MATH TRICKS, BRAIN TWISTERS, AND PUZZLES MATH TRICKS, BRAIN TWISTERS, AND PUZZLES by JOSEPH DEGRAZIA, Ph.D. a -' i - Jfs V-004111p'(cid:1)42(cid:1)7 Illustrated by ARTHUR M. KRiT BELL PUBLISHING COMPANY NEW YORK This book was previously titled Math Is Fun. Copyright MCMXLVIII, MCMLIV by Emerson Books, Inc. All rights reserved. This edition is published by Bell Publishing Company, distributed by Crown Publishers, Inc., by arrangement with Emerson Books, Inc. bcdefgh BELL 1981 EDITION Manufactured in the United States of America Library of Congress Cataloging in Publication Data Degrazia, Joseph, 1883- Math tricks, brain twisters, and puzzles. Earlier ed. published under title: Math is fun. 1. Mathematical recreations. 2. Mathematics- Problems, exercises, etc. I. Title. QA95.D36 1981 793.7'4 80-26941 ISBN 0-517-33649-9 CONTENTS CHAPTER PAGI I. TRIFLES . . . . . . . . . . . 9 II. ON THE BORDERLINE OF MATHEMATICS . . 14 III. FADED DOCUMENTS . . . . . . . . 25 IV. CRYPTOGRAMS . . . . . . . . . 37 V. HOW OLD ARE MARY AND ANN? . . . . 42 VI. WOLF, GOAT AND CABBAGE - AND OTHER ODD COINCIDENCES . . . . . . . . . 46 VII. CLOCK PUZZLES . . . . . . . . . 50 VIII. TROUBLE RESULTING FROM THE LAST WILL AND TESTAMENT . . . . . . . . 53 IX. SPEED PUZZLES . . . . . . . . . 58 X. RAILROAD SHUNTING PROBLEMS . . . 65 XI. AGRICULTURAL PROBLEMS . . . . . . 69 XII. SHOPPING PUZZLES . . . . . . . . 73 XIII. WHIMSICAL NUMBERS . . . . . 78 XIV. PLAYING WITH SqUARES. . . . 83 XV. MISCELLANEOUS PROBLEMS . . . . . . 88 XVI. PROBLEMS OF ARRANGEMENT . . . . . 96 XVII. PROBLEMS AND GAMES . . . . . . . 104 SOLUTIONS . . . . . . . 109-159 5 PREFACE This book is the result of twenty years of puzzle collecting. For these many years I have endeavored to gather everything belonging to the realm of mathematical entertainment from all available sources. As an editor of newspaper columns on scientific entertainment, I found my readers keenly interested in this kind of pastime, and these readers proved to be among my best sources for all sorts of problems, both elementary and intricate. Puzzles seem to have beguiled men in every civilization, and the staples of scientific entertainment are certain historic prob- lems which have perplexed and diverted men for centuries. Besides a number of these, this book contains many problems never before published. Indeed, the majority of the problems have been devised by me or have been developed out of suggestions from readers or friends. This book represents only a relatively small selection from an inexhaustible reservoir of material. Its purpose is to satisfy not only mathematically educated and gifted readers but also those who are on less good terms with mathematics but con- sider cudgeling their brains a useful pastime. Many puzzles are therefore included, especially in the first chapters, which do not require even a pencil for their solution, let alone algebraic formulas. The majority of the problems chosen, however, will appeal to the puzzle lover who has not yet forgotten the ele- ments of arithmetic he learned in high school. And finally, those who really enjoy the beauties of mathematics will find plenty of problems to rack their brains and test their knowl- edge and ingenuity in such chapters as, for example, "Whim- sical Numbers" and "Playing with Squares" The puzzles in this book are classified into groups so that the reader with pronounced tastes may easily find his meat. Those familiar with mathematical entertainment may miss 7 certain all-too-well-known types, such as the famous magic squares. I believe, however, that branches of mathematical entertainment which have long since developed into special sciences belong only in books that set out to treat them exhaustively. Here we must pass them by, if only for reasons of space. Nor have geometrical problems been included. Lack of space has also made it impossible to present every solution fully. In a great many instances, every step of reason- ing, mathematical and other, is shown; in others, only the major steps are indicated; in others still, just the results are given. But in every single class of problems, enough detailed solutions are developed and enough hints and clues offered to show the reader his way when he comes to grips with those problems for which only answers are given without proof. I hope that with the publication of this book I have attained two objectives: to provide friends of mathematics with many hours of entertainment, and to help some of the myriads who since their school days have been dismayed by everything mathematical to overcome their horror of figures. I also take this opportunity of thanking Mr. Andre Lion for the valuable help he has extended me in the compilation of the book. Joseph Degrazia, Ph.D. 8 CHAPTER