Throughout the centuries, from the time of the Sumerian and Babylonian civilizations (roughly 2500 B.C.), one can find no end of mathematical problems and questions — since problems and questions beget more problems and questions in an unending cycle. These problems and questions are the lifeblood of mathematics. Smaller problems lead to larger problems, which in turn lead to substantial mathematical research. For example, consider the mathematics produced during attempts to prove Fermat’s last theorem — which in itself is not an important result, even if true. The following metaphor, attributed to Allen Shields, is particularly apropos: “A mathematical problem is a ‘jackpot’ which gains in value as more of us throw our quarters into it.”

Mathematical problems challenge and interest even those who are outside the profession. Just consider the large number of problem sections found in various journals, magazines, and newspapers throughout the world. Although there is much valuable mathematical information sequestered in the immense problem literature, unfortunately there is no easy way to access this material. It is rarely reviewed and existing indexes are not particularly useful.

In this regard, I am highly critical of the cavalier treatment with which some journals treat their problem sections. For example, consider journals such as The American Mathematical Monthly, Mathematics Magazine, and The College Mathematics Journal. All of these publications contain valuable and interesting problem sections. The yearly problem indexes published by the first two journals only contain a listing of problem numbers and their corresponding proposers and solvers and their page numbers. The College Mathematics Journal does not include any yearly index at all. This is a sorry state of affairs for any journal!

I would like to see (and so, I am sure, would others) at least a return to the indexing system used in the first 19 volumes of The American Mathematical Monthly. Here the indexes also included problem titles and were arranged by various fields, such as Diophantine analysis, algebra, geometry, calculus, mechanics, averages and probability, and miscellaneous. With modern computers and word processing programs, this should be relatively easy to do.

In order to remedy this very sad state of problem indexing, the author, who is an ardent problemist, has taken on the very arduous task of producing a rather complete index system for problems published in a large number of different journals from 1980 through 1984. He also plans to publish similar works for the years 1975 through 1979, 1985 through 1989, etc.

Since it is not easy to classify a problem, the author has sorted each problem by topic (e.g. Geometry/triangles; Analysis/series) and in almost all cases includes the complete statement of each problem. This explicit representation of the sorted problems together with the many other listings included, enables one to locate problems and their solutions (if available) in a relatively easy fashion.

This is a must book for problemists as well as problem editors. I only wish it had been available a long time ago.

Murray S. Klamkin

professor emeritus

University of Alberta