Intriguing Mathematical Problems

220 Pages·1996·2.35 MB·English

by  Oswald Jacoby

Bibliographical Note This Dover edition, first published in 1996, is an
unabridged republication of the work originally published by McGraw-Hill
Book Company, Inc., New York, in 1962 under the title Mathematics for
Pleasure.

Library of Congress Cataloging-in-Publication Data Jacoby, Oswald, 1902-
[Mathematics for pleasure]
Intriguing mathematical problems / Oswald Jacoby with William H. Benson.
p. cm.
Originally published: Mathematics for pleasure. 1st ed. New York: McGraw-
Hill, 1962.
9780486168364
1. Mathematical recreations. I. Benson, William H. II. Title.
QA95.J3 1996
793.7’4—dc20
96-19505
CIP

Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
foreword
By the time I was twenty-one I had become a fair mathematician, an actuary, and
a good cardplayer, and I have had some success in these fields in the years since.
One factor that contributed to these accomplishments was my father’s practice,
starting when I was about six, of posing for me such problems as “If a ball and a
bat cost a dollar and a half between them and the bat costs a dollar more than the
ball, how much does each cost?”
As I grew older I became exposed to more complicated problems, some of
which required mathematical knowledge and all of which demanded the use of
reasoning. In the process I became an avid puzzle fan and an inveterate reader of
books dealing with the lighter side of mathematics and logic.
Naturally enough, I began to want to produce a book of this sort myself—but
one that would be different, challenging, and interesting. I wanted to offer a
book that would avoid problems too easy, too tedious, or too involved in any
great amount of formal mathematics.
Mathematics for Pleasure1 is the result. In its preparation I had the help of the
perfect co-author, William H. Benson, Associate Professor of Mathematics at
Dickinson College—an old friend I had first known when he was Captain
Benson, USN, and I was Commander Jacoby, USNR. In addition to his faculty
for turning out highly accurate work of all sorts, Professor Benson has long
made a hobby of mathematical recreations and ranks with the world’s top
experts on magic squares.
The book is made up of problems, with solutions grouped at the end of each
section. These problems are selected for their interest and challenge and are
much more than a casual collection of problems. Each one was carefully
screened for its ability to stimulate imagination and interest and to allow you to
put your reasoning powers to work. In very few is there a requirement for
advanced mathematical knowledge or prowess, but in all of them you will have
to keep your wits—your reasoning power—going at full steam. While the
problems have been grouped according to type, they have not been arranged in
any particular order of difficulty. So far as actual mathematics is concerned, the
any particular order of difficulty. So far as actual mathematics is concerned, the
average person who hasn’t used his mathematics since school will be surprised
how much of what he learned in the past he still retains.
Some of the problems you will meet in this book are the work of Hubert
Phillips (known also as Caliban) of London, who is unquestionably the world’s
leading problem composer. He has written more than nine books on
mathematical problems and has conducted a well-known and popular British
newspaper feature on problems of this general nature. He kindly made available
to us his complete material.
In addition to Hubert Phillips’ cooperation, we wish to acknowledge with
deep thanks the assistance of Professor John H. Light of Dickinson College; and
of Dr. Edward Cogan, Professor of Mathematics at Sarah Lawrence College, and
Captain Wesley A. Wright, USN (Ret.), who carefully checked all problems,
solutions, and other material.

