Origami, Eleusis, and The Soma Cube
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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 in 1959.)
The New Martin Gardner Mathematical Library
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.
Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games
Origami, Eleusis, and the Soma Cube: Martin Gardner's Mathematical Diversions
Sphere Packing, Lewis Carroll, and Reversi: Martin Gardner's New Mathematical Diversions
Knots and Borromean Rings, RepTiles, and Eight Queens: Martin Gardner's Unexpected Hanging
Klein Bottles, OpArt, and SlidingBlock Puzzles: More of Martin Gardner's Mathematical Games
Sprouts, Hypercubes, and Superellipses: Martin Gardner's Mathematical Carnival
Nothing and Everything, Polyominoes, and Game Theory: Martin Gardner's Mathematical Magic Show
Random Walks, Hyperspheres, and Palindromes: Martin Gardner's Mathematical Circus
Words, Numbers, and Combinatorics: Martin Gardner on the Trail of Dr. Matrix
Wheels, Life, and Knotted Molecules: Martin Gardner's Mathematical Amusements
Knotted Doughnuts, Napier's Bones, and Gray Codes: Martin Gardner's Mathematical Entertainments
Tangrams, Tilings, and Time Travel: Martin Gardner's Mathematical Bewilderments
Penrose Tiles, Trapdoor Ciphers, and the Oulipo: Martin Gardner's Mathematical Tour
Fractal Music, Hypercards, and Chaitin's Omega: Martin Gardner's Mathematical Recreations
The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications: Martin Gardner's Last Mathematical Recreations
Origami, Eleusis, and the Soma Cube
Martin Gardner's Mathematical Diversions
Martin Gardner
CAMBRIDGE UNIVERSITY PRESS
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© 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 2nd SCIENTIFIC AMERICAN Book of Mathematical
Puzzles & Diversions, Simon and Schuster, 1961
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– Origami, Eleusis, and the Soma cube : Martin Gardner's mathematical diversions / Martin Gardner. p. cm. – (The new Martin Gardner mathematical library) Includes bibliographical references and index. ISBN 9780521756105 (hardback) 1. Mathematical recreations. I. Title. II. Series. QA95.G2975 2008 793.74–dc22 2008012534
ISBN 9780521756105 hardback
ISBN 9780521735247 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

ix 
1 The Five Platonic Solids

1 
2 Tetraflexagons

11 
3 Henry Ernest Dudeney: England's Greatest Puzzlist

20 
4 Digital Roots

32 
5 Nine Problems

39 
6 The Soma Cube

51 
7 Recreational Topology

66 
8 Phi: The Golden Ratio

76 
9 The Monkey and the Coconuts

91 
10 Mazes

98 
11 Recreational Logic

106 
12 Magic Squares

117 
13 James Hugh Riley Shows, Inc.

129 
14 Nine More Problems

139 
15 Eleusis: The Induction Game

151 
16 Origami

160 
17 Squaring the Square

173 
18 Mechanical Puzzles

194 
19 Probability and Ambiguity

204 
20 The Mysterious Dr. Matrix

218 
Index

229 
Acknowledgments
Martin Gardner thanks Scientific American for allowing reuse of material from his columns in that magazine, material copyright © 1958 (Chapters 1–7, 9, 17), 1959 (Chapters 8, 10–16, 18, 19), and 1960 (Chapter 20) 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 Egelson), Irving Geis (via heir Sandy Geis), Harold Jacobs, Amy Kasai, Alex Semenoick, 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 of Bernardino Luini, “Boy with a Toy,” Elton Hall Collection is reproduced from “Gibeciere” vol. 1, no. 1 and used by permission. Photograph in Figure 51, courtesy of National Gallery of Art, copyright the Salvador Dali estate, is used by permission. Photograph of Albrecht Durer’s Melancolia I is courtesy of Owen Gingerich. Photograph in figure Y is courtesy of Ed Vogel.
Introduction
Since the appearance of the first Scientific American Book of Mathematical Puzzles & Diversions, in 1959, popular interest in recreational mathematics has continued to increase. Many new puzzle books have been printed, old puzzle books have been reprinted, kits of recreational math materials are on the market, a new topological game (see Chapter 7) has caught the fancy of the country's youngsters, and an excellent little magazine called Recreational Mathematics has been started by Joseph Madachy, a research chemist in Idaho Falls. Chessmen – those intellectual status symbols – are jumping all over the place, from TV commercials and magazine advertisements to Al Horowitz's lively chess corner in The Saturday Review and the knight on Paladin's holster and havegunwilltravel card.
This pleasant trend is not confined to the United States. A classic fourvolume French work, Récréations Mathématiques, by Eduouard Lucas, has been reissued in France in paperback. Thomas H. O’Beirne, a Glasgow mathematician, is writing a splendid puzzle column in a British science journal. A handsome 575page collection of puzzles, assembled by mathematics teacher Boris Kordemski, is selling in Russian and Ukrainian editions. It is all, of course, part of a worldwide boom in math – in turn a reflection of the increasing demand for skilled mathematicians to meet the incredible needs of the new triple age of the atom, spaceship, and computer.
Computers are not replacing mathematicians; they are breeding them. It may take a computer less than 20 seconds to solve a thorny problem, but it may have taken a group of mathematicians many months to program the problem. In addition, scientific research is becoming more and more dependent on the mathematician for important breakthroughs in theory. The relativity revolution, remember, was the work of a man who had no experience in the laboratory. At the moment, atomic scientists are thoroughly befuddled by the preposterous properties of some 30 different fundamental particles, “a vast jumble of odd dimensionless numbers,” as J. Robert Oppenheimer has described them, “none of them understandable or derivable, all with an insulting lack of obvious meaning.” One of these days a great creative mathematician, sitting alone and scribbling on a piece of paper, or shaving, or taking his family on a picnic, will experience a flash of insight. The particles will spin into their appointed places, rank on rank, in a beautiful pattern of unalterable law. At least, that is what the particle physicists hope will happen. Of course the great puzzle solver will draw on laboratory data, but the chances are that he will be, like Einstein, primarily a mathematician.
Not only in the physical sciences is mathematics battering down locked doors. The biological sciences, psychology, and the social sciences are beginning to reel under the invasion of mathematicians armed with strange new statistical techniques for designing experiments, analyzing data, and predicting probable results. It may still be true that if the president of the United States asks three economic advisers to study an important question, they will report back with four different opinions, but it is no longer absurd to imagine a distant day when economic disagreements can be settled by mathematics in a way that is not subject to the usual dismal disputes. In the cold light of modern economic theory, the conflict between socialism and capitalism is rapidly becoming, as Arthur Koestler has put it, as naïve and sterile as the wars in Lilliput over the two ways to break an egg. (I speak only of the economic debate; the conflict between democracy and totalitarianism has nothing to do with mathematics.)
But those are weighty matters, and this is only a book of amusements. If it has any serious purpose at all, it is to stimulate popular interest in mathematics. Such stimulation is surely desirable, if for no other reason than to help the layman understand what the scientists are up to. And they are up to plenty.
I would like to express again my gratitude to the publisher, editors, and staff of Scientific American, the magazine in which these chapters first appeared; to my wife for assistance in many ways; and to the hundreds of friendly readers who continue to correct my errors and suggest new material. I would like also to thank, for her expert help in preparing the manuscript, Nina Bourne of Simon and Schuster.
© Cambridge University Press