Mathematical puzzles for beginners and enthusiasts

255 Pages·16.9 MB·English

by  Geoffrey Mott- Smith

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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