨ 2 <sup>k</sup>`<sup>3</sup> ` 3 k`3 ¨ 2 2k`3 is divisible by 19".

$$\begin{split} 5^{2k+3} \cdot 2^{k+3} + 3^{k+3} \cdot 2^{2k+3} &= \ 5^2 \cdot 2 \left( 5^{2k+1} \cdot 2^{k+2} + 3^{k+2} \cdot 2^{2k+1} \right) \\ &= \ 5^2 \cdot 2 \cdot 3^{k+2} \cdot 2^{2k+1} + 3^{k+3} \cdot 2^{2k+3} \\ &= \ 5^2 \cdot 2 \cdot \left( 5^{2k+1} \cdot 2^{k+2} + 3^{k+2} \cdot 2^{2k+1} \right) \\ &\quad - \left( 5^2 - 3 \cdot 2 \right) 3^{k+2} \cdot 2^{2k+2} .\end{split}$$

The bracket in the first term on the RHS is divisible by 19 (by Ppkq), and the bracket in the second term is equal to 19. Hence both terms on the RHS are divisible by 19, so the RHS is divisible by 19. Therefore the LHS is also divisible by 19, so Ppk ` 1q is true.

6 Ppnq is true for all n ě 0. QED

#### 235.

Note: The proofs of identities such as those in Problem 235, which are given in many introductory texts, ignore the lessons of the previous two problems. In particular,

	- the single statement Ppnq for a particular n, and
	- the "quantified" result to be proved ("for all n ě 1"),

and

• they proceed in the 'wrong' direction (e.g. starting with the identity Ppnq and "adding the same to both sides").

This latter strategy is psychologically and didactically misleading – even though it can be justified logically when proving very simple identities. In these very simple cases, each statement Ppnq to be proved is unusual in that it refers to exactly one configuration, or equation, for each n. And since there is exactly one configuration for each n, the configuration or identity for k`1 can often be obtained by fiddling with the configuration for k. In contrast, in Problem 230, for each value of n, there is a bewildering variety of possible polygons with n vertices, ranging from regular polygons to the most convoluted, re-entrant shapes: the statement Ppnq makes an assertion about all such configurations, and there is no way of knowing whether we can obtain all such configurations for k ` 1 in a uniform way by fiddling with some configuration for k.

Readers should try to write each proof in the intended spirit, and to learn from the solutions – since this style has been chosen to highlight what mathematical induction is really about, and it is this approach that is needed in serious applications.

(a) Let Ppnq be the statement:

$$\frac{n}{2} + 2 + 3 + \dots + n = \frac{n(n+1)}{2} \text{''} \dots$$


$$k\text{ "1 + 2 + 3 + \dots + k = \frac{k\left(k+1\right)}{2}" , ".$$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is an equation, so we start with the LHS of Ppk ` 1q and try to simplify it in an appropriate way to get the RHS of Ppk ` 1):

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{array}{c} 1+2+3+\cdots+k+(k+1) \\ \hline \end{array} \\ &= \begin{array}{c} (1+2+3+\cdots+k)+(k+1) \\ \end{array} . \end{aligned}$$

If we now use Ppkq, which we are supposing to be true, then the first bracket is equal to <sup>k</sup>pk`1<sup>q</sup> 2 , so the complete sum is equal to:

$$\begin{aligned} &=\quad \frac{k(k+1)}{2} + (k+1) \\ &=\quad \frac{(k+1)(k+2)}{2} \\ &=\quad \text{RHS of } \mathbf{P}(k+1). \end{aligned}$$

Hence Ppk ` 1q holds.

If we combine these two bullet points, then we have proved that "Ppnq holds for all n ě 1". QED

(b) Let Ppnq be the statement:

$$\frac{\frac{1}{3 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dotsb + \frac{1}{n \cdot (n+1)}} = 1 - \frac{1}{n+1}\text{''.}$$


$$\frac{\frac{1}{3 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{k \cdot (k+1)}} = 1 - \frac{1}{k+1}\text{''} . $$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is an equation, so we start with the LHS of Ppk ` 1q and try to simplify it in an appropriate way to get the RHS of Ppk ` 1q:

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{array}{c} \frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \dots + \frac{1}{(k+1)(k+2)} \\ &= \begin{bmatrix} \frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \frac{1}{k(k+1)} \end{bmatrix} \end{aligned} $$
 
$$ + \frac{1}{(k+1)(k+2)} . $$

If we now use Ppkq, which we assume is true, then the first bracket is equal to 1 ´ 1 k`1 , so the complete sum is equal to:

$$\begin{aligned} &= \quad \left[1 - \frac{1}{k+1}\right] + \frac{1}{(k+1)(k+2)}\\ &= \quad 1 - \left[\frac{1}{k+1} - \frac{1}{(k+1)(k+2)}\right] \\ &= \quad 1 - \frac{1}{k+2} \\ &= \quad \text{RHS of } \mathbf{P}(k+1). \end{aligned}$$

Hence Ppk ` 1q holds.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

(c) Note: If q " 1, then the LHS is equal to n, but the RHS makes no sense. So we assume q ‰ 1.

Let Ppnq be the statement:

$$q^{14} + q + q^2 + q^3 + \dots + q^{n-1} = \frac{1}{1 - q} - \frac{q^n}{1 - q} \text{ ''.} $$


$$q^{4}1 + q + q^{2} + q^{3} + \cdots + q^{k-1} = \frac{1}{1 - q} - \frac{q^{k}}{1 - q}"\ldots$$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is an equation, so we start with the LHS of Ppk ` 1q and try to simplify it in an appropriate way to get the RHS of Ppk ` 1q:

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{bmatrix} 1+q+q^2+q^3+\cdots+q^k \\ 1+q+q^2+q^3+\cdots+q^{k-1} \end{bmatrix} + q^k. \end{aligned}$$

If we now use Ppkq, which we assume is true, then the first bracket is equal to

$$\frac{1}{1-q} - \frac{q^k}{1-q}$$

so the complete sum is equal to:

$$\begin{aligned} &= \quad \frac{1}{1-q} - \left[ \frac{q^k}{1-q} - q^k \right], \\ &= \quad \frac{1}{1-q} - \frac{q^{k+1}}{1-q} \\ &= \quad \text{RHS of } \mathbf{P}(k+1). \end{aligned}$$

Hence Ppk ` 1q holds.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

(d) Note: The statement to be proved starts with a term involving "0!". The definition

$$n! = 1 \times 2 \times 3 \times \cdots \times n$$

does not immediately tell us how to interpret "0!". The correct interpretation emerges from the fact that several different thoughts all point in the same direction.


Let Ppnq be the statement:

"0 ¨ 0! ` 1 ¨ 1! ` 2 ¨ 2! ` ¨ ¨ ¨ ` pn ´ 1q ¨ pn ´ 1q! " n! ´ 1".


$$2^{\bullet}0 \cdot 0! + 1 \cdot 1! + 2 \cdot 2! + \dots + (k-1) \cdot (k-1)! = k! - 1'' \dots$$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is an equation, so we start with the LHS of Ppk ` 1q and try to simplify it in an appropriate way to get the RHS of Ppk ` 1):

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{bmatrix} 0 \cdot 0! + 1 \cdot 1! + 2 \cdot 2! + \cdots + k \cdot k! \\ \end{bmatrix} \\ &= \begin{bmatrix} 0 \cdot 0! + 1 \cdot 1! + 2 \cdot 2! + \cdots + (k-1) \cdot (k-1)! \end{bmatrix} + k.k!. \end{aligned}$$

If we now use Ppkq, which we assume is true, then the first bracket is equal to k! ´ 1, so the complete sum is equal to:

$$\begin{aligned} &= \quad (k! - 1) + k \cdot k! \\ &= \quad (k+1) \cdot k! - 1 \\ &= \quad (k+1)! - 1 = \text{RHS of } \mathbf{P}(k+1). \end{aligned}$$

Hence Ppk ` 1q holds.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

(e) Let Ppnq be the statement:

"pcos θ ` i sin θq <sup>n</sup> " cos nθ ` i sin nθ "


"pcos θ ` i sin θq <sup>k</sup> " cos kθ ` i sin kθ".

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is an equation, so we start with the LHS of Ppk ` 1q and try to simplify it in an appropriate way to get the RHS of Ppk ` 1q:

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{pmatrix} \cos \theta + i \sin \theta \end{pmatrix}^{k+1} \\ &= \begin{pmatrix} \cos \theta + i \sin \theta \end{pmatrix}^{k} (\cos \theta + i \sin \theta). \end{aligned}$$

If we now use Ppkq, which we assume is true, then the first bracket is equal to pcos kθ ` i sin kθq, so the complete expression is equal to:

$$= \begin{pmatrix} \cos k\theta + i\sin k\theta \end{pmatrix} \begin{pmatrix} \cos \theta + i\sin \theta \end{pmatrix}$$


Hence Ppk ` 1q holds.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

236. Let Ppnq be the statement:

$$1^4(1+2+3+\cdots+n)^2 = 1^3 + 2^3 + 3^3 + \cdots + n^3 \text{ ".} $$


$$1^{\bullet}(1+2+3+\cdots+k)^2 = 1^3 + 2^3 + 3^3 + \cdots + k^{3\bullet}.$$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk`1q is an equation, so we start with one side of Ppk`1q and try to simplify it in an appropriate way to get the other side of Ppk ` 1q. In this instance, the RHS of Ppk ` 1q is the most promising starting point (because we know a formula for the k th triangular number, and so can see how to simplify it):

$$\begin{array}{rcl} \text{RHS of } \mathbf{P}(k+1) &=& 1^3 + 2^3 + 3^3 + \dots + k^3 + (k+1)^3 \\ &=& \left[1^3 + 2^3 + 3^3 + \dots + k^3\right] + (k+1)^3. \end{array}$$

If we now use Ppkq, which we assume is true, then the first bracket is equal to

$$(1+2+3+\cdots+k)^2,$$

so the complete RHS is:

$$\begin{aligned} &=\quad \left(1+2+3+\cdots+k\right)^2+\left(k+1\right)^3\\ &=\quad \left[\frac{k(k+1)}{2}\right]^2+\left(k+1\right)^3\\ &=\quad \frac{1}{4}(k+1)^2\left[k^2+4k+4\right] \\ &=\quad \left[\frac{(k+1)(k+2)}{2}\right]^2\\ &=\quad \left(1+2+3+\cdots+(k+1)\right)^2\\ &=\quad \text{LHS of }\mathbf{P}(k+1). \end{aligned}$$

Hence Ppk ` 1q holds.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

Note: A slightly different way of organizing the proof can sometimes be useful. Denote the two sides of the equation in the statement Ppnq by fpnq and gpnq respectively. Then

• fp1q " 1 <sup>2</sup> " 1 <sup>3</sup> " gp1q; and • simple algebra allows one to check that, for each k ě 1,

$$f(k+1) - f(k) = \left(k+1\right)^3 = g(k+1) - g(k).$$

It then follows (by induction) that fpnq " gpnq for all n ě 1.

