Games and Mathematics Read online

  Games and Mathematics

  The appeal of games and puzzles is timeless and universal. In this unique book, David Wells explores the fascinating connections between games and mathematics, proving that mathematics is not just about calculation but also about imagination, insight and intuition.

  The first part of the book introduces games, puzzles and mathematical recreations, including the Tower of Hanoi, knight tours on a chessboard, Nine Men's Morris and more. The second part explains how thinking about playing games can mirror the thinking of a mathematician, using scientific investigation, tactics and strategy, and sharp observation. Finally, the author considers game-like features found in a wide range of human behaviours, illuminating the role of mathematics and helping to explain why it exists at all.

  This thought-provoking book is perfect for anyone with a thirst for mathematics and its hidden beauty; a good high-school grounding in mathematics is all the background that's required, and the puzzles and games will suit pupils from 14 years.

  David Wells is the author of more than a dozen books on popular mathematics, puzzles and recreations. He has written many articles on mathematics teaching, and a secondary mathematics course based on problem solving. A former British under-21 chess champion and amateur 3-dan at Go, he has also worked as a game inventor and puzzle editor.

  Games and Mathematics

  Subtle Connections

  David Wells

  CAMBRIDGE UNIVERSITY PRESS

  Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City

  Cambridge University Press

  The Edinburgh Building, Cambridge CB2 8RU, UK

  Published in the United States of America by Cambridge University Press, New York

  www.cambridge.org

  Information on this title: www.cambridge.org/9781107024601

  © David Wells 2012

  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

  First published 2012

  Printed and Bound in the United Kingdom by the MPG Books Group

  A catalogue record for this publication is available from the British Library

  Library of Congress Cataloguing in Publication data

  Wells, D. G. (David G.)

  Games and mathematics : subtle connections / David Wells.

  p. cm.

  Includes bibliographical references and index.

  ISBN 978-1-107-02460-1 (hardback)

  1. Games – Mathematical models. 2. Mathematical recreations. 3. Mathematics – Psychological aspects. I. Title.

  QA95.W438 2012

  510 – dc23 2012024343

  ISBN 978-1-107-02460-1 Hardback

  ISBN 978-1-107-69091-2 Paperback

  Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

  Contents

  Acknowledgements

  Part i Mathematical recreations and abstract games

  Introduction

  Everyday puzzles

  1 Recreations from Euler to Lucas

  Euler and the Bridges of Königsberg

  Euler and knight tours

  Lucas and mathematical recreations

  Lucas's game of solitaire calculation

  2 Four abstract games

  From Dudeney's puzzle to Golomb's Game

  Nine Men's Morris

  Hex

  Chess

  Go

  3 Mathematics and games: mysterious connections

  Games and mathematics can be analysed in the head…

  Can you ‘look ahead’?

  A novel kind of object

  They are abstract

  They are difficult

  Rules

  Hidden structures forced by the rules

  Argument and proof

  Certainty, error and truth

  Players make mistakes

  Reasoning, imagination and intuition

  The power of analogy

  Simplicity, elegance and beauty

  Science and games: let's go exploring

  4 Why chess is not mathematics

  Competition

  Asking questions about

  Metamathematics and game-like mathematics

  Changing conceptions of problem solving

  Creating new concepts and new objects

  Increasing abstraction

  Finding common structures

  The interaction between mathematics and sciences

  5 Proving versus checking

  The limitations of mathematical recreations

  Abstract games and checking solutions

  How do you ‘prove’ that 11 is prime?

  Is ‘5 is prime’ a coincidence?

  Proof versus checking

  Structure, pattern and representation

  Arbitrariness and un-manageability

  Near the boundary

  Part ii Mathematics: game-like, scientific and perceptual

  Introduction

  6 Game-like mathematics

  Introduction

  Tactics and strategy

  Sums of cubes and a hidden connection

  A masterpiece by Euler

  7 Euclid and the rules of his geometrical game

  Ceva's theorem

  Simson's line

  The parabola and its geometrical properties

  Dandelin's spheres

  8 New concepts and new objects

  Creating new objects

  Does it exist?

  The force of circumstance

  Infinity and infinite series

  Calculus and the idea of a tangent

  What is the shape of a parabola?

