![]() ![]() Most of the literature references are carried over from the original books, so they are to works published before 1925, although there are scattered references to newer work also. On the plus side, the solutions are generally well-documented with references to the literature, and the books are also extensively cross-referenced when there is a related problem or technique in another section. The indices are skimpy, so often the best way to find something is to go to the relevant section and just skim through all the problems. In a few cases the solution is not given and there is instead a reference where it can be found in the literature.ĭespite the age of the material, I think these books are still very valuable as references, and I have used them several times when solving problems in the American Mathematical Monthly. A little over half of the page count is devoted to solutions, so on the average the solution is only a little longer than the problem statement itself. If one can use it more than once it becomes a method.”)Įach problem has a solution given, although very briefly. (It is the source of the famous saying “An idea which can be used only once is a trick. The Preface goes into quite a lot of detail about how to use the book, and it includes much useful advice about learning mathematics in general. An important distinction from most problem books is that few problems appear in isolation, but are nearly always within a sequence of problems that builds on one idea and explores its consequences. They also give more than a little coverage of topics that were hot back then but have dimmed some today, such as equidistribution of sequences and of schlicht (univalent) functions.Īs in Pólya’s other books, the main concern is discovery and problem solving rather than mathematical facts. They do include many related matters, such as sequences, Riemann integrals, asymptotics, and more, but they omit more modern analysis topics such as functional analysis, linear spaces, and measure theory. By “analysis” they mean what was in the mainstream of analysis in 1925, that is, mostly theory of functions of a single complex variable. These are famous but old problem books, originally published in German in 1925, then lightly revised several times and then published in slightly expanded English editions in 19.
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