#### 237.

(a) T<sup>1</sup> " 1, T<sup>1</sup> ` T<sup>2</sup> " 1 ` 3 " 4, T<sup>1</sup> ` T<sup>2</sup> ` T<sup>3</sup> " 1 ` 3 ` 6 " 10. These may not be very suggestive. But

$$T\_1 + T\_2 + T\_3 + T\_4 = 20 = 5 \times 4,$$

$$T\_1 + T\_2 + T\_3 + T\_4 + T\_5 = 35 = 5 \times 7,$$

and

$$T\_1 + T\_2 + T\_3 + T\_4 + T\_5 + T\_6 = 56 = 7 \times 8$$

may eventually lead one to guess that

$$T\_1 + T\_2 + T\_3 + \dots + T\_n = \frac{n(n+1)(n+2)}{6}.$$

Note 1: This will certainly be easier to guess if you remember what you found in Problem 17 and Problem 63.

Note 2: There is another way to help in such guessing. Suppose you notice that

– adding values for k " 1 up to k " n of a polynomial of degree 0 (such as ppxq " 1) gives an answer that is a "polynomial of degree 1",

$$1 + 1 + \dots + 1 = n,$$

and

– adding values for k " 1 up to k " n of a polynomial of degree 1 (such as ppxq " x) gives an answer that is a "polynomial of degree 2",

$$1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}.$$

Then you might guess that the sum

$$T\_1 + T\_2 + T\_3 + \dots + T\_n$$

will give an answer that is a polynomial of degree 3 in n. Suppose that

$$T\_1 + T\_2 + T\_3 + \cdots + T\_n = An^3 + Bn^2 + Cn + D.$$

We can then use small values of n to set up equations which must be satisfied by A, B, C, D and solve them to find A, B, C, D:

– when n " 0, we get D " 0;


This method assumes that we know that the answer is a polynomial and that we know its degree: it is called "the method of undetermined coefficients". There are various ways of improving the basic method (such as writing the polynomial An<sup>3</sup> ` Bn<sup>2</sup> ` Cn ` D in the form

$$Pn(n-1)(n-2) + Qn(n-1) + Rn + S,$$

which tailors it to the values n " 0, 1, 2, 3 that one intends to substitute).

(b) Let Ppnq be the statement:

"T<sup>1</sup> ` T<sup>2</sup> ` T<sup>3</sup> ` ¨ ¨ ¨ ` T<sup>n</sup> " npn`1qpn`2q 6 ".


$$T\_1 + T\_2 + T\_3 + \dots + T\_k = \frac{k(k+1)(k+2)}{6}.$$

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is an equation, so we start with the LHS of Ppk ` 1q and try to simplify it in an appropriate way to get the RHS of Ppk ` 1q:

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{bmatrix} T\_1 + T\_2 + T\_3 + \cdots + T\_k + T\_{k+1} \\ T\_1 + T\_2 + T\_3 + \cdots + T\_k \end{bmatrix} + T\_{k+1}. \end{aligned}$$

If we now use Ppkq, which we assume is true, then the first bracket is equal to

$$\frac{k(k+1)(k+2)}{6}.$$

so the complete sum is equal to:

$$\begin{aligned} &=\quad \frac{k(k+1)(k+2)}{6} + \frac{(k+1)(k+2)}{2} \\ &=\quad \frac{(k+1)(k+2)(k+3)}{6} \\ &=\quad \text{RHS of } \mathbf{P}(k+1). \end{aligned}$$

Hence Ppk ` 1q holds.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

.

Note: The triangular numbers T1, T2, T3, . . . , Tk, . . . T<sup>n</sup> are also equal to the binomial coefficients ` k`1 2 ˘ . And the sum of these binomial coefficients is another binomial coefficient `<sup>n</sup>`<sup>2</sup> 3 ˘ , so the result in Problem 237 can be written as:

$$
\binom{2}{2} + \binom{3}{2} + \binom{4}{2} + \dots + \binom{n+1}{2} = \binom{n+2}{3} \dots
$$

You might like to interpret Problem 237 in the language of binomial coefficients, and prove it by repeated use of the basic Pascal triangle relation (Pascal (1623–1662)): ˜ ¸ ˜ ¸ ˜ ¸

$$
\binom{k}{r} + \binom{k}{r+1} = \binom{k+1}{r+1}.
$$

$$
\binom{n+2}{3} = \binom{n+1}{2} + \binom{n+1}{3}
$$

Start by rewriting

$$\mathbf{\color{red}{238.}}$$

(a) We know from Problem 237(b) that

$$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n(n+1) = \frac{n(n+1)(n+2)}{3}.$$

Also

$$\begin{aligned} \left(1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + n(n+1) \right) &= \begin{aligned} \left(1 \cdot (1+1) + 2 \cdot (2+1) + 3 \cdot (3+1) \right) &= \left(1 \cdot (2+1) + 2 \cdot (2+1) + 3 \cdot (3+1) \right) \\ &+ \cdots + n \cdot (n+1) \\ &= \left(1^2 + 1 \right) + \left(2^2 + 2 \right) + \left(3^2 + 3 \right) \\ &+ \cdots + \left(n^2 + n \right) \\ &= \left(1^2 + 2^2 + 3^2 + \cdots + n^2 \right) \\ &+ \left(1 + 2 + 3 + \cdots + n \right) . \end{aligned} $$

Therefore

$$\begin{aligned} \left(1^2 + 2^2 + 3^2 + \dots + n^2\right)^2 &= \frac{n(n+1)(n+2)}{3} - \frac{n(n+1)}{2} \\ &= -\frac{n(n+1)(2n+1)}{6} .\end{aligned}$$

(b) Guess:

$$1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + 3 \cdot 4 \cdot 5 + \dots + n(n+1)(n+2) = \frac{n(n+1)(n+2)(n+3)}{4}.$$

The proof by induction is entirely routine, and is left for the reader.

$$\begin{aligned} \begin{array}{rcl} 1\cdot 2\cdot 3 + 2\cdot 3\cdot 4 + \cdots + n(n+1)(n+2) &=& 1\cdot (1+1)(1+2) + 2\cdot (2+1)(2+2) \\ &+ \cdots + n\cdot (n+1)(n+2) \\ &=& (1^3 + 3\cdot 1^2 + 2\cdot 1) \\ &+ (2^3 + 3\cdot 2^2 + 2\cdot 2) \\ &+ \cdots + (n^3 + 3n^2 + 2n) \\ &=& (1^3 + 2^3 + \cdots + n^3) \\ &+ 3(1^2 + 2^2 + \cdots + n^2) \\ &+ 2(1+2+\cdots+n). \end{array} \end{aligned}$$

Therefore

$$\begin{aligned} \left(1^3 + 2^3 + 3^3 + \dots + n^3\right) &= \frac{n(n+1)(n+2)(n+3)}{4} \\ &= 3\left[\frac{n(n+1)(2n+1)}{6}\right] \\ &= n(n+1) \\ &= \left[\frac{n(n+1)}{2}\right]^2 .\end{aligned}$$

#### 239.

(a) Let fpxq be any such polynomial. If fpakq " 0, then we know (by the Remainder Theorem) that fpxq has px ´ akq as a factor. Since the a<sup>k</sup> are all distinct, and fpakq " 0 for each k, 0 ď k ď n ´ 1, we have

$$f(x) = (x - a\_0)(x - a\_1)(x - a\_2) \cdots (x - a\_{n-1}) \cdot g(x)$$

for some polynomial gpxq. And since we are told that fpxq has degree n, gpxq has degree 0, so is a constant. Hence every such polynomial of degree n has the form

C ¨ px ´ a0qpx ´ a1qpx ´ a2q ¨ ¨ ¨ px ´ a<sup>n</sup>´<sup>1</sup>q.

Since fpanq " b, we can substitute to find C:

$$C = \frac{b}{(a\_n - a\_0)(a\_n - a\_1)(a\_n - a\_2)\cdots(a\_n - a\_{n-1})}.$$

(b) Let Ppnq be the statement:

"Given any two labelled sets of n ` 1 real numbers a0, a1, a2, . . . , an, and b0, b1, b2, . . . , bn, where the a<sup>i</sup> are all distinct (but the b<sup>i</sup> need not be), there exists a polynomial f<sup>n</sup> of degree n, such that fnpa0q " b0, fnpa1q " b1, fnpa2q " b2, . . . , fnpanq " bn."

' When n " 0, we may choose f0pxq " b<sup>0</sup> to be the constant polynomial. Hence Pp0q is true.

' Suppose that Ppkq is true for some particular (unspecified) k ě 0; that is, we know that, for this particular k:

"Given any two labelled sets of k ` 1 real numbers a0, a1, a2, . . . , ak, and b0, b1, b2, . . . , bk, where the a<sup>i</sup> are all distinct (but the b<sup>i</sup> need not be), there exists a polynomial f<sup>k</sup> of degree k, such that fkpa0q " b0, fkpa1q " b1, fkpa2q " b2, . . . , fkpakq " bk."

We wish to prove that Ppk ` 1q must then be true.

Now Ppk ` 1q is the statement:

"Given any two labelled sets of pk `1q `1 real numbers a0, a1, . . . , a<sup>k</sup>`<sup>1</sup>, and b0, b1, b2, . . . , b<sup>k</sup>`<sup>1</sup>, where the a<sup>i</sup> are all distinct (but the b<sup>i</sup> need not be), there exists a polynomial f<sup>k</sup>`<sup>1</sup> of degree k ` 1, such that

$$f\_{k+1}(a\_0) = b\_0, \ f\_{k+1}(a\_1) = b\_1, \ f\_{k+1}(a\_2) = b\_2, \ \dots, \ f\_{k+1}(a\_{k+1}) = b\_{k+1}. "$$

So to prove that Ppk ` 1q holds, we must start by considering

any two labelled sets of pk ` 1q ` 1 real numbers a0, a1, a2, . . . , a<sup>k</sup>`<sup>1</sup>, and b0, b1, b2, . . . , b<sup>k</sup>`<sup>1</sup>, where the a<sup>i</sup> are all distinct (but the b<sup>i</sup> need not be).

We must then somehow construct a polynomial function f<sup>k</sup>`<sup>1</sup> of degree k ` 1 with the required property.

