The public face of mathematics is sometimes grey and utilitarian: math is a useful tool, a haphazard collection of recipes and algorithms, a necessary prerequisite to understanding science. The private face is much more beautiful: Mathematics as Queen, not servant, of Science. Math is a uniquely aesthetic discipline; mathematicians use words like beauty, depth, elegance, and power to describe excellent ideas. In order to truly enjoy mathematics, one must learn to appreciate the beauty of elegant arguments, and then learn to construct them.
In A Mathematical Mosaic, I hope to get across a little of what it is that mathematicians actually do. Most "big ideas"and recurring themes in mathematics come up in surprisingly simple problems or puzzles that are accessible with relatively little background. Many of them have been collected here, along with an inkling of how they relate to the frontiers of modem research. These themes will also be traced backwards; often ideas centuries old take on new meaning and relevance as mathematical understanding advances. And if in the process of looking at these important ideas we get a little playful and irreverent, well, that is also in keeping with the nature of mathematics.
This book does not purport to teach problem solving, although it might communicate what is interesting and exciting about grappling with a problem. Instead, the reader should be left with a large chunk of mathematics to digest slowly.
This is not a book intended to ever be "finished with or completed. Instead, it should be read bit by bit. Like all mathematics, it should be read with pencil in hand, with more time spent in deep thought or frantic scribbling than in actual reading. Mathematics is not a spectator sport!
You will also notice that, like any mosaic, this book comes in many small pieces of different sorts. It is loosely arranged in order of difficulty, and the headers often indicate the general theme of the article. You might enjoy just starting with one section which especially interests you; from there, you can skip to other sections which link with your earlier choices. You will soon discover that ideas from one field carry over into many others. And if you don't end up at a section that amazes and perplexes you, I will be sorely disappointed.
Many sections end with some Food for Thought. More than just exercises, these excursions are intended to be starting points for independent thinking. The material beforehand will help, but often only tangentially. I have deliberately included very few solutions. Indeed, often there is no single "correct" solution, and sometimes the problems are so open-ended that there is no way to completely "finish them. Don't try to solve every one; choose one or two that catch your eye and think about them on and off for a few days. Success should not be defined by how many problems you solve, but by how many new ideas you have. And remember: patience is a virtue.
Every so often throughout the book, references are mentioned. This is not to intimidate you into reading many weighty tomes. Instead, they are there to provide some ideas for further reading in case you find some field particularly interesting.
I have also included short profiles of interesting and talented young people who have been involved in mathematics for many years. I have chosen them fairly randomly from the large number of fascinating people I have had the good fortune to meet through my involvement in mathematics education. Some of them are going on in mathematics (and indeed one has already made his name in the field), while others are pursuing their interests in related subjects. Through these profiles, I hope to share their unique perspectives on the joys of mathematics.
You will notice that these young people have all done extremely well in mathematical competitions; collectively they have won eleven Gold Medals at International Mathematical Olympiads. In noting their achievements, I certainly do not mean to imply that competitions are the only gateway to mathematics, or that they are a necessary prerequisite to success in the field. My selection merely reflects the fact that my own involvement with young mathematicians has grown out of my association with mathematics competitions.
There is no background required to begin enjoying this book. You should be able to pick up many of the ideas as you go along. When you truly get stuck, ask a friend or a teacher for help. You might be surprised at how much you learn! For some of the later sections, a familiarity with more advanced ideas such as mathematical induction, indirect arguments, set theory, complex numbers, and even calculus will be useful or even necessary, but the vast majority can be read with nothing more than early high school mathematics and a little chutzpah.
But enough of this talk. On to the mathematics!
Cambridge, Mass.
January, 1996