New Mathematical Diversions

MARTIN GARDNER MARTIN GARDNER NEW MATHEMATICAL MATHEMATICALN~N RTIN GARDNER More a from Martin Gardner as they appeared in Scientific American with a Postscript and new Bibliography from the Author and over 100 drawings and diagrams The Mathematical Association of America Washington, D.C. @ 1995 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 95-76293 ISBN 0-88385-517-8 Printed in the United States of America Current Printing (last digit): lo987654321 SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications JAMES W. DANIEL, Chair Spectrum Editorial Board ROGER HORN, Editor ARTHUR T. BENJAMIN DIPA CHOUDHURY HUGH M. EDGAR RICHARD K. GW DAN KALMAN LESTER H. LANGE MARY R. PARKER A. WAYNE ROBERTS EDWARD R. SCHEINERMAN SANFORD SEGAL SPECTRUM SERIES The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new rnathe- matical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teachers of mathematics, mathematical amateurs, and researchers. All the Math That's Fit to Print, by Keith Devlin Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher From Zero to Infinity, by Constance Reid I Want to be a Mathematician, by Paul R. Halmos Journey into Geometries, by Marta Sved The Last Problem, by E. T. Bell (revised and updated by Underwood Dudley) The Lighter Side of Mathematics: Proceedings of the Eug2ne Strens Memorial Conference on Recreational Mathematics & its History, edited by Richard K. Guy and Robert E. Woodrow Lure of the Integers, by Joe Roberts Mathematical Carnival, by Martin Gardner Mathematical Circus, by Martin Gardner Mathernatical Cranks, by Underwood Dudley Mathernatical Magic Show, by Martin Gardner Mathematics: Queen and Servant of Science, by E. T. Bell Memorabilia Mathematics, by Robert Edouard Moritz New Mathematical Diversions, by Martin Gardner Numerical Methods that Work, by Forman Acton Out of the Mouths of Mathematicians, by Rosemary Schmalz Polyominoes, by George Martin The Search for E. T. Bell, also known as John Taine, by Constance Reid Shaping Space, edited by Marjorie Senechal and George Fleck Student Research Projects in Calculus, by Marcus Cohen, Edward D. Gaughan, Arthur Knoebel, Douglas S. Kurtz, and David Pengelley The Trisectors, by Underwood Dudley The Words of Mathematics, by Steven Schwartzman Mathematical Association of America 1529 Eighteenth Street, NW Washington, DC 20036 800-33 1-1MAA FAX 202-265-2384 CONTENTS INTRODUCTION 1. The Binary System [Answers on page 221 2. Group Theory and Braids [Answers on page 331 3. Eight Problems [Answers on page 391 4. The Games and Puzzles of Lewis Carroll [Answers on page 531 5. Paper Cutting [Answers on page 691 6. Board Games [Answers on page 801 7. Packing Spheres [Answers on page 901 8. The Transcendental Number Pi [Answers on page 1001 9. Victor Eigen : Mathemagician [Answers on page 1121 10. The Four-Color Map Theorem [Answers on page 122 1 11. Mr. Apollinax Visits New York [Answers on page 1321 12. Nine Problems [Answers on page 1401 13. Polyominoes and Fault-Free Rectangles [Answers on page 1581 8 Contents 14. Euler's Spoilers: The Discovery of an Order-10 Graeco-Latin Square [Answers on page 1711 15. The Ellipse [Answers on page 1831 16. The 24 Color Squares and the 30 Color Cubes [Answers on page 1951 17. H.S.M. Coxeter [Answers on page 2091 18. Bridg-it and Other Games [Answers on page 2161 19. Nine More Problems [Answers on page 2251 20. The Calculus of Finite Differences [Answers on page 2441 POSTSCRIPT BIBLIOGRAPHY INTRODUCTION "A good mathematical joke," wrote the British mathematician John Edensor Littlezoood (in the i?~troductionto his Mathemati- cian's Miscellany), "is better, and better mathematics, than a dozen mediocre papers." This is a book of mathematical jokes, if "joke" is taken in a sense broad enough to include any kind of mathematics that is mixed zc~itha strong element of fun. Most mathematicians relish such play, though of course they keep it zoithin reasonable bounds. There is a fascination about recreational mathematics that can, for some persons, become a kind of drug. Vladimir Nabokov's great chess novel, The Defense, is about such a man. He permitted chess (one form of mathematical play) to dominate his mind so completely that he finally lost contact with the real world and ended his miserable life-game zoith zohat chess problemists call a suimate or self-mate. He jumped out of a zoindow. It is consistent with the steady disintegratiotl of Nabokov's chess master that as a boy he had been a poor student, even in mathematics, at the same time that he had been "extraordinarily engrossed in a collec- tion of