  9 Convergent and divergent series

  The pioneers

  The harmonic series diverges

  Weird objects and mysterious situations

  A practical use for divergent series

  10 Mathematics becomes game-like

  Euler's relation for polyhedra

  The invention-discovery of groups

  Atiyah and MacLane disagree

  Mathematics and geography

  11 Mathematics as science

  Introduction

  Triangle geometry: the Euler line of a triangle

  Modern geometry of the triangle

  The Seven-Circle Theorem, and other New Theorems

  12 Numbers and sequences

  The sums of squares

  Easy questions, easy answers

  The prime numbers

  Prime pairs

  The limits of conjecture

  A Polya conjecture and refutation

  The limitations of experiment

  Proof versus intuition

  13 Computers and mathematics

  Hofstadter on good problems

  Computers and mathematical proof

  Computers and ‘proof’

  Finally: formulae and yet more formulae

  14 Mathematics and the sciences

  Scientists abstract

  Mathematics anticipates science and technology

  The success of mathematics in science

  How do scientists use mathematics?

  Methods and technique in pure and applied mathematics

  Quadrature: finding the areas under curves

  The cycloid

  Science inspires mathematics

  15 Minimum paths: elegant simplicity

  A familiar puzzle

  Dev
eloping Heron's theorem

  Extremal problems

  Pappus and the honeycomb

  16 The foundations: perception, imagination, insight

  Archimedes' lemma and proof by looking

  Chinese proofs by dissection

  Napoleon's theorem

  The polygonal numbers

  Problems with partitions

  Invented or discovered? (Again)

  17 Structure

  Pythagoras' theorem

  Euclidean coordinate geometry

  The average of two points

  The skew quadrilateral

  18 Hidden structure, common structure

  The primes and the lucky numbers

  Objects hidden behind a veil

  Proving consistency

  Transforming structure, transforming perception

  19 Mathematics and beauty

  Hardy on mathematics and chess

  Experience and expectations

  Beauty and Brilliancies in chess and mathematics

  Beauty, analogy and structure

  Beauty and individual differences in perception

  The general versus the specific and contingent

  Beauty, form and understanding

  20 Origins: formality in the everyday world

  The psychology of play

  The rise and fall of formality

  Religious ritual, games and mathematics

  Formality and mathematics

  Hidden mathematics

  Style and culture, style in mathematics

  The spirit of system versus problem solving

  Visual versus verbal: geometry versus algebra

  Women, games and mathematics

  Mathematics and abstract games: an intimate connection

  References

  Index

  Acknowledgements

  The illustration of the tower of Hanoi on page 17 is reproduced with permission from Book of Curious and Interesting Puzzles [Wells 1992/2006: 66] published by Dover publications, Inc. New York.

  The diagram of the 21-point cubic (page 128) is from The Penguin Dictionary of Curious and Interesting Geometry [Wells 1991: 43] and is reprinted by permission of John Sharp, the illustrator of that book, who also produced the image of Fatou dust on page 212.

  The figure of the Al Mani knight tour (page 15) can be found at www.mayhematics.com/t/history/1a.htm and elsewhere.

  Part I

  Mathematical recreations and abstract games

  Introduction

  Abstract games, traditional puzzles and mathematics are closely related. They are often extremely old, they are easily appreciated across different cultures, unlike language and literature, and they are hardly affected by either history or geography. Thus the ancient Egyptian game of Mehen which was played on a spiral board and called after the serpent god of that name, disappeared from Egypt round about 2900–2800 BCE according to the archaeological record but reappeared in the Sudan in the 1920s. Another game which is illustrated in Egyptian tomb paintings is today called in Italian, morra, ‘the flashing of the fingers’ which has persisted over three thousand years without change or development. Each player shows a number of fingers while shouting his guess for the total fingers shown. It needs no equipment and it can be played anywhere but it does require, like many games, a modest ability to count [Tylor 1879/1971: 65].

  As, of course, do dice games. Dice have been unearthed at the city of Shahr-i Sokhta, an archaeological site on the banks of the Helmand river in southeastern Iran dating back to 3000 BCE and they were popular among the Greeks and Romans as well as appearing in the Bible.

