Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi
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For 25 of his 90 years, Martin Gardner wrote “Mathematical Games and Recreations,” a monthly column for Scientific American magazine. These columns have inspired hundreds of thousands of readers to delve more deeply into the large world of mathematics. He has also made significant contributions to magic, philosophy, debunking pseudoscience, and children's literature. He has produced more than 60 books, including many best sellers, most of which are still in print. His Annotated Alice has sold more than a million copies. He continues to write a regular column for the Skeptical Inquirer magazine. (The photograph is of the author at the time of the first edition.)
The New Martin Gardner Mathematical Library
Editorial Board
From 1957 through 1986 Martin Gardner wrote the “Mathematical Games” columns for Scientific American that are the basis for these books. Scientific American editor Dennis Flanagan noted that this column contributed substantially to the success of the magazine. The exchanges between Martin Gardner and his readers gave life to these columns and books. These exchanges have continued and the impact of the columns and books has grown. These new editions give Martin Gardner the chance to bring readers up to date on newer twists on old puzzles and games, on new explanations and proofs, and on links to recent developments and discoveries. Illustrations have been added and existing ones improved, and the bibliographies have been greatly expanded throughout.
1 Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner’s First Book of Mathematical Puzzles and Games
2 Origami, Eleusis, and the Soma Cube: Martin Gardner’s Mathematical Diversions
3 Sphere Packing, Lewis Carroll, and Reversi: Martin Gardner’s New Mathematical Diversions
4 Knots and Borromean Rings, RepTiles, and Eight Queens: Martin Gardner’s Unexpected Hanging
5 Klein Bottles, OpArt, and SlidingBlock Puzzles: More of Martin Gardner’s Mathematical Games
6 Sprouts, Hypercubes, and Superellipses: Martin Gardner’s Mathematical Carnival
7 Nothing and Everything, Polyominoes, and Game Theory: Martin Gardner’s Mathematical Magic Show
8 Random Walks, Hyperspheres, and Palindromes: Martin Gardner’s Mathematical Circus
9 Words, Numbers, and Combinatorics: Martin Gardner on the Trail of Dr. Matrix
10 Wheels, Life, and Knotted Molecules: Martin Gardner’s Mathematical Amusements
11 Knotted Doughnuts, Napier’s Bones, and Gray Codes: Martin Gardner’s Mathematical Entertainments
12 Tangrams, Tilings, and Time Travel: Martin Gardner’s Mathematical Bewilderments
13 Penrose Tiles, Trapdoor Ciphers, and the Oulipo: Martin Gardner’s Mathematical Tour
14 Fractal Music, Hypercards, and Chaitin’s Omega: Martin Gardner’s Mathematical Recreations
15 The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications: Martin Gardner’s Last Mathematical Recreations
Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi
Martin Gardner’s First Book of Mathematical Puzzles and Games
Martin Gardner
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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Information on this title: www.cambridge.org/9780521756150
© Mathematical Association of America 2008
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 2008
First edition published as The SCIENTIFIC AMERICAN Book of Mathematical Puzzles & Diversions, Simon and Schuster, 1959
Printed in the United States of America
A catalog record for this publication is available from the British Library.
Library of Congress Cataloging in Publication Data
Gardner, Martin, 1914– Hexaflexagons, probability paradoxes, and the Tower of Hanoi : Martin Gardner’s first book of mathematical puzzles and games / Martin Gardner. p. cm. – (The new Martin Gardner mathematical library) Includes bibliographical references and index. ISBN 9780521756150 (hardback) 1. Mathematical recreations. I. Title. II. Series. QA95.G247 2008 793.74 – dc22 2008012533
ISBN 9780521756150 hardback
ISBN 9780521735254 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or thirdparty Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
Contents
Acknowledgments

viii 
Introduction to the First Edition

ix 
Preface to the Second Edition

xiii 
1 Hexaflexagons

1 
2 Magic with a Matrix

16 
3 Nine Problems

24 
4 Ticktacktoe

37 
5 Probability Paradoxes

48 
6 The Icosian Game and the Tower of Hanoi

63 
7 Curious Topological Models

73 
8 The Game of Hex

82 
9 Sam Loyd: America's Greatest Puzzlist

94 
10 Mathematical Card Tricks

109 
11 Memorizing Numbers

115 
12 Nine More Problems

123 
13 Polyominoes

137 
14 Fallacies

157 
15 Nim and Tac Tix

166 
16 Left or Right?

