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ENTERTAINING MATHEMATICAL PUZZLES

ENTERTAINING MATHEMATICAL PUZZLES
Martin Gardner
Illustrated by Anthony Ravielli

DOVER PUBLICATIONS, INC. NEW YORK

Text copyright © 1961 by Martin Gardner. Illustrations copyright © 1961 by Anthony Ravielli. All rights reserved under Pan American and International Copyright Conventions. Published in Canada by General Publishing Company, Ltd., 30Lesmill Road, Don Mills, Toronto, Ontario.

ThiS Dover edition, first published in 1986, is an unabridged and slightly corrected republication of the work first published by Thomas Y Crowell Company, New York, in 1961 under the title MatheTfUltical Puzzles. Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

Library of CongressCataloging-in-Publication Data
GardneI; Martin, 1914Entertaining mathematical puzzles. Reprint. Originally published: Mathematical puzzles. New York: Crowell, 1961. With minor corrections. Bibliography: p. 1. Mathematical recreatiOnS. 1. Title. QA95.G29 1986 793.7'4 86-16505 ISBN 0-486-25211-6

FOR JIMMY

Introduction
In selecting material for this collection I have done my best to find puzzlesthat are unusual and entertaining, that call for only the most elementary knowledge of mathematics, but at the same time provide stimulating glimpses into higher levels of mathematical thinking. The puzzles (many of which appeared in my column "On the Light Side" that ran in Science World) have been grouped into sections, each dealing with a different area of mathematics. Brief comments at thebeginning of each section suggest something of the nature and importance of the kind of mathematics one must use in tackling the puzzles of that section. In the answers, I have tried to go into as much detail as space permits in explaining how each problem is solved, and pointing out some of the inviting paths that wind away from the problems into lusher areas of the mathematical jungle. Perhaps inplaying with these puzzles you will discover that mathematics is more delightful than you expected. Perhaps this will make you want to study the subject in earnest, or less hesitant about taking up the study of a science for which a knowledge of advanced mathematics will eventually be required. Surely no one today can doubt the enormous pracvii

tical value of mathematics. Without its use as atool, the discoveries and achievements of modem science would have been impossible. But many people do not realize that mathematicians actually enjoy mathematics. Take my word for it, there is as much satisfaction in knocking over an interesting problem with a well-aimed thought as there is in knocking over ten wooden pins with a well-aimed bowling ball. In one of L. Frank Baum's funniest fantasies,The Emerald City of Oz, Dorothy (together with the Wizard and her uncle and aunt) visit the city of Fuddlecumjig in the Quadling section of Oz. Its remarkable inhabitants, the Fuddles, are made of pieces of painted wood cleverly fitted together like three-dimensional jigsaw puzzles. As soon as an outsider approaches they scatter in a heap of disconnected pieces on the Hoor so that the visitorwill have the pleasure of putting them together again. As Dorothy's party leaves the city, Aunt Em remarks: "Those are certainly strange people, but I really can't see what use they are, at all." "Why, they amused us for several hours," replies the Wizard. "That is being of use to us, I'm sure." "I think they're more fun than playing solitaire or mumbletypeg," Uncle Henry adds. "For my part, I'm gladwe visited the Fuddles." I hope that you will resist mightily the temptation to look at the answer before you try seriously to work
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a problem. And I hope that when you finish with these puzzles you will be glad, like Uncle Henry, to have been befuddled by them.
M arlin Gardner

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Contents
Introduction
Part I
vii

ARITHMETIC PUZZLES
The Colored Socks Weighty Problem The...