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SI0.0 GEOFFREY MOTT-SMITH MATHEMATICAL PUZZLES FOR BEGINNERS & ENTHUSIASTS for recreation and mental stimulus this book will prove a diverting pastime/ as well as a challenge to your ability to concentrate and apply logic to your reasoning MATHEMATICAL PUZZLES for Beginners and Enthusiasts By Geoffrey Mott-Smith SECOND REVISED EDITION DOVER PUBLICATIONS, INC. COPYRIGHT 1946 BY THE BLAKISTON COMPANY COPYRIGHT 1954 BY DOVER PUBLICATIONS, INC. This new Dover edition is an unabridged republication of the first edition with revisions and corrections. Library of Congress Catalog Card Number: 55-3389 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York 14, N. Y. PREFACE If you have had fun in solving puzzles, if you like to enter- tain your friends with puzzles and mental gymnastics, this is your kind of book. Here you will find easy puzzles, hard puzzles, puzzles useful and amusing, puzzles for beginners and puzzles for old- timers, puzzles to challenge your logic, your ingenuity, your knowledge. Some of these puzzles are old favorites; many of them are new ones invented by the author. The primary object of the book is to entertain. To solve many of the puzzles, you need no knowledge of mathematics other than simple arithmetic. Other puzzles require a knowledge of ele- mentary algebra and plane geometry. Some of the puzzles are solved for the reader in the text, in order to show how to attack more complex puzzles of the same type. Answers to all puzzles are given on pages 139-235, and here the full method of solu- tion is explained for all the more difficult puzzles. The chapters of the book are arranged in the order in which they should be read by anyone whose schooling in mathematics is not fresh in his mind. The first chapter contains easy arithmetic puzzles, most of which can and should be solved without recourse to pencil and paper. The second chapter takes up puzzles based on logic, to sharpen the reader's ingenuity. The third and fourth chapters present a variety of types of puzzles, ranging from easy to difficult, which can be solved by simple algebra. Geometry is introduced by dissection puzzles, some of which are solved by theoretic considerations and some by simple trial and error. Other aspects of geometry are touched upon in the sixth chapter. The seventh and eighth chapters dig into the properties of digits and integers; they contain the hardest puzzles in the book. Related puzzles of decimation are given a separate chapter following. The tenth and eleventh chapters concern permutations and combina- tions and probability, a rather specialized field, so that funda- mental formulas are given for the guidance of the beginner. The last two chapters analyze some number and board games of a mathematical character and others with elements of mathematical V vi PREFACE exactitude. While games seem far removed from the formal study of mathematics, they afford an excellent opportunity for the exer- cise of ingenuity in analysis. Within each chapter the puzzles are best attacked in the given order, since in some cases a puzzle depends for its solution upon some previous puzzle in the chapter. In the Appendix are given tables of primes, squares, and so on, together with explanations of how to extract square and cube root. Besides being generally useful to the puzzle addict, these tables are needed for the solution of a few problems in this book. If the reader wishes to delve further into the theory of mathe- matical puzzles, he should consult the works of Sam Loyd, H. E. Dudeney, and W. W. Rouse Ball. Sam Loyd (1847-1910) was a genius in the invention of puzzles of all sorts. Besides being one of the great pioneers in the composition of chess problems, he invented many of the forms in which puzzles are now cast. His works were printed mostly in periodicals, but several compendiums of his puzzles have been published. H. E. Dudeney (1857-1931) was an English mathematician who interested himself in puzzles, and published several collections of his own inventions. He was the first to solve a number of classi- cal problems. The reader of his works must be prepared to find very easy and very difficult puzzles intermixed without warning. Another English mathematician of the same period, W. W. Rouse Ball, published in 1892 his Mathematical Recreations, one of the definitive works on classical problems and the theory of their solution. The books of these pioneers are out of print, but second-hand copies are fairly easy to obtain, and the books are of course avail- able in many libraries. Thanks are due to Albert H. Morehead, Rubin Atkin, Lewi Tonks, and L. F. Lafleur for valuable suggestions incorporated in the text, and to my son John for his able assistance in the prepara- tion of the manuscript. G. M-S. CONTENTS PREFACE v Part One MATHEMATICAL PUZZLES I. EASY ARITHMETICAL PUZZLES . . .. 3 II. PUZZLES OF INFERENCE AND INTERPRE- TATION 11 III. ALGEBRAIC PUZZLES—GROUP ONE . .. 20 IV. ALGEBRAIC PUZZLES—GROUP TWO . . 30 V. DISSECTION OF PLANE FIGURES . . .. 40 VI. GEOMETRICAL PUZZLES 54 VII. PROPERTIES OF DIGITS 67 VIII. PUZZLES ABOUT INTEGERS 79 IX. DECIMATION PUZZLES 94 X. PERMUTATIONS AND COMBINATIONS . 98 XI. PROBLEMS OF PROBABILITY 109 XII. NUMBER GAMES 115 XIII. BOARD GAMES 120 Part Two SOLUTIONS SOLUTIONS AND ANSWERS TO THE PUZZLES IN PART ONE 139 APPENDIX HOW TO EXTRACT SQUARE ROOT 236 HOW TO EXTRACT CUBE ROOT 238 TABLE OF SQUARE ROOTS 239 vii viii CONTENTS TABLE OF POWERS OF 2 239 TABLE OF SQUARE NUMBERS 24O TABLE OF PRIME NUMBERS 24I TABLE OF TRIANGULAR NUMBERS 242 GLOSSARY 243 Part One MATHEMATICAL PUZZLES I. Easy Arithmetical Puzzles 1. HOW HIGH !S A POLE? How high is a pole that casts a shadow 21 feet long, if a 6-foot man standing beside the pole casts a shadow 4 J/2 feet long? How deep is a well, if a rope that just reaches from bottom to top can be wrapped exactly 12 times around the cylindrical drum of a windlass, the drum being 7 inches in diameter? How many sheep jump over a fence in an hour if 10 sheep jump over a fence in 10 minutes? 2. DOMINO SETS. In a domino set that runs up to double- six, there are 28 bones (pieces). In a set that runs up to double- nine, there are 55 bones. How many bones are there in a domino set that runs up to double-twelve? 3. MARK-DOWN. A clothing dealer trying to dispose of an overcoat cut in last year's style marked it down from its original price of $30 to $24. Failing to make a sale he reduced the price still further to $19.20. Again he found no takers, so he tried an- other price reduction and this time sold it. What was the selling price, if the last mark-down was consistent with the others? 3

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