Oswald Jacoby
Table of Contents
Title Page
Copyright Page
foreword
1 - fun with numbers
2 - fun with letters
3 - the odds: explorations in probability
4 - where inference and reasoning reign
5 - the answers are whole numbers
1
fun with numbers
When man first began to count—probably well before he began to write—It was
only natural that he use the facilities he had. There were five fingers on each
hand, so at first he counted to five and not long after that to ten. If he wanted to
move into the realm of higher numbers he could use his ten toes and get as far as
twenty. After that he had to start all over again—which is why almost every
language has a single-word name for each number up to twenty but starts with
double and triple words from then on.
The use of the toes for counting never became very popular, even though man
was counting nicely before he stopped going barefoot. It was easier to think of
higher numbers as so many tens and units until the important number of ten tens
was reached. A name was needed for this, and still another for ten tens of tens,
so from very early times most languages have had words or symbols for hundred
and thousand.
Written numeration (in its broadest sense) goes back to prehistoric man. The
cuneiform baked-clay tablets of the Sumerians and the Babylonians, as well as
Egyptian records on papyrus, prove that as long as five thousand years ago these
peoples had a usable system of numbers. The Egyptian system is considered a
decimal system: it had names and symbols for 10, 100, 1000, 10,000, 100,000,
and 1,000,000; it did not, however, employ the principle of position or use a
zero.
The Sumerians and Babylonians used a sexagesimal or 60-base system that
did include the principle of position, and late in the game (about 200 B.C.)
introduced a symbol for zero.
By the time they had reached the height of their culture, the Mayas of Central
America had a calendar and a highly developed 20-base system of numbers that
included zero and the principle of position.
The Greek system used the entire Greek alphabet as a series of symbols
representing numbers in arithmetical operations. The Roman system, which also
representing numbers in arithmetical operations. The Roman system, which also
used specific letters and letter combinations to represent numbers, was an
improvement but remained cumbersome and inefficient. Even with the abacus,
which made addition and subtraction feasible, multiplication and division as we
know these processes were not possible. Before arithmetic could gain any real
momentum a better system was necessary.
During the early centuries of the Christian era the Hindus of India furnished
the solution by formulating the principle of position and by the use of ten
symbols to represent zero and the first nine digits. With these two developments,
larger numbers could be expressed simply in terms of the position of each
symbol. Thus 24 represents two tens and four units; 2304 represents two tens of
tens of tens plus three tens of tens plus four units, or
2(10)3 + 3(10)2 + 0(10) + 4.
Despite the immediately obvious advantages of this system, not until around
the beginning of the thirteenth century did it reach Western Europe (by way of
Arabia), and its adoption there was not more or less complete until the beginning
of the sixteenth century.
The system, good as it was, still left a problem with fractions—a problem that
was solved by the introduction during the late sixteenth and early seventeenth
centuries of the decimal system. After the decimal system began to be used, it
became simple to represent two tens plus eight units plus one tenth plus six
hundredths, for example, merely by writing 28.16. Just as each place to the left
of the decimal point represents a multiplication by ten, so each place to the right
represents a division by ten.

Both the Romans and the Egyptians had a word or symbol for ten, hundred, and
thousand; we have words for much larger numbers. One million is one thousand
thousands and is written 1,000,000; one billion is one thousand millions and is
written 1,000,000,000; and one trillion is one thousand billions and is written
1,000,000,000,000. While Arabic numerals and the decimal point take care of
reasonably large and reasonably small numbers, this does not mean that we can
understand or visualize what a large number stands for. For instance,
376,987,543,843,217 is a fifteen-digit number. It can be referred to as three
hundred seventy-six trillion, nine hundred eighty-seven billion, five hundred
forty-three million, eight hundred forty-three thousand, two hundred seventeen.
Beyond that its meaning is questionable. We also have words for even larger
numbers, but unless we can express these larger numbers in terms of smaller
numbers, they mean nothing to us.
And today we have increasing need for large numbers. Consider these: There
are 31,536,000 seconds in one year; if a man could count at the rate often per
second, it would take him more than three years to count to a billion.
Astronomers have to deal with vast numbers, since the known universe covers a
considerable amount of space. In order to simplify matters they use a unit known
as the light year, the distance light travels in one year. Since light travels at the
rate of a trifle over 186,000 miles per second, the number of miles in a light year
is a thirteen-digit number slightly smaller than six trillion. Our national debt is a
twelve-digit number, but our national wealth is conservatively estimated to be a
thirteen-digit number. And 635,013,559,600 is a fairly well-known twelve-digit
number; it represents the number of possible bridge hands a player can hold.
Most of the classic problems that deal with large numbers are concerned with
the fact that when you multiply enough small numbers together you come up
with a very large number.
Tradition has it that the inventor of the game of chess was invited by his
emperor to name his own reward. “All I ask,” the inventor is supposed to have
said, “is to be awarded one grain of wheat for the first square on the chessboard,
two grains for the second, four for the third, eight for the fourth, and so on. His
Majesty laughed heartily and agreed at once to the apparently modest request.
He changed his mind around the twenty-first square and had the man hanged.
The emperor had to come up with 1,048,576 grains of wheat for that square and,
although it was within his limits, he realized he would before too long exhaust
his wheat resources. Just in case you don’t know how many grains of wheat
would eventually have been required, the answer is 264 − 1, or
18,446,744,073,709,551,615.

But for large numbers or small, when you solve the problems that follow in this
section of the book, only the principles of arithmetic that are known to all of us
will be involved—with one possible exception: the concept of prime numbers. If
you consider the basic arithmetical operations of addition, subtraction,
multiplication, and division when applied to whole numbers (or integers), you
will notice immediately that the sum, difference, or product of any two integers
is also an integer. However, if you divide one integer by another, the result is not