Because we are supposing that Ppkq is known to be true, we can focus on the first k ` 1 numbers in each of the two lists – a0, a1, a2, . . . , ak, and b0, b1, b2, . . . , b<sup>k</sup> – and we can then be sure that there is a polynomial f<sup>k</sup> of degree k such that

$$f\_k(a\_0) = b\_0, f\_k(a\_1) = b\_1, f\_k(a\_2) = b\_2, \dots, f\_k(a\_k) = b\_k.$$

The next step is slightly indirect: we make use of the polynomial f<sup>k</sup>`<sup>1</sup> which we are still trying to construct, and focus on the polynomial

$$f(x) = f\_{k+1}(x) - f\_k(x)$$

satisfying

$$f(a\_0) = f(a\_1) = \dots = f(a\_k) = 0,\\ f(a\_{k+1}) = b\_{k+1} - f\_k(a\_{k+1}) = b \quad \text{(say)}.$$

Part (a) guarantees the existence of such a polynomial fpxq of degree k ` 1 and tells us exactly what this polynomial function fpxq is equal to. Hence we can construct the required polynomial f<sup>k</sup>`<sup>1</sup>px) by setting it equal to fpxq ` fkpxq, which proves that Ppk ` 1q is true.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

#### 240.

(a) 5 cents cannot be made; 6 " 3 ` 3; 7 " 3 ` 4; 8 " 4 ` 4; 9 " 3 ` 3 ` 3; etc.

Guess: Every amount ą N " 5 can be paid directly (without receiving change).

(b) Let Ppnq be the statement:

"n cents can be paid directly (without change) using 3 cent and 4 cent coins".


To prove Ppk ` 1q we consider the statement Ppk ` 1q:

"k ` 1 cents can be paid directly".

We know that Ppkq is true, so we know that "k cents can be paid directly".

– If a direct method of paying k cents involves at least one 3 cent coin, then we can replace one 3 cent coin by a 4 cent coin to produce a way of paying k ` 1 cents.

Hence we only need to worry about a situation in which the only way to pay k cents directly involves no 3 cent coins at all – that is, paying k cents uses only 4 cent coins. But then there must be at least two 4 cent coins (since k ě 6), and we can replace two 4 cent coins by three 3 cent coins to produce a way of paying k ` 1 cents directly. Hence


#### 241.

(a) z <sup>2</sup> ´ z ` 1 " 0, so z " 1˘ ? ´3 2 (these are the two primitive sixth roots of unity). 6 z <sup>2</sup> " ´1˘ ? ´3 2 (the two primitive cube toots of unity), and

$$z^2 + \frac{1}{z^2} = -1.$$


As soon as one starts calculating z 2 and <sup>1</sup> <sup>z</sup><sup>2</sup> , it becomes clear that it is time to p-a-u-s-e and think. ˆ ˙2 ˆ ˙

$$
\left(z + \frac{1}{z}\right)^2 = \left(z^2 + \frac{1}{z^2}\right) + 2,
$$

so whenever z ` 1 <sup>z</sup> " k is an integer,

$$z^2 + \left(\frac{1}{z}\right)^2 = k^2 - 2$$

is also an integer.

(e) Let Ppnq be the statement:

"if z has the property that z ` 1 z is an integer, then z <sup>m</sup> ` 1 <sup>z</sup><sup>m</sup> is also an integer for all m, 0 ď m ď n".


"if z has the property that z ` 1 z is an integer, then z <sup>m</sup> ` 1 <sup>z</sup><sup>m</sup> is also an integer for all m, 0 ď m ď k".

We wish to prove that Ppk ` 1q must then be true.

If z ` 1 z is an integer, then, by Ppkq,

> "z <sup>m</sup> ` 1 <sup>z</sup><sup>m</sup> is also an integer for all m, 0 ď m ď k".

So to prove that Ppk ` 1q holds, we only need to show that

"z <sup>k</sup>`<sup>1</sup> ` 1 zk`1 is an integer".

By the Binomial Theorem:

$$\begin{aligned} \left(z+\frac{1}{z}\right)^{k+1} &= \quad \left(z^{k+1}+\frac{1}{z^{k+1}}\right)+\binom{k+1}{1}\left(z^{k-1}+\frac{1}{z^{k-1}}\right) \\ &+\binom{k+1}{2}\left(z^{k-3}+\frac{1}{z^{k-3}}\right)+\cdots \end{aligned}$$

The LHS is an integer (since z ` 1 z is an integer), and (by Ppkq) every term on the RHS is an integer except possibly the first. Hence the first term is the difference of two integers, so must also be an integer.

Hence Ppk ` 1q is true.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

Note: If <sup>k</sup> ` <sup>1</sup> " <sup>2</sup><sup>m</sup> is even, the expansion of ` z ` 1 z ˘<sup>k</sup>`<sup>1</sup> has an odd number of terms, so the RHS of the above re-grouped expansion ends with the term ` 2m m ˘ ¨ z <sup>m</sup> ¨ ` 1 z ˘m , which is also an integer.

#### 242.

Note: In the solution to Problem 241 we included the condition on z as part of the statement Ppnq.

In Problem 242 the result to be proved has a similar background hypothesis – "Let p be a prime number". It may make the induction clearer if, as in the statement of the Problem, this hypothesis is stated before starting the induction proof.

Let p be any prime number. We let Ppnq be the statement:

"n <sup>p</sup> ´ n is divisible by p ".


"k <sup>p</sup> ´ k is divisible by p".

We wish to prove Ppk ` 1q – that is,

"pk ` 1q <sup>p</sup> ´ pk ` 1q is divisible by p"

must then be true. Using the Binomial Theorem again we see that

$$(k+1)^p - (k+1) \quad = \left[k^p + \binom{p}{p-1}k^{p-1} + \binom{p}{p-2}k^{p-2} + \dots + \binom{p}{1}k + 1\right]$$

$$-(k+1)$$

$$= \left(k^p - k\right) + \left[\binom{p}{p-1}k^{p-1} + \binom{p}{p-2}k^{p-2} + \dots + \binom{p}{1}k\right].$$

By Ppkq, the first bracket on the RHS is divisible by p; and in each of the other terms each of the binomial coefficients ` p r ˘ , 0 ă r ă p,


Hence each term in the second bracket is a multiple of p. So the RHS (and hence the LHS) is divisible by p.

Hence Ppk ` 1q is true.

If we combine these two bullet points, we have proved that "Ppnq holds for all n ě 1". QED

#### 243.

$$10.037037037\dots \text{ (for ever) } = \frac{37}{1000} + \frac{37}{1000\,000} + \frac{37}{1000\,000\,000} + \dotsb \text{ (for ever) } \dotsb$$

This is a geometric series with first term a " 37 <sup>1000</sup> and common ratio r " 1 <sup>1000</sup> , and so has sum a 1 ´ r " 37 999 " 1 27 .

#### 244.

(a) Each person receives in total:

$$\frac{1}{4} + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^3 + \left(\frac{1}{4}\right)^4 + \dots \text{ (for every) } = \frac{1}{3}$$

(here a " 1 <sup>4</sup> " r). (b) You have

$$1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \dotsb \text{ (for every } = \frac{2}{3}\text{)}$$
 (here  $a = 1$ ,  $r = -\frac{1}{2}$ ); I have 
$$\frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \dotsb \text{ (for every } = \frac{1}{3}$$
 (here  $a = \frac{1}{2}$ ,  $r = -\frac{1}{2}$ ).

245. The trains are 120 km apart, and the fly travels at 50 km/h towards Train B, which is initially 120 km away and travelling at 30 km/h.

The relative speed of the fly and Train B is 80 km/h, so it takes <sup>3</sup> hours before they meet. In this time Train A and Train B have each travelled 45 km, so they are now 30 km apart. The fly then turns right round and flies back to Train A.

The relative speed of the fly and Train A is then also 80 km/h, so it takes just hours (or 22.5 minutes) for the fly to return to Train A. Train A and Train B have each travelled <sup>45</sup> km in this time, so they are now <sup>30</sup> km apart. The fly then turns round and flies straight back to Train B.

Train B is <sup>30</sup> km away and the relative speed of the fly and Train B is again 80 km/h, so the journey takes <sup>3</sup> hours (or 5.625 minutes).

Continuing in this way, we see that the fly takes

$$\frac{3}{2} + \frac{3}{8} + \frac{3}{32} + \frac{3}{128} + \dotsb \text{ (for every) } = 2 \text{ hours.}$$

Hence the fly travels 100 km before being squashed.

Note: The two trains are approaching each other at 60 km/h, so they crash in exactly 2 hours – during which time the fly travels 100 km.

#### 246.

(a)(i) 3 ą 2 ; therefore

$$\frac{1}{3^2} < \frac{1}{2^2},$$
 so 
$$\frac{1}{2^2} + \frac{1}{3^2} < \frac{2}{2^2} = \frac{1}{2}.$$
 $\pi^2 > c^2 > \pi^2 \quad c^2 > c^2 > \frac{1}{c}$ 

(ii) 7 ą 6 ą 5 ą 4 ; therefore

> ă ă ă , ` ` ` ă " .

so

(b) The argument in part (a) gives an upper bound for each bracketed expression in the sum

$$
\left(\frac{1}{1^2}\right) + \left(\frac{1}{2^2} + \frac{1}{3^2}\right) + \left(\frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \frac{1}{7^2}\right) + \left(\frac{1}{8^2} + \dots + \frac{1}{15^2}\right) + \dotsb
$$

Replacing each bracket by its upper bound, we see that the sum is

$$\begin{aligned} &< \quad \frac{1}{1^2} + \frac{2}{2^2} + \frac{4}{4^2} + \frac{8}{8^2} + \cdots \\ &= \quad 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \text{ (for every)} \\ &= \quad 2. \end{aligned}$$

(c) The finite partial sums

$$S\_n = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2}$$


$$1 = S\_1 < S\_2 < S\_3 < \cdots < S\_n < S\_{n+1} < \cdots < 2.1$$

It follows that there is some (unknown) number S ď 2 to which the partial sums converge as n Ñ 8, and we take this (unknown) exact value S to be the exact value of the endless sum

$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} + \dots \text{ (for every)}$$

To see, for example, that S ą , notice that

$$\begin{aligned} S\_{21} &> \quad S\_4\\ &= \quad \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} \\ &= \quad 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} \\ &> \quad \frac{17}{12}. \end{aligned}$$

Note 1: The claim that

"an increasing sequence of partial sums Sn, all less than 2, must converge to some number S ď 2"

is a fundamental property of the real numbers – called completeness.

Note 2: Just as one can obtain better and better lower bounds for S – like " ă S", so one can improve the upper bound "S ă 2". For example, if in part (b) we avoid replacing the third term <sup>1</sup> by <sup>1</sup> , we get a better upper bound "S ă ", which is <sup>5</sup> less than 2.

247.

(a) Let Ppnq be the statement:

" 1 <sup>1</sup><sup>2</sup> ` 1 <sup>2</sup><sup>2</sup> ` 1 <sup>3</sup><sup>2</sup> ` ¨ ¨ ¨ ` <sup>1</sup> <sup>n</sup><sup>2</sup> ď 2 ´ 1 n ".


