Mathematical Excursions: Side Trips along Paths Not Generally Traveled in Elementary Courses in Mathematics

158 Pages·1957·2.89 MB·English

by  Helen Abbot Merrill

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MATHEMATICAL
EXCURSIONS
Side trips along paths not generally
traveled in elementary courses
in Mathematics
BY
HELEN ABBOT MERRILL
DOVER PUBLICATIONS, INC.
COPYBlGBT, 1933,
By HELEN A. MERRILL
This new Dover edition first published
in 195i, is an unabridged and unaltered
TC'J>ublication of the first edition.
Printed in the l:nited States of America.
CONTENTS
CIIAPTEB PAOli
I. ON DIVIDING 1
II. DIFFERENT WAYS OF WRITING NUMBERS 13
III. MULTIPLYING WITHOUT THE MULTIPLICA-
TION TABLE 23
IV. MOSTLY ON SQUARES 32
V. THE CHARM OF DECIMALS 44
VI. Is Tms FORMULA TRUE ~ 62
VII. MAGIC SQUARES 67
VIII. A FEw REMARKS ON MEASURING AND ON
INCOMMENSURABLE NUMBERS 77
IX. SOME FACTS ABOUT 7r • 82
X. GEOMETRICAL ARITHMETIC 92
XI. ODDITIES OF NUMBERS 108
XII. EQUATIONS WITH MANY ANSWERS 119
XIII. DRAWING A STRAIGHT LINE WITHOUT A
RULER 128
XIV. THE IMPOSSIBLE IN MATHEMATICS 137
ANSWERS TO SoME OF THE PROBLEMS 144
[ V ]
This edition is dedicated to the memory of
Helen Abbott Merrill
ACKNOWLEDGMENT
THE problems in this book have been collected over
a long period of years with no thought of making
public use of them. It is clear to all who gather such
material for private use that an acknowledgment of
one's debt in detail is quite impossible. The standard
books on mathematical recreations in English, French,
German, and Italian have been drawn on, as well as the
more elementary magazines.
Those who are trying to help our young folks to see
not only that Mathematics is a useful tool and a fine
mental discipline, but that hard work may be rare good
fun, and that the subject widens out into fields of ever
growing wonder and fascination, have generally learned
what a large role entertaining problems may play.
For much help given to me in my teaching and in
the preparation of this book I desire here to express
my gratitude to a long line of authors whose names
must remain unmentioned.
[ vii ]

INTRODUCTION
THERE is something about a puzzle which appeals
to almost everyone, young or old. Perhaps it is the
challenge to our thinking powers, the feeling that we
must not be conquered by so small a thing. Perhaps
our curiosity is aroused to see what mode of attack
will succeed, by what clever device a puzzle may be
solved, especially if its real nature is skilfully con
cealed.
Any good text-book in Arithmetic or Algebra or
Geometry is sure to contain some stimulating problems,
but many more such problems, interesting and amusing,
as well as instructive, lie a little off the beaten track.-
Our course through the early years of mathematical
study is apt to be rather clearly mapped out, resembling
the plan of a European tour - so many days for
London, so many days for Oxford, etc.; quite the
right sights to see, but many tourists come home with
no notion of the delightful sights which they have not
seen for lack of time or lack of guidance.
This little book is meant to play the part that some
of the more detailed or specialized guide books play for
the tourist. You have made a trip through Arith
metic, but here are some sights you may have missed,
something new about counting or division or decimals.
Here are a few side trips, away from the main traveled
[ ix ]
INTRODUCTION
roads, into some of the fields that border on your path
through Algebra and Geometry. They are not the
chief sights of such a journey, but they may add some
pleasure and profit to your trip.
N ext to an abundance of poetry stored in our minds,
I believe that there are few things that add more to
our ability to divert and enjoy ourselves than a good
supply of mathematical puzzles.
Perhaps you have heard that Lewis Carroll, author
of "Alice in Wonderland," was a mathematician. He
was a poor sleeper, and used to amuse himself when
wakeful at night by working out problems in his head.
Some of these midnight amusements were published in
a little book called "Pillow Problems." We might be
made wider awake rather than soothed by thinking
out problems in the dark, but it is a fine thing to have
some on hand for entertainment while one waits for a
train or a dentist's appointment, or is shut in with a
cold.
A professor of ZoOlogy once told me that, if she were
shut up in prison and could have only one book, she
thought she would ask for a book of Geometry originals,
because they would give her entertainment and keep
her brain from getting dull.
Most of the mathematical books which are not text
books are written for rather learned people, but this
book is not written for the learned. All the mathe
matical knowledge that it calls for is some Algebra and
Geometry, enough to show you that those subjects
can be very entertaining. One pleasant feature of
Mathematics is that we do not have to know a great
[ x ]
INTRODUCTION
deal about it in order to get amusement from it, and a
still more pleasant one is that the farther we go the
more we find to surprise and entertain us.
There are many different kinds of problems in this
book. Anyone who has an interest in such subjects
is sure to find something here to prove diverting, to
furnish mental exercise, and to give a little notion of
what may be found farther on along the mathematical
road.
As I take up the various topics I mean to talk to
you very informally, as if you were right here with me,
putting questions to you which I hope you will try to
answer for yourselves before you go on to see what I
say in answer.
Perhaps some of the problems will prove to be posers.
An occasional hint has been given, but it seems a pity
to give too many and so to spoil your pleasure in
wrestling with these puzzles. Answers to many of the
problems are given, but on a different page, so that
you need not see them unless you wish.
These pages contain only a few samples of what is
to be found in the inexhaustible storehouse of Mathe
matics.
[ xi J