I TRIFLES We shall begin with some tricky little puzzles which are just on the borderline between serious problems and obvious jokes. The mathematically inclined reader may perhaps frown on such trifles, but he should not be too lofty about them, for he may very well fall into a trap just because he relies too much on his arithmetic. On the other hand, these puzzles do not depend exclusively on the reader's simplicity. The idea is not just to pull his leg, but to tempt him mentally into a blind alley unless he watches out. A typical example of this class of puzzle is the Search for the missing dollar, a problem-if you choose to call it one-which some acute mind contrived some years ago and which since then has traveled around the world in the trappings of prac- tically every currency. A traveling salesman who had spent several nights in a little upstate New York hotel asks for his bill. It amounts to $30 which he, being a trusting soul, pays without more ado. Right after the guest has left the house for the railroad station the desk clerk realizes that he had overcharged his guest $5. So he sends the bellboy to the station to refund the overcharge to the guest. The bellhop, it turns out, is far less honest than his supervisor. He argues: "If I pay that fellow only $3 back he will still be overjoyed at getting something he never expected- and I'll be richer by $2. And that's what he did. Now the question is: If the guest gets a refund of $3 he had paid $27 to the hotel all told. The dishonest bellhop has kept $2. That adds up to $29. But this monetary transaction started with $30 being paid to the desk clerk. Where is the 30th dollar? 9 Unless you realize that the question is misleading you will search in vain for the missing dollar which, in reality, isn't missing at all. To clear up the mess you do not have to be a certified public accountant, though a little bookkeeping knowledge will do no harm. This is the way the bookkeeper would proceed: The desk clerk received $30 minus $5, that is, $25; the bellhop kept $2; that is altogether $27 on one side of the ledger. On the other side are the expenses of the guest, namely $30 minus $3, also equalling $27. So there is no deficit from the bookkeeper's angle, and no dollar is missing. Of course, if you mix up receipts and expenses and add the guest's expenses of $27 to the dishonest bellhop's profit of $2, you end up with a sum of $29, and a misleading question. The following are further such puzzles which combine a little arithmetic with a dose of fun. 1. How much is the bottle? Rich Mr. Vanderford buys a bottle of very old French brandy in a liquor store. The price is $45. When the store owner hands him the wrapped bottle he asks Mr. Vanderford to do him a favor. He would like to have the old bottle back to put on display in his window, and he would be willing to pay for the empty bottle. "How much?" asks Mr. Vanderford. "Well," the store owner answers, "the full bottle costs $45 and the brandy costs $40 more than the empty bottle. Therefore, the empty bottle is .. ." "Five dollars," interrupts Mr. Vander- ford, who, having made a lot of money, thinks he knows his figures better than anybody else. "Sorry, sir, you can't figure," says the liquor dealer and he was right. Why? 2. Bad day on the used-car market. A used-car dealer complains to his friend that today has been a bad day. He has sold two cars, he tells his friend, for $750 each. One of the sales yielded him a 25 per cent profit. On the other one he took a loss of 25 per cent. "What are you worry- ing about?" asks his friend. "You had no loss whatsoever." 10 "On the contrary, a substantial one," answers the car dealer. Who was right? 3. The miller's fee. In a Tennessee mountain community the miller retains as his fee one-tenth of the corn the mountaineer farmers deliver for grinding. How much corn must a farmer deliver to get 100 pounds of cornmeal back, provided there is no loss? 4. Two watches that need adjusting. Charley and Sam were to meet at the railroad station to make the 8 o'clock train. Charley thinks his watch is 25 min- utes fast, while in fact it is 10 minutes slow. Sam thinks his watch is 10 minutes slow, while in reality is has gained 5 min- utes. Now what is going to happen if both, relying on their timepieces, try to be at the station 5 minutes before the train leaves? 5. Involved family relations. A boy says, "I have as many brothers as sisters." His sister says, "I have twice as many brothers as sisters." How many brothers and sisters are there in this family? 6. An ancient problem concerning snails. You may have come across the ancient problem of the snail which, endeavoring to attain a certain height, manages during the day to come somewhat closer to its objective, though at a 11