David Wells - Games and Mathematics: Subtle Connections

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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, So 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

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 : 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 29002800 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 /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 .

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 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.

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