$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \quad \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{k^2} + \frac{1}{(k+1)^2} \\ &= \quad \left[ \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{k^2} \right] + \frac{1}{(k+1)^2} \\ &\lesssim \quad \left[ 2 - \frac{1}{k} \right] + \frac{1}{(k+1)^2} \\ &= \quad 2 - \left[ \frac{1}{k} - \frac{1}{(k+1)^2} \right] \\ &< \quad 2 - \frac{1}{k+1} .\end{aligned}$$

Hence Ppk ` 1q holds.

6 Pp1q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true. 6 Ppnq is true, for all n ě 1. QED

(b) Let Ppnq be the statement:

$$\begin{array}{cccc} \text{``} \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{n^2} & \sim 1.68 - \frac{1}{n} \text{''} .\\ \text{• Then} & & & , \text{ } \dots \text{ } \dots \text{ } \dots \text{ } \dots \end{array}$$

$$\text{LHS of } \mathbf{P(4)} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} = 1.423611111\dots \dots \text{ , }$$

and RHS of Pp4q " 1.43. Hence Pp4q is true.

' Suppose we know that Ppkq is true for some k ě 4. We want to prove that Ppk ` 1q holds.

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \quad \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{k^2} + \frac{1}{(k+1)^2} \\ &= \quad \left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots + \frac{1}{k^2}\right) + \frac{1}{(k+1)^2} \\ &< \quad \left[1.68 - \frac{1}{k}\right] + \frac{1}{(k+1)^2} \\ &= \quad 1.68 - \left[\frac{1}{k} - \frac{1}{(k+1)^2}\right] \\ &< \quad 1.68 - \frac{1}{k+1} .\end{aligned}$$

Hence Ppk ` 1q holds.

6 Pp1q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true.

6 Ppnq is true, for all n ě 1. QED

#### 248.

(a)(i) n! " n ˆ pn ´ 1q ˆ pn ´ 2q ˆ ¨ ¨ ¨ ˆ 3 ˆ 2 ˆ 1 ě 2 ˆ 2 ˆ 2 ˆ ¨ ¨ ¨ ˆ 2 ˆ 1 " 2 n´1 whenever n ě 1. 6 1 <sup>n</sup>! ď ` 1 2 ˘<sup>n</sup>´<sup>1</sup> for all n ě 1. 6 1 0! ` 1 1! ` 1 2! `¨ ¨ ¨` <sup>1</sup> <sup>n</sup>! ď 1` " 1 ` 1 <sup>2</sup> ` ` 1 2 ˘<sup>2</sup> ` ¨ ¨ ¨ ` ` 1 2 ˘<sup>n</sup>´<sup>1</sup> ı ă 3 for all n ě 0. (ii) As we go on adding more and more terms, each finite sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!}$$

is bigger than the previous sum. Since every finite sum

$$
\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} < 3,
$$

the sums increase, but never reach 3, so they accumulate closer and closer to a value "e" ď 3. Moreover, this value "e" is certainly larger than the sum of the first two terms <sup>1</sup> 0! ` 1 1! " 2, so 2 ă e ď 3.

$$\text{(b)} \text{(i)} \text{ Let } \mathbf{P}(n) \text{ be the statement:}$$

"

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} \lesssim 3 - \frac{1}{n \cdot n!}"\text{.}$$


$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{array}{c} \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{(k+1)!} \\ &= \begin{bmatrix} \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{k!} \end{bmatrix} + \frac{1}{(k+1)!} \\ &\leqslant \ 3 - \frac{1}{k \cdot k!} + \frac{1}{(k+1)!} \\ &= \ 3 - \frac{1}{k(k+1)!} \\ &< \ 3 - \frac{1}{(k+1) \cdot (k+1)!} \end{aligned}$$

Hence Ppk ` 1q holds.

6 Pp1q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true. 6 Ppnq is true, for all n ě 1. QED

(ii) [The reasoning here uses the constant "3" while ignoring the refinement "3 ´ 1 n.n! ", and so sounds exactly like part (a)(ii).] As we add more terms, each finite sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!}$$

is bigger than the previous sum. Since every finite sum

$$
\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} < 3,
$$

the partial sums increase, but never reach 3; so they accumulate closer and closer to a value "e" ď 3. Moreover, this value "e" is certainly larger than the sum of the first three terms <sup>1</sup> 0! ` 1 1! ` 1 2! " 2.5, so 2.5 ă e ď 3.

(c) Note: Examine carefully the role played by the number "3" in the above induction proof in (b)(ii). It is needed precisely to validate the statement Pp1q: since <sup>1</sup> 0! ` 1 1! " 3´ 1 <sup>1</sup>ˆ1! ". But the number "3" plays no active part in the second induction step, and could be replaced by any other number we choose. The exact value "e" of the infinite series is not really affected by what happens

when n " 1. Suppose we ask: "What number C<sup>2</sup> should replace "3" if we only want to prove that

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} \le C\_2 - \frac{1}{n \cdot n!}, \text{ for all } n \gg 2?$$

The only part of the induction proof where C<sup>2</sup> then matters is when we try to check that Pp2q holds; so we must choose the smallest possible C<sup>2</sup> to satisfy

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} \leqslant C\_2 - \frac{1}{2.2!} : $$

that is, C<sup>2</sup> " 2.75.

(i) Let Ppnq be the statement:

" 1 0! ` 1 1! ` 1 2! ` ¨ ¨ ¨ ` <sup>1</sup> <sup>n</sup>! ď 2.75 ´ 1 n¨n! ".


$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \quad \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{(k+1)!} \\ &= \quad \left[ \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{k!} \right] + \frac{1}{(k+1)!} \\ &\lesssim \quad 2.75 - \frac{1}{k \cdot k!} + \frac{1}{(k+1)!} \\ &= \quad 2.75 - \frac{1}{k(k+1)!} \\ &< \quad 2.75 - \frac{1}{(k+1)(k+1)!} \end{aligned}$$

Hence Ppk ` 1q holds.

6 Pp2q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true. 6 Ppnq is true, for all n ě 2. QED (ii) As we add more terms, each finite sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!}$$

is bigger than the previous sum. Since every finite sum

$$
\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} < 2.75,
$$

the finite sums increase, but never reach 2.75, so they accumulate closer and closer to a value "e" ď 2.75. Moreover, this value "e" is certainly larger than the sum of the first four terms

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} > 2.66,$$

so 2.66 ă e ď 2.75.

(d)(i) Let Ppnq be the statement:


$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{array}{c} 1 \\ 0 \end{array} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{(k+1)!} \\ &= \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{k!} \end{aligned} + \frac{1}{(k+1)!} \\ &\quad \leqslant \quad 2.7222\dots \text{(for every)} - \frac{1}{k \cdot k!} + \frac{1}{(k+1)!} \\ &= \quad 2.7222\dots \text{(for every)} - \frac{1}{k(k+1)!} \\ &< \quad 2.7222\dots \text{(for every)} - \frac{1}{(k+1)(k+1)!} \end{aligned}$$

Hence Ppk ` 1q holds.

6 Pp3q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true. 6 Ppnq is true, for all n ě 3. QED

(ii) As we add more terms, each finite sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!}$$

is bigger than the previous sum.

Since every finite sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} < 2.7222\dots \text{ (for every)},$$

the finite sums increase, but never reach 2.7222 ¨ ¨ ¨ (for ever), so they accumulate closer and closer to a value "e" ď 2.7222 ¨ ¨ ¨ (for ever). Moreover, this value "e" is certainly larger than the sum of the first five terms

$$
\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} > 2.708,
$$

so 2.708 ă e ď 2.7222 ¨ ¨ ¨ (for ever).

Note: This process of refinement can continue indefinitely. But we only have to go one further step to pin down the value of "e" with surprising accuracy. The next step uses the same proof to show that

 $2^{\frac{n}{11}} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} \lesssim 2.7185 - \frac{1}{n \cdot n!}, \text{ for all } n \gg 4^{\text{" $\dots$ }}$ 

and to conclude that the endless sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots \text{ (for every)}$$

has a definite value "e" that lies somewhere between 2.716 and 2.71875. We could then repeat the same proof to show that

$$\frac{1}{n!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} \leqslant 2.718333\cdots \text{ (for every)} - \frac{1}{n \cdot n!}, \text{ for all } n \gg 5, \dots$$

and use the lower bound 2.7177 . . . from the first seven terms to conclude that the endless sum

$$\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \dots + \frac{1}{n!} + \dots \text{ (for every)}$$

has a definite value "e" that lies somewhere between 2.7177 and 2.718333 ¨ ¨ ¨ (for ever). And so on.

249. Let Ppnq be the statement:

$$\sqrt[\text{``}\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{n}} \gg \sqrt{n}\text{''} \dots$$

• LHS of Pp1q " 1 " RHS of Pp1q. Hence Pp1q is true.

• Suppose we know that Ppkq is true for some k ě 1. We want to prove that Ppk ` 1q holds.

$$\begin{array}{rcl} \text{LHS of } \mathbf{P}(k+1) &=& \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{k+1}} \\ &=& \left(\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots + \frac{1}{\sqrt{k}}\right) + \frac{1}{\sqrt{(k+1)}} \\ &\gg& \sqrt{k} + \frac{1}{\sqrt{k+1}} \\ &\gg& \sqrt{k+1} \quad \left(\text{since } \frac{1}{\sqrt{k+1}} \gg \frac{1}{\sqrt{k+1} + \sqrt{k}}\right). \end{array}$$

Hence Ppk ` 1q holds.

6 Pp1q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true. 6 Ppnq is true, for all n ě 1. QED

250. Let a, b be real numbers such that a ‰ b, and a ` b ą 0.

One of a, b is then the greater, and we may suppose this is a – so that a ą b. If a ą b ą 0, then a <sup>n</sup> ą b <sup>n</sup> ą 0 for all n; if b ă 0, then a ` b ą 0 implies that a " |a| ą |b|, so a <sup>n</sup> ą b n for all n.

Let Ppnq be the statement:

$$\frac{\omega a^n + b^n}{2} \gg \left(\frac{a+b}{2}\right)^n, \dots$$


$$\begin{aligned} \text{RHS of } \mathbf{P}(k+1) &= \quad \left(\frac{a+b}{2}\right)^{k+1} \\ &= \quad \frac{a+b}{2} \cdot \left(\frac{a+b}{2}\right)^k \\ &\lesssim \quad \frac{a+b}{2} \cdot \frac{a^k + b^k}{2} \quad \text{(by } \mathbf{P}(k)) \\ &= \quad \frac{a^{k+1} + b^{k+1}}{4} + \frac{ab^k + ba^k}{4} \\ &< \quad \frac{a^{k+1} + b^{k+1}}{2} \quad \text{(since } (a^k - b^k)(a - b) > 0). \end{aligned}$$

Hence Ppk ` 1q holds.