  The earliest puzzles or board games and those found in ‘primitive’ societies tend to be fewer and simpler than more recent creations yet we can understand and appreciate them despite the vast differences in every other aspect of culture.

  ‘Culture’ is undoubtedly the right word: puzzles and games are not trivia, mere pleasant pastimes which offer fun and amusement but serious features of all human societies without exception – and they lead eventually to mathematics. String figures are a perfect example. They have been found in northern America among the Inuit, among the Navajo and Kwakiutl Indians, in Africa and Japan and among the Pacific islands and the Maori and Australian aborigines [Averkieva & Sherman 1992]. This is not necessarily evidence of ancient exchanges between cultures. It could just be that people everywhere tend to fiddle with bits of string – and the results can be very pleasing, like the Jacob's ladder in Figure 1.

  Figure 1 Jacob's ladder

  String figures are extremely abstract. Although usually made on two hands, or sometimes the hands and feet or with four hands, Jacob's ladder would be recognisably ‘the same’ if it were fifty feet wide and made from a ship's hawser, yet these abstract playful objects can also be useful. The earliest record of a string figure is the plinthios (Figure 2), described in a fourth-century Greek miscellany. It was recommended for supporting a fractured chin, and much resembles the Jacob's ladder figure [Probert 1999].

  Figure 2 Plinthios string figure

  No surprise then that string figures are more than an anthropological curiosity, that they are mathematically puzzling, related to everyday knots – including braiding, knitting, crochet and lace-work – and to one of the most recent branches of mathematics, topology.

  The oldest written puzzle plausibly goes back to Ancient Egypt:

  There are seven houses each containing seven cats. Each cat kills seven mice and each mouse would have eaten seven ears of spelt. Each ear of spelt would have produced seven hekats of grain. What is the total of all these?

  This curiosity, paraphrased here, is problem 79 in the Rhind papyrus which was written about 1650 BC. Nearly 3000 years later in his Liber Abaci (1202), Fibonacci posed this problem:

  Seven old women are travelling to Rome, and each has seven mules. On each mule there are seven sacks, in each sack there are seven loaves of bread, in each loaf there are seven knives, and each knife has seven sheaths. The question is to find the total of all of them.

  It is tempting to suppose that these puzzles are related. If they are, could there be a historical connection across 5000 years with this riddle from the British eighteenth century Mother Goose collection?

  As I was going to St. Ives,

  I met a man with seven wives.

  Each wife had seven sacks,

  Each sack had seven cats,

  Each cat had seven kits;

  Kits, cats, sacks and wives,

  How many were going to St. Ives?

  Another widespread puzzle concerns a man, a wolf, a goat and a cabbage, to be transported across a river in a small boat, never leaving the wolf alone with the goat or the goat alone with the cabbage. It first appeared in a collection attributed to the medieval scholar Alcuin of York (735–804), Propositions to Sharpen the Young [Alcuin of York 1992].

  Tartaglia (1500–1557) who famously solved the cubic equation and then gave the solution to Cardan who scandalously broke his solemn agreement not to publish it, printed a version featuring three brides and their jealous husbands who have to cross a river in a boat that will only take two people. If no bride can be accompanied by another women's future husband, how many trips are required?

  Essentially the same puzzle is also found in Africa, in Ethiopia, in the Cape Verde Islands, in Cameroon, and among the Kpelle of Liberia and elsewhere. Since these African versions are often logically distinct from the Western, they may well be entirely independent: the difficulty of transporting uncongenial items across a river used to be universal [Ascher 1990]. Mathematics and riddles ‘Mathematics has much in common with riddling, and with humour. Everything in mathematics has many meanings. Every diagram and every figure, every sum and every equation, can be “seen” in different ways. Every sentence, in English or in algebra, can be variously read and interpreted.

  …Mathematics, riddles and, humor have something else in common. They share similar emotions. Humor, of course, is quick. No one
laughs at a joke which has taken an hour to work out, and a joke that has to be explained is an embarrassment to the comedian and the audience. Riddles are harder work. Mathematics – and science – are harder still, but even more enjoyable.

  …This book is about solving mathematical riddles by “seeing”, sometimes with the eyes, sometimes without.’