THE SECOND SCIENTIFCC - L MERICAN BOOK 0 F MATHEMATICAL PUZZLES AND DIVERSIONS 1 MARTIN GARDNER THE SECOND SCIENTIFIC AMERICAN BOOK OF Mathematical Puzzles & Diversions The 2nd SCIENTIFIC AMERICAN Book of ILLUSTRATED WITH DRAWINGS AND DIAGRAMS MARTIN GARDNER Mathematical Puzzles A NEW SELECTION : from Origami to Recreational Logic, from Digital Roots and Dudeney Puzzles to the Diabolic Square, from the Golden Ratio to the Generalized Ham Sandwich Theorem. With mathematical commentaries by Mr. Gardner, ripostes from readers of Scientific American, references for further reading and, of course, solutions. With a new Postscript by the author THE UNIVERSITY OF CHICAGO PRESS Material previously published in Scienti,fic American is copyright O 1958,1959,1960 by Scientific American, Inc. Most of the drawings and diagrams appear by courtesy of Scientijic American, in whose pages they were originally published. The University of Chicago Press, Chicago 60637 Copyright O 1961,1987 by Martin Gardner All rights reserved. Published 1961 University of Chicago Press Edition 1987 Printed in the United States of America Library of Congress Cataloging in Publication Data Gardner, Martin, 1914- The 2nd Scientific American book of mathematical puzzles & diversions. Reprint. Originally published as v. 2 of The Scientific American book of mathematical puzzles & diversions. New York : Simon and Schuster, 1961. Bibliography: p. 1. Mathematical recreations. I. Scientific American. 11. Title. 111. Title: Second Scientific American book of mathematical puzzles & diversions. QA95.Gl6 1987 793.7'4 87-10760 ISBN 0-22628253-8 (pbk.) For J. H. G. who likes to tackle puzzles big enough to walk upon CONTENTS INTRODUCTION The Five Platonic Solids [Answers on page 221 Henry Ernest Dudeney: England's Greatest Puxxlist [Answers on page 411 Digital Roots [Answers on page 601 Nine Problems [Answers on page 561 The Soma Cube [Answers on page '771 Recreational Topology [Answers on page 881 Phi: The Golden Ratio [Answers on page 1021 The Monkey and the Coconuts [Answers on page 1101 Recreational Logic [Answers on page 1271 Magic Squares James Hugh Riley Shows, Inc. [Answers on page 1491 Nine More Problems [Answers on page 1561 Eleusis : The Induction Game [Answers on page 1'721 8 Contents 16. Origami [Answers on page 1841 17. Squaring the Square 186 18. Mechanical Puzzles [Answers on page 2181 19. Probability and Ambiguity [Answers on page 2291 20. The Mysterious Dr. Matrix 233 [Answers on page 2421 REFERENCES FOR FURTHER READING 245 POSTSCRIPT 253 INTRODUCTION SINCE THE APPEARANCE of the first Scientific American Book of Mathematical Puzzles & Diversions, in 1959, popular in- terest in recreational mathematics has continued to increase. Many new puzzle books have been printed, old puzzle books 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. Chess- men - those intellectual status symbols - are jumping all over the place, from TV commercials and magazine adver- tisements to A1 Horozoitz's lively chess corner in The Satur- day Review and the knight on Paladin's holster and have- gun-will-travel card. This pleasant trend is not confined to the U.S. A classic four-volume French work, Rkcrkations Mathkmatiques, by Edouard Lucas, has been reissued in France in paperbacks. Thomas H. OJBeirne, a Glasgozu mathematician, is writing a splendid puzzle column in a British science journal. In the U.S.S.R. a handsome 575-page collection of puzzles, assem- bled by mathematics teacher Boris Kordemski, is selling in Russian and Ukrainian editions. It is all, of course, part of a world-wide boom in math -in turn a reflection of the in- creasing demand for skilled mathematicians to meet the in- credible needs of the nezo triple age of the atom, spaceship and computer. The computers are not replacing mathematicians; they 10 Introduction are breeding them. It may take a computer less than twenty seconds to solve a thorny problem, but it mau have taken a group of mathematicians many months to program the prob- lem. In addition, scientific research is becoming more and more dependent on the mathematician for important break- throughs in theory. The relativity revolzction, remember, zuns 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 thirty digerent fun- damental particles; "a vast jumble of odd dimensionless numbers,'' as J. Robert Oppenheimer has described them, "none of them understandable or derivable, all zoith an in- sulting 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, zvill experience a flash of insight. The particles zoill spin into their appointed places, rank on rank, in a beautiful pattern of unalterable law. At least, that is what the particle physi- cists hope will happen. Of course the great puzzle solver toill draw on laboratory data, but the chances are that he zuill be, like Einstein, primarily a mathematician. Not only in the physical sciences is mathematics battering dozun locked doors. The biological sciences, psychology and the social sciences are beginning to reel under the invasion of mathematicians armed with strange new statistical tech- niques for designing experiments, analyzing data, 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 zoill 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 zuay that is not subject to the uszial dismal disputes. In the cold light of modern economic theory the conflict between [...]