6 Pp1q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true.

6 Ppnq is true, for all n ě 1. QED

251. Let x be any real number ě ´1. If x " ´1, then p1 ` xq <sup>n</sup> " 0 ě 1 ´ n " 1 ` nx, for all n ě 1. Thus we may assume that x ą ´1, so 1 ` x ą 0. Let Ppnq be the statement: "p1 ` xq <sup>n</sup> ě 1 ` nx".


$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{pmatrix} 1+x \end{pmatrix}^{k+1} \\ &= \begin{pmatrix} 1+x \end{pmatrix} \cdot \begin{pmatrix} 1+x \end{pmatrix}^{k} \\ &\geqslant \begin{aligned} \begin{pmatrix} 1+x \end{pmatrix} \cdot \begin{pmatrix} 1+kx \end{pmatrix} & \begin{Bmatrix} \mathbf{y} \ \mathbf{P}(k), \ \text{since } 1+x > 0 \end{Bmatrix} \\ &= \begin{array}{rcl} 1+(k+1)x+kx^2 \\ \end{array} \\ &\geqslant \begin{array}{rcl} 1+(k+1)x \end{array} \end{aligned}$$

Hence Ppk ` 1q holds.

6 Pp1q holds; and whenever Ppkq is known to be true, Ppk ` 1q is also true. 6 Ppnq is true, for all n ě 1. QED

252. The problem is discussed after the statement of Problem 252 in the main text.

#### 253.

	- (iii) Let Ppnq be the statement:
		- " 1 <sup>1</sup> ` 1 <sup>2</sup> ` 1 <sup>3</sup> ` ¨ ¨ ¨ ` <sup>1</sup> <sup>2</sup>n´<sup>1</sup> ă n". Then
		- ' Pp2q is true by (i), since

$$
\frac{1}{1} + \frac{1}{2} + \frac{1}{3} < 1 + \left(\frac{1}{2} + \frac{1}{2}\right) = 2.
$$

' Suppose that Ppkq is true for some k ě 2.

$$\begin{aligned} \text{LHS of } \mathbf{P}(k+1) &= \begin{bmatrix} \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2^k - 1} \end{bmatrix} \\ &+ \begin{bmatrix} \frac{1}{2^k} + \dots + \frac{1}{2^{k+1} - 1} \end{bmatrix}. \end{aligned}$$

The first bracket is ă k (by Ppkq); and each of the 2<sup>k</sup> terms in the second bracket is ď 1 <sup>2</sup><sup>k</sup> , so the whole bracket is ď 1.

Hence the LHS of Ppk ` 1q ă k ` 1, so Ppk ` 1q is true.

Hence Ppnq is true for all n ě 2.

	- (ii) 5, 6, 7 ă 8; hence <sup>1</sup> 5 , 1 6 , 1 7 are all ą 1 8 . 6 1 <sup>5</sup> ` 1 <sup>6</sup> ` 1 <sup>7</sup> ` 1 <sup>8</sup> ą 1 <sup>8</sup> ` 1 <sup>8</sup> ` 1 <sup>8</sup> ` 1 <sup>8</sup> " 1 2 .

.

(iii) Let Ppnq be the statement:

" 1 <sup>1</sup> ` 1 <sup>2</sup> ` 1 <sup>3</sup> ` ¨ ¨ ¨ ` <sup>1</sup> <sup>2</sup><sup>n</sup> ą 1 ` n 2 ".

Then

' Pp2q is true by (i), since

$$\begin{aligned} \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} &= -1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right) \\ &> -1 + \frac{1}{2} + \left(\frac{1}{4} + \frac{1}{4}\right) \\ &= -1 + 2 \times \frac{1}{2} .\end{aligned}$$

' Suppose that Ppkq is true for some k ě 2. LHS of <sup>P</sup>p<sup>k</sup> ` <sup>1</sup>q " " 1 <sup>1</sup> ` 1 <sup>2</sup> ` 1 <sup>3</sup> ` ¨ ¨ ¨ ` <sup>1</sup> 2k ‰ ` " 1 <sup>2</sup>k`<sup>1</sup> ` ¨ ¨ ¨ ` <sup>1</sup> 2k`1 ı . The first bracket is ą 1 ` k 2 (by Ppkq); and each of the 2<sup>k</sup> terms in the second bracket is ě 1 2k`1 , so the whole bracket is ě 1 2 . Hence the LHS of Ppk ` 1q ą 1 ` k`1 2 , so Ppk ` 1q is true. Hence Ppnq is true for all n ě 2.

#### 254.

(a) We use induction. Let Ppnq be the statement:

"n identical rectangular strips of length 2 balance exactly on the edge of a table if the successive protrusion distances (first beyond the edge of the table, then beyond the leading edge of the strip immediately below, and so on) are the terms

$$\frac{1}{n}, \frac{1}{n-1}, \frac{1}{n-2}, \dots, \frac{1}{3}, \frac{1}{2}, \frac{1}{1}$$

of the finite harmonic series in reverse order."


Let k ` 1 identical strips be arranged as described in the statement Ppk ` 1q. The statement Ppkq guarantees that the top k strips would exactly balance if the leading edge of the bottom strip were in fact the edge of the table; hence the combined centre of gravity of the top k strips is positioned exactly over the leading edge of the bottom strip.

Let M be the mass of each strip; since the leading edge of the bottom strip is distance <sup>1</sup> k`1 beyond the edge of the table, the top k strips produce a combined moment about the edge of the table equal to kM ˆ 1 .

k`1 The centre of gravity of the bottom strip is distance 1 ´ 1 <sup>k</sup>`<sup>1</sup> " k k`1 from the edge of the table in the opposite direction; hence it contributes a moment about the edge of the table equal to M ˆ ´ ´ k k`1 ¯ .

6 the total moment of the whole stack about the edge of the table is equal to zero, so the centre of gravity of the combined stack of k ` 1 strips lies exactly over the edge of the table. Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 1.

(b) Problem 253(b)(iii) now guarantees that a stack of 2<sup>n</sup> strips can protrude a distance ą 1 ` n 2 beyond the edge of the table.

Note: Ivars Petersen's Mathematical Tourist blog contains an entry in 2009

http://mathtourist.blogspot.com/2009/01/overhang.html

which explores how one can obtain large overhangs with fewer strips if one is allowed to use strips to counterbalance those that protrude beyond the edge of the table.

### 255.

(a)(i) Let Ppnq be the statement:

"s2p´<sup>2</sup> ă s2<sup>p</sup> ă s2q`<sup>1</sup> ă s2q´<sup>1</sup> for all p, q such that 1 ď p, q ď n".

' Pp1q is true (since s<sup>0</sup> is the empty sum, so

$$0 = s\_0 < s\_2 = \frac{1}{2} < s\_3 = \frac{5}{6} < s\_1 = 1.$$

' Suppose that Ppkq is true for some k ě 1. Then most of the inequalities in the statement Ppk`1q are part of the statement Ppkq; the only outstanding inequalities which remain to be proved are:

$$s\_{2k} \prec s\_{2k+2} \prec s\_{2k+3} \prec s\_{2k+1}.$$

which are true, since

$$s\_{2k+3} = s\_{2k+2} + \frac{1}{2k+3} = s\_{2k+1} - \frac{1}{2k+2} + \frac{1}{2k+3}$$

and

$$s\_{2k+2} = s\_{2k} + \frac{1}{2k+1} - \frac{1}{2k+2}.$$

Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 2.


256. Let Ppnq be the statement:

"f<sup>k</sup> has at least k distinct prime factors".


Then f<sup>k</sup>`<sup>1</sup> " fkpf<sup>k</sup> `1q. The first factor f<sup>k</sup> has at least k distinct prime factors. And the second factor f<sup>k</sup> ` 1 ą f<sup>k</sup> ą 1, so has at least one prime factor. Moreover HCFpfk, f<sup>k</sup> ` 1q " 1, so the second bracket has no factor in common with fk. Hence f<sup>k</sup>`<sup>1</sup> has at least k ` 1 distinct prime factors, so Ppk ` 1q is true.

Hence Ppnq is true for all n ě 1.

Note: This problem [suggested by Serkan Dogan] gives a different proof of the result (Problem 76(d)) that the list of prime numbers goes on for ever.

#### 257.

	- ' 0 points leave the line in pristine condition namely a single interval so Pp0q is true.

' Suppose that Ppkq is true for some k ě 0. Consider an arbitrary straight line divided by k ` 1 points A0, A1, . . . , Ak. Then the k points A1, . . . , A<sup>k</sup> divide the line into k ` 1 intervals (by Ppkq). The additional point A<sup>0</sup> is distinct from A1, . . . , A<sup>k</sup> and so must lie inside one of these k ` 1 intervals, and divides it in two – giving pk ` 1q ` 1 " k ` 2 intervals altogether. Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 0.

(b) (i) We want a function R satisfying

$$R\_0 = 1, \ R\_1 = 2, \ R\_2 = 4, \ R\_3 = 7.$$

If we notice that in part (a)

$$n+1 = 1 + \binom{n}{1},$$

then we might guess that

$$R\_n = 1 + \binom{n}{1} + \binom{n}{2}.$$

(ii) Let Ppnq be the statement:

"n distinct straight lines in the plane divide the plane into at most

$$f(n) = 1 + \binom{n}{1} + \binom{n}{2}$$

regions".

' 0 lines leave the plane in pristine condition – namely a single region – so Pp0q is true, provided that ˜ ¸ ˜ ¸

$$1 + \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 2 \end{pmatrix} = 1.$$

' Suppose that Ppkq is true for some k ě 0.

Consider the plane divided by k ` 1 straight lines m0, m1, . . . , mk. Then the k lines m1, . . . , m<sup>k</sup> divide the plane into at most

$$R\_k = 1 + \binom{k}{1} + \binom{k}{2}$$

regions (by Ppkq).

The additional line m<sup>0</sup> is distinct from m1, . . . , m<sup>k</sup> and so meets each of these lines in at most a single point – giving at most k points on the line m0. These points divide m<sup>0</sup> into at most k ` 1 intervals, and each of these intervals corresponds to a cut-line, where the line m<sup>0</sup> cuts one of the regions created by the lines m1, m2, . . . , m<sup>k</sup> into two pieces – giving at most

$$\begin{aligned} R\_k + (k+1) &= -1 + \binom{k}{1} + \binom{k}{2} + k + 1\\ &= -1 + \binom{k+1}{1} + \binom{k+1}{2} \\ &= -R\_{k+1} \end{aligned}$$

regions altogether. Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 0.

(c) (i) We want a function S satisfying

$$S\_0 = 1, \ S\_1 = 2, \ S\_2 = 4, \ S\_3 = 8, \ S\_4 = 15, \ \dots$$

After thinking about the differences between successive terms in part (b), we might guess that ˜ ¸ ˜ ¸ ˜ ¸ ˜ ¸

$$S\_n = \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \binom{n}{3}.$$

(ii) Let Ppnq be the statement:

"n distinct planes in 3-space divide space into at most

$$S\_n = \binom{n}{0} + \binom{n}{1} + \binom{n}{2} + \binom{n}{3}$$

regions".