177 
Index

189 
Acknowledgments
Martin Gardner thanks Scientific American for allowing reuse of material from his columns in that magazine, material Copyright © 1956 (Chapter 1), and 1957 (Chapters 2–13), and 1958 (Chapters 14–16) by Scientific American, Inc. He also thanks the artists who contributed to the success of these columns and books for allowing reuse of their work: James D. Egelson (via heirs Jan and Nicholas Egleson), Irving Geis (via heir Sandy Geis), Harold Jacobs, Amy Kasai, and Bunji Tagawa (via Donald Garber for the Tagawa Estate). Artists names are cited where these were known. All rights other than use in connection with these materials lie with the original artists.
Photograph in Figure 48 is courtesy of the Museum of Fine Arts, Boston, 2008. Used by permission.
Introduction to the First Edition
The element of play, which makes recreational mathematics recreational, may take many forms: a puzzle to be solved, a competitive game, a magic trick, paradox, fallacy, or simply mathematics with any sort of curious or amusing fillip. Are these examples of pure or applied mathematics? It is hard to say. In one sense recreational mathematics is pure mathematics, uncontaminated by utility. In another sense it is applied mathematics, for it meets the universal human need for play.
Perhaps this need for play is behind even pure mathematics. There is not much difference between the delight a novice experiences in cracking a clever brain teaser and the delight a mathematician experiences in mastering a more advanced problem. Both look on beauty bare – that clean, sharply defined, mysterious, entrancing order that underlies all structure. It is not surprising, therefore, that it is often difficult to distinguish pure from recreational mathematics. The fourcolor map theorem, for example, is an important theorem in topology, yet discussions of the theorem will be found in many recreational volumes. No one can deny that paper flexagons, the subject of this book's opening chapter, are enormously entertaining toys; yet an analysis of their structure takes one quickly into advanced group theory, and articles on flexagons have appeared in technical mathematical journals.
Creative mathematicians are seldom ashamed of their interest in recreational mathematics. Topology had its origin in Euler's analysis of a puzzle about crossing bridges. Leibniz devoted considerable time to the study of a pegjumping puzzle that recently enjoyed its latest revival under the trade name of Test Your HighQ. David Hilbert, the great German mathematician, proved one of the basic theorems in the field of dissection puzzles. Alan Turing, a pioneer in modern computer theory, discussed Sam Loyd's 15puzzle (here described in Chapter 9) in an article on solvable and unsolvable problems. I have been told by Piet Hein (whose game of Hex is the subject of Chapter 8) that when he visited Albert Einstein he found a section of Einstein's bookshelf devoted to books on recreational mathematics. The interest of those great minds in mathematical play is not hard to understand, for the creative thought bestowed on such trivial topics is of a piece with the type of thinking that leads to mathematical and scientific discovery. What is mathematics, after all, except the solving of puzzles? And what is science if it is not a systematic effort to get better and better answers to puzzles posed by nature?
The pedagogic value of recreational mathematics is now widely recognized. One finds an increasing emphasis on it in magazines published for mathematics teachers, and in the newer textbooks, especially those written from the “modern” point of view. Introduction to Finite Mathematics, for example, by J. G. Kemeny, J. Laurie Snell, and Gerald L. Thompson, is livened by much recreational material. These items hook a student's interest as little else can. The high school mathematics teacher who reprimands two students for playing a surreptitious game of ticktacktoe instead of listening to the lecture might well pause and ask: “Is this game more interesting mathematically to these students than what I am telling them?” In fact, a classroom discussion of ticktacktoe is not a bad introduction to several branches of modern mathematics.
In an article on “The Psychology of Puzzle Crazes” (Nineteenth Century Magazine, December 1926) the great English puzzlist Henry Ernest Dudeney made two complaints. The literature of recreational mathematics, he said, is enormously repetitious, and the lack of an adequate bibliography forces enthusiasts to waste time in devising problems that have been devised long before. I am happy to report that the need for such a bibliography has at last been met. Professor William L. Schaaf, of Brooklyn College, compiled an excellent fourvolume bibliography, titled Recreational Mathematics, which can be obtained from the National Council of Teachers of Mathematics. As to Dudeney's other complaint, I fear that it still applies to current books in the field, including this one, but I think readers will discover here more than the usual portion of fresh material that has not previously found its way between book covers.
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