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Discovering Mathematics: The Art of Investigation

A. Gardiner
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Ioana Mihaila
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According to the author’s own testimony, as presented in the preface of the book, Discovering Mathematics … is written with the purpose of guiding the reader into investigating some interesting and challenging — but at the same time elementary — mathematics. While guiding the reader’s work through a sequence of well-chosen exercises and question, the author is also hoping to familiarize him with typical mathematical processes and techniques.

To this purpose, the book consists of two parts: the first contains four “short investigations,” and the second has two longer ones. The problems presented in the first part of the book serve as a warm-up for the longer ones, which expand on some of the questions in the beginning, but take them to new levels of generalization, while at the same time developing longer sequences of problems that can be solved more easily by moving from one to another. Topics covered include some familiar mathematical games such as the game of Nim and variations, arithmetical properties of numbers, area and volume formulas and properties of lattices, all presented in relation with the underlying idea of using recurrences for making and solving generalizations.

I must confess that I am myself a big fan of sets of problems that develop ideas and as such make the solving of the harder questions easier. Choosing the right problems and the right order for this type of sets is difficult, but it leaves the solver not only with a solution, but also with a gratifying feeling of understanding the thought process involved in breaking the mystery. From this point of view, Discovering Mathematics… does a wonderful job.

On the other hand, the reader should be forewarned that this is neither a recreational mathematics book, nor a textbook… but rather a problem book. The exercises fall in a sequence with hardly a comment in between, and there is not much choice for progressing through the book aside from spending the time to solve each problem or reading the rather concise solutions. This is because the author, in his intent to coerce the reader into really doing the work, formulates many of the questions by asking the reader to generalize the result obtained in the previous exercise or compare the answer from the previous exercise to something else. If you didn’t really do the previous exercise (maybe because you thought it wasn’t very interesting and wanted to skip ahead), well… there’s no way to use that answer. For those patient enough to work through all the problems, both hints and answers are provided in each section.

Due to the manner in which the book is written, I think the book will have its greatest appeal to people who are already wholeheartedly convinced of the beauty of mathematics. For such an audience, the interesting problems and the sequenced presentation will provide many hours of problem solving fun.

Ioana Mihaila ( is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is also interested in mathematics competitions.

PART I: Short Investigations
Advice to the Reader
1. Analyzing Games
2. From Rhyme to Reason
3. Thinking Things Through
4. Asking Questions
PART II: Extended Investigations
Investigation 1: Flips
5. Introducing 9-flips
6. The Art of Guessing
7. Other Flips
8. Flips in Other Bases
9. Counting 9-flips with n Digits
10. Recurrence Relations
11. The End of the Road
Investigation 2: The Postage Stamp Problem
12. Getting Stuck In
13. Coming Unstuck: The Search for Proof
14. One Picture is Worth a Thousand Words
15. The Final Licking
16. The Coin Problem