' 0 planes leave our 3D space in pristine condition – namely a single region – so Pp0q is true – provided that

$$
\begin{pmatrix} 0 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} + \begin{pmatrix} 0 \\ 2 \end{pmatrix} + \begin{pmatrix} 0 \\ 3 \end{pmatrix} = 1.
$$

' Suppose that Ppkq is true for some k ě 0. Consider 3D divided by k ` 1 planes m0, m1, . . . , mk. Then the k planes m1, . . . , m<sup>k</sup> divide 3D into at most

$$S\_k = \binom{k}{0} + \binom{k}{1} + \binom{k}{2} + \binom{k}{3}$$

regions (by Ppkq).

The additional plane m<sup>0</sup> is distinct from m1, . . . , m<sup>k</sup> and so meets each of these planes in (at most) a line – giving rise to at most k lines on the plane m0. This arrangement of lines on the plane m<sup>0</sup> divides m<sup>0</sup> into at most

$$R\_k = 1 + \binom{k}{1} + \binom{k}{2}$$

regions, and each of these regions on the plane m<sup>0</sup> is the "cut" where the plane m<sup>0</sup> cuts an existing region into two pieces – giving rise to at most

$$\begin{aligned} \label{eq:1} S\_k + R\_k &=& \left[ \begin{pmatrix} k \\ 0 \end{pmatrix} + \begin{pmatrix} k \\ 1 \end{pmatrix} + \begin{pmatrix} k \\ 2 \end{pmatrix} + \begin{pmatrix} k \\ 3 \end{pmatrix} \right] + \left[ 1 + \begin{pmatrix} k \\ 1 \end{pmatrix} + \begin{pmatrix} k \\ 2 \end{pmatrix} \right] \\ &=& \binom{k+1}{0} + \binom{k+1}{1} + \binom{k+1}{2} + \binom{k+1}{3} \\ &=& S\_{k+1} \end{aligned}$$

regions altogether. (There is no need for any algebra here: one only needs to use the Pascal triangle condition.)

Hence Ppk ` 1q is true whenever Ppkq is true.

Hence Ppnq is true for all n ě 0.

258. Notice that, given a dissection of a square into k squares, we can always cut one square into four quarters (by lines through the centre, and parallel to the sides), and so create a dissection with k ` 3 squares.

Let Ppnq be the statement:

"Any given square can be cut into m (not necessarily congruent) smaller squares, for each m, 6 ď m ď n".

• Let n " 6. Given any square of side s (say). We may cut a square of side <sup>2</sup><sup>s</sup> 3 from one corner, leaving an L-shaped strip of width <sup>s</sup> 3 , which we can then cut into 5 smaller squares, each of side <sup>s</sup> 3 . Hence Pp6q is true.

Let n " 7. Given any square, we can divide the square first into four quarters; then divide one of these smaller squares into four quarters to obtain a dissection into 7 smaller squares. Hence Pp7q is true.

Let n " 8. Given a square of side s (say). We may cut a square of side <sup>3</sup><sup>s</sup> 4 from one corner, leaving an L-shaped strip of width <sup>s</sup> 4 , which we can then cut into 7 smaller squares, each of side <sup>s</sup> 4 . Hence Pp8q is true.

• Suppose that Ppkq is true for some k ě 8. Then k ´ 2 ě 6, so any given square can be dissected into k ´ 2 smaller squares (by Ppkq). Taking this dissection and dividing one of the smaller squares into four quarters gives a dissection of the initial square into k´2`3 squares. Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 6.

259. Let Ppnq be the statement:

"Any tree with n vertices has exactly n ´ 1 edges".


Consider an arbitrary tree T with k ` 1 vertices.

[Idea: We need to find some way of reducing T to a tree T <sup>1</sup> with k vertices. This suggests "removing an end vertex". So we must first prove that "any tree T has an end vertex".]

Definition The number of edges incident with a given vertex v is called the valency of v.

Lemma Let S be a finite tree with s ą 1 vertices. Then S has a vertex of valency 1 – that is an "end vertex".

Proof Choose any vertex v0. Then v<sup>0</sup> must be connected to the rest of the tree, so v<sup>0</sup> has valency at least 1.

If v<sup>0</sup> is an "end vertex", then stop; if not, then choose a vertex v<sup>1</sup> which is adjacent to v0.

If v<sup>1</sup> is an "end vertex", then stop; if not, choose a vertex v<sup>2</sup> ‰ v<sup>0</sup> which is adjacent to v1.

If v<sup>2</sup> is an "end vertex", then stop; if not, choose a vertex v<sup>3</sup> ‰ v<sup>1</sup> which is adjacent to v2.

Continue in this way.

All of the vertices v0, v1, v2, v3, . . . must be different (since any repeat v<sup>m</sup> " v<sup>n</sup> with m ă n would define a cycle

$$v\_m, v\_{m+1}, v\_{m+2}, \dots, v\_{n-1}, v\_n = v\_m$$

in the tree S). Since we know that the tree is finite (having precisely s vertices), the process must terminate at some stage. The final vertex v<sup>e</sup> is then an "end vertex", of valency 1. QED

If we apply the Lemma to our arbitrary tree T with k ` 1 vertices, we can choose an "end vertex" v and remove both it and the edge e incident with it to obtain a tree T <sup>1</sup> having k vertices. By Ppkq we know that T <sup>1</sup> has exactly k ´ 1 edges, so when we reinstate the edge e, we see that T has exactly pk ´ 1q ` 1 edges, so Ppk ` 1q is true.

Hence Ppnq is true for all n ě 1.

### 260.

Note: All the polyhedra described in this solution are "spherical" by virtue of having their vertices located on the unit sphere.

	- (ii) A square based pyramid with its apex at the North pole.

$$V - E + F = 2.$$

Now each edge has exactly two end vertices, so 2E counts the exact number of ordered pairs pv, eq, where e is an edge, and v is a vertex "incident with e".

On the other hand, in a spherical polyhedron, each vertex v has valency at least 3; so each vertex v occurs in at least 3 pairs pv, eq of this kind. Hence 2E ě 3V . In the same way, each edge e lies on the boundary of exactly 2 faces, so 2E counts the exact number of ordered pairs pf, eq, where e is an edge of the face f.

On the other hand, in a spherical polyhedron, each face f has at least 3 edges; so each face f occurs in at least 3 pairs pf, eq of this kind. Hence 2E ě 3F. If E " 7, then 14 ě 3V , and 14 ě 3F; now V and F are integers, so V ď 4 and

F ď 4. Hence V ` F ď 8. However V ` F " E ` 2 " 9. This contradiction shows that no such polyhedron exists. QED

(c) We show by induction how to construct certain "spherical" polyhedra, with at most one non-triangular face. Let Ppnq be the statement:

"There exists a spherical polyhedron with at most one non-triangular face, and with e edges for each e, 8 ď e ď n".

' We know that there exists a such a spherical polyhedron with n " 6 edges – namely the regular tetrahedron (with four faces, which are all equilateral triangles).

We know there is no such polyhedron with n " 7 edges (by part (b)).

When n " 8, there is no spherical polyhedron with n " 8 edges and all faces triangular (since we would then have 16 " 2E " 3F, as in part (b)). However, there exists a spherical polyhedron with n " 8 edges and just one non-triangular face – namely the square based pyramid with its apex at the North pole.

When n " 9, we can join three points on the equator to the North and South poles to produce a triangular bi-pyramid (the dual of a triangular prism), with all faces triangular, and with n " 9 edges.

When n " 10, there is no spherical polyhedron with n " 10 edges and with all faces triangles (since we would then have to have 20 " 2E " 3F, as in part (b)); but there exists a spherical polyhedron with n " 10 edges and just one face which is not an equilateral triangle – namely the pentagonal based pyramid with its apex at the North pole.

This provides us with a starting point for the inductive construction. In particular Pp8q, Pp9q, and Pp10q are all true.

' Suppose that Ppkq is true for some k ě 10. The only part of the statement Ppk ` 1q that remains to be demonstrated is the existence of a suitable polyhedron with k ` 1 edges.

Since k ě 10, we know that k ´ 2 ě 8, so (by Ppkq) there exists a polyhedron with all its vertices on the unit sphere, with at most one non-triangular face, and with e " k ´ 2 edges. Take this polyhedron and remove a triangular face ABC. Now add a new vertex X on the sphere, internal to the spherical triangle ABC, and add the edges XA, XB, XC and the three triangular faces XAB, XBC, XCA, to produce a spherical polyhedron with e " pk´2q`3 " k`1 edges, and with at most one non-triangular face. Hence Ppk`1q is true.

Hence Ppnq is true for all n ě 8.

261. To prove a result that is given in the form of an "if and only if" statement, we have to prove two things: "if", and "only if".

We begin by proving the "only if" part:

"a map can be properly coloured with two colours only if every vertex has even valency".

Let M be a map that can be properly coloured with two colours. Let v be any vertex of the map M.

The edges e1, e2, e3, . . . incident with v form parts of the boundaries of the sequence of regions around the vertex v (with e1, e<sup>2</sup> bordering one region; e2, e<sup>3</sup> bordering the next; and so on). Since we are assuming that the regions of the map M can be "properly coloured" with two colours, the succession of regions around the vertex v can be properly coloured with just two colours. Hence the colours of the regions around the vertex v must alternate (say black-white-black- . . . ). And since the map is finite, this sequence must return to the start – so the number of such regions at the vertex v (and hence the number of edges incident with v – that is, the valency of v) must be even.

We now prove the "if" part:

"a map can be properly coloured with two colours if every vertex has even valency".

Suppose that we have a map M in which each vertex has even valency. We must prove that any such map M can be properly coloured using just two colours. Let Ppnq be the statement:

"any map with m edges, in which each vertex has even valency, can be properly coloured with two colours whenever m satisfies 1 ď m ď n,".


Most of the contents of the statement Ppk ` 1q are already guaranteed by Ppkq. To prove that Ppk ` 1q is true, all that remains to be proved is that

any map with exactly k`1 edges, in which every vertex has even valency, can be properly coloured using just two colours.

Consider an arbitrary map M with k ` 1 edges, in which each vertex has even valency.

[Idea: We need to find some way of reducing the map M to a map M<sup>1</sup> with ď k edges, in which every vertex still has even valency.]

Since k ě 1, the map M has at least 2 edges. Choose any edge e of M, with (say) vertices u1, u<sup>2</sup> as its endpoints, and with regions R, S on either side of e.

