One of the things that I am most grateful for in my mathematical life is Dover Publications. There have been several occasions when they rescued from oblivion an old book that I liked so much that it spent more time hidden in my house than on its shelf in the library collection. One such jewel is The Stanford Mathematics Problem Book.
This is a book for those who read and used all the MAA contest problem books and couldn’t wait for the next volume. The problems it contains are of the same nature: clever questions aimed at high school students. They represent the entire collection of problems used in the Stanford examination for high school seniors, from 1946 to 1965. The introduction of the book gives a brief history of the competition and its purpose.
Similar to the more familiar AMC problems collected in the MAA series, the purpose of the Stanford examination problems is to test the students’ originality rather than practical knowledge. There is however an essential difference in the way the AMC and Stanford competitions were run, and thus in the way the problems were written. While the AMC problems are “multiple choice”, the ones included in Pólya and Kilpatrick’s book are all “open answer”. As such, no more than 5 problems were included in a given examination. This gives the problem solver and the contest problem writer a different, and I would argue better, perspective.
There are two qualities that make the book a wonderful resource. One is that fact that each of the twenty examinations contains a balanced mixture of elementary yet challenging questions from different areas of mathematics. The second is the structuring of the book in three parts: Problems, Hints and Solutions. In this manner, the solver-reader can get as little or as much help as he wants/needs on any particular problem.
In conclusion, it is wonderful to have this book back in print. And for the modest amount it costs, one can afford to leave the library copy for others to enjoy too.
Ioana Mihaila (firstname.lastname@example.org) is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is interested in mathematics competitions.