MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3. Spheres and Hyperspheres Answers on page 43 4. Patterns of Induction Answers on page 55 5. Elegant Triangles Answers on page 64 6. Random Walks and Gambling Answers on page 74 7. Random Walks on the Plane and in Space Answers on page 86 8. Boolean Algebra Answers on page 99 9. Can Machines Think? Answers on page 110 MATHEMATICALCIRCUS Cyclic Numbers Answers on page 121 Eccentric Chess and Other Problems Answers on page 130 Dominoes Answers on page 150 Fibonacci and Lucas Numbers Answers on page 167 lSimplicity Answers on page 180 'The Rotating Round Table and Other Problems Answers on page 191 ,Solar System Oddities Answers on page 21 4 ,Mascheroni Constructions Answers on page 228 The Abacus Answers on page 241 .Palindromes: Words and Numbers Answers on page 251 Dollar Bills Answers on page 260 Bibliography Preface LAAA'L'I'Y-DEEEZ AND GENNNNNTLEMEN: il'elcorne to the Greatest Mathematical Show on Earth! Coine and see the lrlost captivating and nlind-boggling puzzles ever derived by llunlan ingenuity! Marvel at the mysterious and fascinating parade of patterns in numbers, in words, in geometry, and in nature! Thrill to exotic and exciting paradoxes, to unbelievable feats of irlerltal gymnastics! il'atch the three rings in Chapters 1, 3, 8, 15, and 1t! This is it-the REAL SOURCE of all the Best Anlusements, no~v Larger than Ever, in a new edition. Martin Gardner is once again the skillfl~lringirlaster of a fast-paced varietj sho~l-.There's something here for everyone; indeed, there are dozens of things here for everyone. The t~.\~entychapters of this book are nicely bal- anced between all sorts of stinlulatiiig ideas, suggested by doxvn-to-earth objects like niatchsticks and dollar bills as well as by faraway objects like planets and infinite random walks. i\'e learn about ancient devices for arithmetic and about irloderil explanations of artificial intelligence. Tllere are feasts here for the eyes and hands as well as for the brain. The 300 colunlns Martin Gardner has written about mathematical recreations are treasures of the world, anal- ogous to the symphonies of Joseph Haydii or the paint- ings of Hieronynlous Bosch. For nlany years I have xii MATHEMATICALCIRCUS kept them close at hand in my working library as sources of information and inspiration. First I saved the original pages, torn from my copies of Scientific American where the material first appeared. Later, as collections of columns began to appear in book form, I avidly acquired each book, appreciating the extra anecdotes and facts that had been added. I hope that someday these gems become among the first collections of writings to be made available on line, when technology permits books to be distributed on cassettes in digital form. What makes these little essays so special? There are many reasons, probably more than I can think of, but I suppose the principal one is that Martin's own enthusi- asm shines through his gracious writing. He has a special knack for describing mathematical ideas with a mini- mum of jargon, so that the beauty of the concepts can be appreciated by people of all ages, in all walks of life. His writing is accessible to my parents and to my children; yet the mathematics he describes is sufficiently concen- trated that professionals like myself can still learn much from it. P. T. Barnum correctly observed that people like to be hoodwinked once in awhile, and Martin the Magician is full of tricks and amusing swindles. But the important thing is that he is scrupulously fair. He painstakingly checks all of his facts and provides excellent historical background. These essays are masterpieces of scholar- ship as well as exposition; they are thoroughly reliable and carefully researched. On several occasions I have made what I thought was a comprehensive study of some topic, while Martin was independently preparing a col- umn about the same thing. Invariably I would find that he had selected all the choicest tidbits I knew and he would also have uncovered a few nuggets I had missed. So hurry, hurry, hurry-step right up into the Big Top: A stupendous show is about to begin! Gather a bagful of peanuts and take your seats. The band is ready to start the overture. On with the show! Introduction Sometimes these cogitations still amaze The troubled midnight and the noon's repose. -T. S. Eliot THECHAPTERS OF THIS BOOK originally appeared in Scientific American as monthly columns with the heading "Mathematical Games." At times mathematicians ask me what I take the phrase to mean. It is not easy to reply. The word "game" was used by Ludwig Wittgenstein to illustrate what he meant by a "family word" that cannot be given a single definition. It has many meanings that are linked together somewhat like the members of a human family, meanings that became linked as the language evolved. One can define "mathematical games" or "recreational mathematics" by saying it is any kind of mathematics with a strong play element, but this is to say little, because "play," "recreation," and "game" are roughly synonymous. In the end one has to fall back oh such dodges as defining poetry as what poets write, or jazz as what jazz musicians play. Recreational math is the kind of math that recreational mathematicians enjoy. Although I cannot define a mathematical game any better than I can a poem, I do maintain that, whatever it is. it is the best way to capture the interest of young people in teaching ele- mentary mathematics. A good mathematical puzzle, paradox, or magic trick can stimulate a child's imagination much faster than a practical application (especially if the application is remote from the child's experience), and if the "game" is chosen care- xiii xiv Infroducfion fully it can lead almost effortlessly into significant mathematical ideas. Not only children but adults too can become obsessed by a puzzle that has no foreseeable practical use, and the history of mathematics is filled with examples of work on such puzzles, by professionals and amateurs alike, that led to unexpected conse- quences. In his book Mathematics: Queen and Servant of Sci- ence, Eric Temple Bell cites the early work on classifying and enumerating knots as something that once seemed little more than puzzle play, but later became a flourishing branch of topology: So the problems of knots after all were more than mere puz- zles. The like is frequent in mathematics, partly because mathe- maticians have sometimes rather perversely reformulated seri- ous problems as seemingly trivial puzzles abstractly identical with the difficult problems they hoped but failed to solve. This low trick has decoyed timid outsiders who might have been scared off by the real thing, and many deluded amateurs have made substantial contributions to mathematics without suspect- ing what they were doing. An example is T. P. Kirkman's (1806- 1895) puzzle of the fifteen schoolgirls (1850) given in books on mathematical recreations. Some mathematical puzzles really are trivial and lead nowhere. Yet both types have something in common, and no one has ex- pressed this better than the eminent mathematician Stanislaw Ulam in his autobiography, Adventures of a Mathematician: With all its grandiose vistas, appreciation of beauty, and vi- sion of new realities, mathematics has an addictive property which is less obvious or healthy. It is perhaps akin to the action of some chemical drugs. The smallest puzzle, immediately rec- ognizable as trivial or repetitive, can exert such an addictive influence. One can get drawn in by starting to solve such puz- zles. I remember when the Mathematical hlonthly occasion- ally published problems sent in by a French geometer concern- ing banal arrangements of circles, lines and triangles on the plane. "Belanglos," as the Germans say, but nevertheless these figures could draw you in once you started to think about how to solve them, even when realizing all the time that a solution could hardly lead to more exciting or more general topics. This Introduction xv is much in contrast to what I said about the history of Fermat's theorem, which led to the creation of vast new algebraical con- cepts.