Suppose first that u<sup>1</sup> " u2, so the boundary of the region R (say) consists only of the edge e. Hence e is a loop, and S is the only region neighbouring R. The edge e contributes 2 to the valency of u1; so if we delete the edge e, we obtain a map M<sup>1</sup> in which every vertex again has even valency, in which the regions R and S have been amalgamated into a region S 1 . Since M<sup>1</sup> has just k edges, M<sup>1</sup> can be properly coloured with just two colours. If we now reinstate the edge e and the region R, we can give S the same colour as S 1 (in the proper colouring of M<sup>1</sup> ) and give R the opposite colour to S 1 to obtain a proper colouring of the map M with just two colours.

Hence we may assume that u<sup>1</sup> ‰ u2, so that e is not the complete boundary of R. We may then slowly shrink the edge e to a point – eventually fusing the old vertices u1, u<sup>2</sup> together to form a new vertex u 1 , where two new regions R 1 , S <sup>1</sup> meet. The result is then a new map M<sup>1</sup> , in which all other vertices are unchanged (and so have even valency), and in which

$$\text{valency}(u') = (\text{valency}(u\_1) - 1) + (\text{valency}(u\_2) - 1)$$

which is also even.

Hence every vertex of the new map M<sup>1</sup> has even valency. Moreover, M<sup>1</sup> has at most k edges, so (by Ppkq) we know that the map M<sup>1</sup> can be properly coloured with just two colours. And in this colouring of M<sup>1</sup> , there are an odd number of colour changes as one goes from R 1 to S 1 through the other regions that meet around the old vertex u<sup>1</sup> of M, so S 1 receives the opposite colour to R 1 . The guaranteed proper two-colouring of M<sup>1</sup> therefore extends back to give a proper two-colouring of the original map M. Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 1.

262. Let Ppnq be the statement:

"The 2<sup>n</sup> sequences of length n consisting of 0s and 1s can be arranged in a cyclic list such that any two neighbouring sequences (including the last and the first) differ in exactly one coordinate position."

• When n " 2, the required cycle is obvious:

$$00 \to 10 \to 11 \to 01 \quad (\to 00).$$

So Pp2q is true.

• The general construction is perhaps best illustrated by first showing how Pp2q leads to Pp3q.

The above cycle for sequences of length 2 gives rise to two disjoint cycles for sequences of length 3:

– first by adding a third coordinate "0":

$$000 \to 100 \to 110 \to 010 \quad (\to 000)$$

– then by adding a third coordinate "1":

001 Ñ 101 Ñ 111 Ñ 011 pÑ 001q.

Now eliminate the final join in each cycle (010 Ñ 000 and 011 Ñ 001) and instead link the two cycles together by first reversing the order of the first cycle, and then inserting the joins 000 Ñ 001 and 011 Ñ 010 to form a single cycle. In general, suppose that Ppkq is true for some k ě 1. Then we construct a single cycle for the 2<sup>k</sup>`<sup>1</sup> sequences of length k ` 1 as follows:

Take the cycle of the 2<sup>k</sup> sequences of length k guaranteed by Ppkq, and form two disjoint cycles of length 2<sup>k</sup>


Then link the two cycles into a single cycle of length 2<sup>k</sup>`<sup>1</sup> , by eliminating the final step

$$v\_1 v\_2 \cdot \cdot \cdot v\_k 0 \to 00 \cdot \cdot \cdot 00$$

in the first cycle, and

$$v\_1 v\_2 \cdot \cdot \cdot v\_k 1 \to 00 \cdot \cdot \cdot 01$$

in the second cycle, reversing the first cycle, and inserting the joins

00 ¨ ¨ ¨ 00 Ñ 00 ¨ ¨ ¨ 01 and v1v<sup>2</sup> ¨ ¨ ¨ vk1 Ñ v1v<sup>2</sup> ¨ ¨ ¨ vk0

to produce a single cycle of the required kind joining all 2<sup>k</sup>`<sup>1</sup> sequences of length k ` 1. Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 2.

Note: The significance of what we call Gray codes was highlighted in a 1953 patent by the engineer Frank Gray (1887–1969) – where they were called reflected binary codes (since the crucial step in their construction above involves taking two copies of the previous cycles, reversing one of the cycles, and then producing half of the required cycle by traversing the first copy before returning backwards along the second copy). Their most basic use is to re-encode the usual binary counting sequence

$$11 \to 10 \to 11 \to 100 \to 101 \to 110 \to 111 \to 1000 \to 1001 \to 1010 \to \cdots, \text{ and } \text{card}(10) = \text{card}(10)$$

where a single step can lead to the need to change arbitrarily many binary digits (e.g. the step from 3 " "11" to 4 " "100" changes 3 digits, and the step from 7 " "111" to 8 ""1000" changes 4 digits, etc.) – a requirement that is inefficient in terms of electronic "switching", and which increases the probability of errors. In contrast, the Gray code sequence changes a single binary digit at each step. However, the physical energy which is saved through reducing the amount of "switching" in the circuitry corresponds to an increase in the need for unfamiliar mathematical formulae, which re-interpret each vector in the Gray code as the ordinary integer in the counting sequence to which it corresponds.

#### 263.

(a) The whole construction is inductive (each label derives from an earlier label). So let Ppnq be the statement:

"if HCFpr, sq " 1 and 2 ď r `s ď n, then the positive rational <sup>r</sup> s occurs once and only once as a label, and it occurs in its lowest terms".

' By construction the root is given the label <sup>1</sup> 1 , so <sup>1</sup> 1 occurs. And it cannot occur again, since the numerator and denominator of each parent vertex are both positive, neither i nor j can ever be 0. Hence Pp2q is true.

Notice that the basic construction:

"if <sup>i</sup> j is a 'parent' vertex, then we label its 'left descendant' as <sup>i</sup> i`j , and its 'right descendant' <sup>i</sup>`<sup>j</sup> j "

guarantees that, since we start by labelling the root with the positive rational 1 1 , all subsequent 'descendants' are positive.

Moreover, if any 'descendant' were suddenly to appear not "in lowest terms", then either


Since we begin by labelling the root <sup>1</sup> 1 , where HCFp1, 1q " 1, it follows that all subsequent labels are positive rationals in lowest terms.


Hence Ppnq is true for all n ě 2.

(b) The fact that the labels are left-right symmetric is also an inductive phenomenon. We note that the one fully "left-right symmetric" label, namely <sup>1</sup> 1 , occurs in the only fully "left-right symmetric" position – namely at the root.

All other labels occur in reciprocal pairs: <sup>r</sup> s and <sup>s</sup> r , where we may assume that r ą s. The fact that these occur as labels of "left-right symmetric" vertices derives from the fact that

r s is the 'right descendant' of <sup>r</sup>´<sup>s</sup> s and

s r is the 'left descendant' of <sup>s</sup> r´s .

So if we know that the earlier reciprocal pair reciprocal pair <sup>r</sup>´<sup>s</sup> s and <sup>s</sup> r´s occur as labels of symmetrically positioned vertices, then it follows that the same is true of the descendant reciprocal pair <sup>r</sup> s and <sup>s</sup> r . We leave the reader to write out the proof by induction – for example, using the statement

"Ppnq: if r, s ą 0, and 2 ď r ` s ď n, then the reciprocal pair <sup>r</sup> s , s r occur as labels of vertices at the same level below the root, with the two labelled vertices being mirror images of each other about the vertical mirror through the root vertex."

264. The intervals in this problem may be of any kind (including finite or infinite). Each interval has two "endpoints", which are either ordinary real numbers, or ˘8 (signifying that the interval goes off to infinity in one or both directions).

Let Ppnq be the statement:

"if a collection of n intervals on the x-axis has the property that any two intervals overlap in an interval (of possibly zero length – i.e. a point), then the intersection of all intervals in the collection is a non-empty interval".

When n " 2, the hypothesis of Pp2q is the same as the conclusion. So Pp2q is true. Suppose that Ppkq is true for some k ě 2. We seek to prove that Ppk ` 1q is true. So consider a collection of k ` 1 intervals with the property that any two intervals in the collection intersect in a non-empty interval. If this collection includes one interval that is listed more than once, then the required conclusion follows from Ppkq. So we may assume that the intervals in our collection are all different.

Among the k`1 intervals, consider first those with the largest right hand endpoint. If there is only one such interval, denote it by I0; if there is more than one interval with the same largest right hand endpoint, let I<sup>0</sup> be the interval among those with the largest right hand endpoint that has the largest left hand endpoint. In either case, put I<sup>0</sup> aside for the moment, leaving a collection S of k intervals with the required property.

By Ppkq we know that the intervals in the collection S intersect in a non-empty interval I, with left hand endpoint a and right hand endpoint b (say).

We have to show that the intersection I X I<sup>0</sup> is non-empty.

The proof that follows works if the endpoint b is included in the interval I. The slight adjustment needed if b is not included in the interval I is left to the reader.

Since the right hand endpoint of I<sup>0</sup> is the largest possible, and since points between a and b belong to all the intervals of S, we can be sure that the right hand endpoint of I<sup>0</sup> is ě b.

Moreover, for each point x ą b, we know that there must be some interval I<sup>x</sup> in the collection S which does not stretch as far to the right as x. Since, by hypothesis, the intersection I<sup>0</sup> XI<sup>x</sup> is non-empty, the left hand endpoint of I<sup>0</sup> lies to the left of every such point x, so I<sup>0</sup> must overlap the interval I, whence it follows that I X I<sup>0</sup> is a non-empty interval as required.

Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 2.

265. If one experiments a little, it should become clear


These three observations essentially determine the answer – namely that tank T should be joined to the other tanks in increasing order of their initial contents. For a proof by induction, let Ppnq be the statement:

"given n tanks containing a1, a2, a3, . . . , a<sup>n</sup> litres respectively, where

$$a\_1 < a\_2 < a\_3 < \cdots < a\_n,$$

if T is the tank containing the smallest amount a<sup>1</sup> litres, then the optimal sequence for linking the other n ´ 1 tanks to tank T (optimal in the sense that it transfers the maximum amount of water to tank T) is the sequence that links T successively to the other tanks in increasing order of their initial contents".


$$a\_1 < a\_2 < a\_3 < \cdots < a\_{k+1},$$

and where T is the tank which initially contains a<sup>1</sup> litres.

Suppose that in the optimal sequence of k successive joins to the other k tanks (that is, that transfers the largest possible amount of water to tank T), the succession of joins is to join T first to tank T2, then to tank T3, and so on up to tank T<sup>k</sup>`<sup>1</sup> (where tank T<sup>m</sup> is not necessarily the tank containing a<sup>m</sup> litres). There are now two possibilities: either


By the very first bullet point, in order to guarantee the optimal overall outcome of the final linking with tank T<sup>k</sup>`<sup>1</sup> the amount in tank T after it has been linked to tank T<sup>k</sup> must be "as large as it can possibly be" (given the amounts in the tanks T, T2, T3, . . . , Tk). Hence statement Ppkq applies to the initial sequence of k ´ 1 joins (of T to T2, then to T3, and so on up to Tk), and guarantees that these tanks must be in increasing order of their initial contents. In particular, the last tank in this sequence, Tk, must be the one containing a<sup>k</sup>`<sup>1</sup> litres. But if we denote by a litres the amount in tank T just before it links with tank T<sup>k</sup> (containing a<sup>k</sup>`<sup>1</sup> litres), then the last two linkings, with b " a<sup>k</sup>`<sup>1</sup> and c " a<sup>m</sup> contradict the second bullet point at the start of this solution. Hence case (ii) cannot occur.