... Cut the rectangle along the broken lines S t a r t a s shown in 9a, then fold the two center squares back and to the left Fold back the column on the extreme right The cardboard should now appear as shown in 9c Again fold back the column on the right The single square projecting on the left is now folded forward and to the right This brings all six of the "1" squares to the front Fasten together the. .. outline of each center pentagon with the point of a knife so that the pentagon flaps fold easily in one direction Place the patterns together a s shown a t right in The Five Platonic Solids 19 the illustration so that the flaps of each pattern fold toward the others Weave a rubber band alternately over and under the projecting ends, keeping the patterns pressed flat When you release the pressure, the dodecahedron... underlay the structure of the traditional The Five Platonic Solids TETRAHEDRON / HEXAHEDRON F I G 1 DODECAHEDRON The five Platonic solids The cube and octahedron a r e "duals" in t h e sense t h a t if the centers of all pairs of adjacent faces on one a r e connected by s t r a i g h t lines, t h e lines form the edges of the other The dodecahedron and icosahedron a r e dually related in the same way The. .. Today there is scarcely a single puzzle book that does not contain (often without credit) dozens of brilliant mathematical problems that had their origin in Dudeney's fertile imagination He was born in the English village of Mayfield in 1857 Thus he was 16 years younger than Sam Loyd, the American puzzle genius I do not know whether the two men ever met, but in the 1890s they collaborated on a series of. .. problem of the spider and the fly H e n r y E r n e s t Lludeney: E n g l a n d ' s Greatest Puzzlist 37 is a t the middle of a n end wall, one foot from the ceiling The fly is a t the middle of the opposite end wall, one foot above the floor, and too paralyzed with fear to move What is the shortest distance the spider must crawl in order to reach the fly? The problem is solved by cutting the room... Renaissance The analysis of the Platonic solids provides the climactic final book of Euclid's Elements Johannes Kepler believed throughout his life that the orbits of the six planets known in his day could be obtained by nesting the five solids in a certain order within the orbit of Saturn Today the mathematician no longer views the Platonic solids with mystical reverence, but their rotations are studied... dual The Five Platonic Solids I5 four elements: fire, earth, air and water The dodecahedron was obscurely identified with the entire universe Because these notions were elaborated in Plato's Timaeus, the regular polyhedrons came to be known as the Platonic solids The beauty and fascinating mathematical properties of these five forms haunted scholars from the time of Plato through the Renaissance The. .. for the English magazine Tit-Bits, and later they arranged to exchange puzzles for their magazine and newspaper columns This may explain the large amount of duplication in the published writings of Loyd and Dudeney Of the two, Dudeney was probably the better mathematician Loyd excelled in catching the public fancy with manufactured toys and advertising novelties None of Dudeney’s creations had the. .. 10 units to the path, making a total of 35 The Christmas message conveyed by the letters is "Noel" (no "L") C H A P T E R TWO w Tetrajlexagons H EXAFLEXAGONS a r e diverting six-sided paper structures that can be "flexed" t o bring different surfaces into view They a r e constructed by folding a strip of paper a s explained in the first Scientific American Book of Mathematical Puzzles and Diversions. .. familiar with the binary system To facilitate finding the three positions in which you must hold the solid, simply mark in some way the three corners which must be pointed toward you a s you face the spectator There a r e other interesting ways of numbering the faces of an octahedral die I t is possible, for example, to arrange the digits 1 through 8 in such a manner that the total of the four faces . THE SECOND SCIENTIFCC - L MERICAN BOOK 0 F MATHEMATICAL PUZZLES AND DIVERSIONS 1 MARTIN GARDNER THE SECOND SCIENTIFIC AMERICAN BOOK OF Mathematical Puzzles & Diversions. American book of mathematical puzzles & diversions. Reprint. Originally published as v. 2 of The Scientific American book of mathematical puzzles & diversions. New York : Simon. p. 1. Mathematical recreations. I. Scientific American. 11. Title. 111. Title: Second Scientific American book of mathematical puzzles & diversions. QA95.Gl6 1987 793.7'4