problems entitled Merry Mathematics, in the fantastical misbehavior of numbers and the zoayzoard frolic of geometric lines, in everything that the schoolbook lacked." The moral is: Enjoy mathematical play, if you have the mind and taste for it, but don't enjoy it too much. Let it provide occa- sional holidays. Let it stimulate your interest in serious science and mathematics. Bzit keep it under firm control. And if you can't keep it under control, you can take some com- fort from the point of Lord Dunsany's story "The Chess-Player, the Financier, and Another." A financier recalls a friend named Smoggs zoho toas on the road to becoming a brilliant financier until he got sidetracked by chess. "It came gradually at first: he used to play chess with a man during the lunch hour, tohen he and I both zoorked for the same firm. And after a while he began 10 Introduction to beat the fellozo. And then he joined a chess club, and some kind of fascination seemed to come over him; something, like drink, or more like poetry or music . he could have been a financier. They say it's no harder than chess, though chess leads to nothing. I never saw such brains zcvxsted." "There are men like that," agrees the prison zuarden. "It's a pity . ." And he locks the financier back in his cell for the night. My thanks again to Scientific American for permission to re- print these twenty columns. As in the tzoo previous book collec- tions, the columns have been expanded, errors corrected and much nezu material added that zoas sent to me by readers. I am grateful, also, to my wife for help in proofing; to my editor, Nina Bourne; above all, to that still-growing band of readers, scattered throughout the nation and the zuorld, zohose zoelcome letters have so great- ly enriched the material reprinted here. Evoly met TO L. R. AHCROF, emitero meno. CHAPTER ONE I3 The Binary System A red ticket showed between wiper and windshield; I carefully tore it into two, four, eight pieces. - Vladimir Nabokov, Lolita THE NUMBER SYSTEM now in use throughout the civilized world is a decimal system based on successive powers of 10. The digit at the extreme right of any number stands for a multiple of 100, or 1. The second digit from the right indicates a multiple of 101; the third digit, a multiple of 102, and so on. Thus 777 ex- presses the sum of (7 X 100) + (7 X 101) + (7 X lo2). The widespread use of 10 as a number base is almost certainly due to the fact that we have ten fingers; the very word "digit" reflects this. If Mars is inhabited by humanoids with twelve fingers, it is a good bet that Martian arithmetic uses a notation based on 12. The simplest of all number systems that make use of the posi- tions of digits is the binary system, based on the powers of 2. Some primitive tribes count in binary fashion, and ancient Chi- nese mathematicians knew about the system, but it was the great German mathematician Gottfried Wilhelm von Leibniz who seems to have been the first to develop the system in any detail. For Leibniz, it symbolized a deep metaphysical truth. He regarded 0 as an emblem of nonbeing or nothing; 1 as an emblem of being or substance. Both are necessary to the Creator, because a cosmos containing only pure substance would be indistinguishable from the empty cosmos, devoid of sound and fury and signified by 0. Just as in the binary system any integer can be expressed by a suitable placing of 0's and l's, so the mathematical structure of 14 The Binary System the entire created world becomes possible, Leibniz believed, as a consequence of the primordial binary split between being and nothingness. From Leibniz's day until very recently the binary system was little more than a curiosity, of no practical value. Then came the computers! Wires either do or do not carry a current, a switch is on or off, a magnet is north-south or south-north, a flip-flop mem- ory circuit is flipped or flopped. For such reasons enormous speed and accuracy are obtained by constructing computers that can process data coded in binary form. "Alas !" writes Tobias Dantzig in his book Number, the Language of Science, "what was once hailed as a monument to monotheism ended in the bowels of a robot." Many mathematical recreations involve the binary system: the game of Nim, mechanical puzzles such as the Tower of Hanoi and the Rings of Cardan, and countless card tricks and "brainteasers." Here we shall restrict our attention to a familiar set of "mind- reading" cards, and a closely related set of punch-cards with which several remarkable binary feats can be performed.

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