Hence Ppk ` 1q is true.

Hence Ppnq is true for all n ě 2.

#### 266.

Note: Like all practical problems, this one requires an element of initial "modelling" in order to make the situation amenable to mathematical analysis.

'Residue' clings to surfaces; so the total amount of 'residue' will depend on


Since we are given no information about quantities, we may fix the amount of residue remaining in the 'empty' flask at "1 unit", and the amount of solvent in the other flask as "s units".

If we add the solvent, we get a combined amount of 1 ` s units of solution – which we may assume (after suitable shaking) to be homogeneous, with the chemical concentration reduced to "1 part in 1 ` s".

The first modelling challenge arises when we try to make mathematical sense of what remains at each stage after we empty the flask. The internal surface area of the flask, to which any diluted residue may adhere, is fixed. If we make the mistake of thinking of the original chemical as "thick and sticky" and the solvent as "thin", then the viscosity of the diluted residue will change relative to the original, and will do so in ways that we cannot possibly know. Hence the only reasonable assumption (which may or may not be valid in a particular instance) is to assume that the viscosity of the original chemical is roughly the same as the viscosity of the chemical-solvent mixture. This then suggests that, on emptying the diluted mixture, roughly the same amount (1 unit) of diluted mixture will remain adhering to the walls of the flask. So we will be left with 1 unit of residue, with a concentration of <sup>1</sup> 1`s . In particular, if s " 1, using all the solvent at once reduces the amount of toxic chemical residue to <sup>1</sup> 2 unit (with the other <sup>1</sup> 2 unit consisting of solvent).

But what if we use only half of the solvent first, empty the flask, and then use the other half? Adding <sup>s</sup> 2 units of solvent (and shaking thoroughly) produces 1 ` s 2 units of homogeneous mixture, with a concentration of 1 part per 1 ` s 2 . When we empty the flask, we expect roughly 1 unit of residue with this concentration – so just <sup>2</sup> 2`s units of the chemical, with <sup>s</sup> 2`s units of solvent.

If we then add the other <sup>s</sup> 2 units of solvent, this produces 1 ` s 2 units of mixture with a concentration of 1 part per ` 1 ` s 2 ˘2 . When we empty the flask, we expect roughly 1 unit of residue with concentration 1 part per ` 1 ` s 2 ˘2 . In particular, if s " 1, this strategy reduces the amount of toxic chemical in the 1 unit of residue to <sup>4</sup> 9 units. Since <sup>4</sup> <sup>9</sup> ă 1 2 this two-stage strategy seems more effective than the previous "all at once" strategy.

Suppose that we use a four-stage strategy – using first one quarter of the solvent, then another quarter, and so on. We then land up with roughly 1 unit of residue with concentration 1 part per ` 1 ` s 4 ˘4 . In particular, if s " 1, we land up with the amount of toxic chemical in the 1 unit of residue equal to <sup>256</sup> <sup>625</sup> units, and <sup>256</sup> <sup>625</sup> ă 4 9 . More generally, if we use ` 1 n ˘ th of the solvent, n times, the final amount of toxic chemical in the 1 unit of residue is equal to ` 1 ` s n ˘´<sup>n</sup> . And as n gets larger and larger, this expression gets closer and closer to e ´s . In particular, when s " 1, this strategy leaves a final amount of chemical in the 1 unit of residue approximately equal to <sup>1</sup> <sup>e</sup> " 0.367879 ¨ ¨ ¨ .

Note: The situation here is similar to that faced by a washing machine designer, who wishes to remove traces of detergent from items that have been washed, without using unlimited amounts of water. The idea of having a "fixed amount of solvent" corresponds to the goal of "water efficient" rinsing. However, the washing machine cycle, or programme, clearly cannot repeat the rinsing indefinitely (as would be required in the limiting case above).

267.

(i) If ? 2 " m n , then

$$2n^2 = m^2\tag{\*}$$

Hence m<sup>2</sup> is even. It follows that m must be even.

Note: It is a fact that, if m " 2k is even, then m<sup>2</sup> " 4k 2 is also even. But this is completely irrelevant here. In order to conclude that "m must be even", we have to prove:

```
Claim m cannot be odd.
Proof Suppose m is odd.
6 m " 2k ` 1 for some integer k.
But then
```

$$m^2 = (2k+1)^2 = 4k^2 + 4k + 1$$

would be odd, contrary to "m<sup>2</sup> must be even". Hence m cannot be odd. QED

(ii) Since m is even, we may write m " 2m<sup>1</sup> for some integer m<sup>1</sup> . Equation (˚) in (i) above then becomes n <sup>2</sup> " 2pm<sup>1</sup> q 2 , so n 2 is even. Hence, as in the Note above, n must be even, so we can write n " 2n 1 for some integer n 1 .

(iii) If ? 2 " m n , then m " 2m<sup>1</sup> , and n " 2n 1 are both even. 6 ? 2 " m <sup>n</sup> " 2m<sup>1</sup> <sup>2</sup>n<sup>1</sup> " m<sup>1</sup> n1 .

In the same way, it follows that m<sup>1</sup> and n 1 are both even, so we may write m<sup>1</sup> " 2m<sup>2</sup> , n <sup>1</sup> " 2n 2 for some integers m<sup>2</sup> , n 2 .

Continuing in this way then produces an endless decreasing sequence of positive denominators

$$n > n' > n'' > \dots > 0.$$

contrary to the fact that such a sequence can have length at most n ´ 1 (or in fact, at most 1 ` log<sup>2</sup> n).

#### 268.

(i) If a ă b and c ą 0, then ac ă bc.

If 0 ă ? <sup>2</sup> <sup>ď</sup> 1, then (multiplying by ? 2) it follows that 2 ď ? 2 ď 1, which is false. Hence ? 2 ą 1. ?

We now know that 1 ă 2, so multiplying by ? 2 gives ? ? 2 ă 2. Hence 1 ă <sup>2</sup> <sup>ă</sup> 2. In particular, ? 2 cannot be written as a fraction with denominator 1, so Pp1q is true.

(ii) Suppose Ppkq is true for some k ě 1. Most of the statement Ppk ` 1q is implied by <sup>P</sup>pkq: all that remains to be proved is that ? 2 cannot be written as a fraction with denominator n " k ` 1.

Suppose ? 2 " m n , where n " k ` 1 and m are positive integers. Then m " 2m<sup>1</sup> and n " 2n 1 are both even (as in Problem 267). So ? 2 " m<sup>1</sup> <sup>n</sup><sup>1</sup> with n <sup>1</sup> ď k, contrary to Ppkq. Hence Ppk ` 1q holds. 6 Ppnq is true for all n ě 1.

#### 269.

(a) Suppose that S is not empty. Then by the Least Element Principle the set S must contain a least element k: that is, a smallest integer k which is not in the set T. Then k ‰ 1 (since we are told that T contains the integer 1). Hence k ą 1.

Therefore k ´ 1 is a positive integer which is smaller than k. So k ´ 1 is not an element of S, and hence must be an element of T. But if k´1 is an element of T, then we are told that pk´1q`1 must also be a member of T. This contradiction shows that S must be empty, so T contains all positive integers.

(b) Suppose that T does not have a smallest element. Clearly 1 does not belong to the set T (or it would be the smallest element of T). Hence 1 must be an element of the set S.

Now suppose that, for some k ě 1, all positive integers 1, 2, 3, . . . , k are elements of S, but k ` 1 is not an element of S. Then k ` 1 would be an element of T, and would be the smallest element of T, which is not possible. Hence S has the property that

"whenever k ě 1 is an element of S, we can be sure that k ` 1 is also an element of S."

The Principle of Mathematical Induction then guarantees that these two observations (that 1 is an element of S, and that whenever k is an element of S, so is k ` 1) imply that S contains all the positive integers, so that the set T must be empty – contrary to assumption.

Hence T must have a smallest element.

#### 270.


$$
\underline{OP'} = \underline{OQ} - \underline{QP'} = \underline{OQ} - \underline{QP} = m - n,
$$

and

$$
\underline{OQ}' = \underline{QR} - \underline{Q'R} = \underline{QR} - \underline{Q'P'} = \underline{QR} - \underline{QP'} = n - (m - n).
$$


$$2n \ge m - n \ge (2n - m) - (m - n) = 3n - 2m \ge \dots \ge 0.$$

Zarathustra's last, most vital lesson: "Now do without me." George Steiner (1929– )

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Teaching Mathematics is nothing less than a mathematical manifesto. Arising in response to a limited National Curriculum, and engaged with secondary schooling for those aged 11–14 (Key Stage 3) in particular, this handbook for teachers will help them broaden and enrich their students' mathematical education. It avoids specifying how to teach, and focuses instead on the central principles and concepts that need to be borne in mind by all teachers and textbook authors—but which are little appreciated in the UK at present. This study is aimed at anyone who would like to think more deeply about the discipline of 'elementary mathematics', in England and Wales and anywhere else. By analysing and supplementing the current curriculum, Teaching Mathematics provides food for thought for all those involved in school mathematics, whether as aspiring teachers or as experienced professionals. It challenges us all to reflect upon what it is that makes secondary school mathematics educationally, culturally, and socially important.

# **The Essence of Mathematics Through Elementary Problems**

## **ALEXANDRE BOROVIK AND TONY GARDINER**

It is increasingly clear that the shapes of *reality* – whether of the natural world, or of the built environment – are in some profound sense *mathemati cal*. So it would benefi t students and educated adults to understand what makes mathemati cs itself 'ti ck', and why its shapes, patt erns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the fi rst part, and explore the extent to which elementary mathemati cs allows us all to understand something of the nature of mathemati cs *from the inside*.

*The Essence of Mathemati cs* consists of a sequence of 270 problems – with commentary and full soluti ons. The reader is assumed to have a reasonable grasp of school mathemati cs. More importantly, s/he should want to understand something of mathemati cs *beyond the classroom*, and be willing to engage with (and to refl ect upon) challenging problems that highlight the essence of the discipline.

The book consists of six chapters of increasing sophisti cati on (Mental Skills; Arithmeti c; Word Problems; Algebra; Geometry; Infi nity), with interleaved commentary. The content will appeal to students considering further study of mathemati cs at university, teachers of mathemati cs at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathemati cs *really* works.

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