While on my vacation to Ireland, I brought along a book called "Solving Mathematical Problems: A Personal Perspective" by Terence Tao. The book was written when Tao was a 15 year old mathematics prodigy competing in international mathematics competitions. Today at 36, he is one of the world's top mathematicians.
I have really enjoyed reading this book. I work with a lot of folks on developing their problem solving abilities. In the past, I have used the books by Polya (here is one and another) on problem solving heuristics. While these books are good, I really like the tone of "Solving Mathematical Problems." Tao does a very good job of describing the experimental aspects of problem solving.
Many folks new to problem solving will see a fully formed solution and do not understand that there may have been quite a bit of trial and error involved in coming up with that solution. Thus, the most common question I get from these newbies is "how do I know where to start"? Tao addresses this question by including with his solutions a discussion of alternative approaches and why he chose not to use them. I really appreciate this style of presentation -- all in the voice of a 15-year old. It is conversational and pleasant to read.
The book is focused on the kind of problems encountered in mathematics competitions:
- Analytic geometry
- Euclidean geometry
- Number theory
- Diophantine Equations
- Sequences and Series
It is a small book, but filled with mathematical gems. If you decide to check it out, I think you will find it worthwhile.