Bruce Humphries, Inc., Boston, 1934

**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 skillfully concealed.

Any good textbook in Arithmetic or Algebra or Geometry is sure to contain some stimulating problems, buy 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 Arithmetic, 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 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.

Next 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 Zoölogy 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 textbooks are written for rather learned people, but this book is not written for the learned. All the mathematical knowledge that it calls for is 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 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 Mathematics.

- On Dividing
- Different Ways of Writing Numbers
- Multiplying without the Multiplication Table
- Mostly on Squares
- The Charms of Decimals
- Is this Formula True?
- Magic Squares
- A Few Remarks on Measuring and on Incommensurable Numbers
- Some Facts about π
- Geometrical Arithmetic
- Oddities of Numbers
- Equations with Many Answers
- Drawing a Straight Line without a Ruler
- The Impossible in Mathematics