by Wells, David
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.’