He worked on a great variety of mathematical topics, including series, number theory, combinatorics, and probability.
In his later days, he spent considerable effort on trying to characterize the general methods that we use to solve problems, and to describe how problemsolving should be taught and learned. He wrote three book on the subject: How to Solve It, Mathematics of Plausible Reasoning Volume I: Induction and Analogy in Mathematics, and Mathematics of Plausible Reasoning Volume II: Patterns of Plausible Reasoning.
In How to Solve It, Polya provides general heuristics for solving problems of all kinds, not simply mathematical ones. The book includes advice for teaching students mathematics and a miniencyclopedia of heuristic terms. It was translated into several languages and has sold over a million copies. Russian physicist Zhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Polya's famous book.
In Mathematics of Plausible Reasoning Volume I, Polya discusses inductive reasoning in mathematics, by which he means reasoning from particular cases to the general rule. (He also includes a chapter on the technique called mathematical induction, but that technique is not his main theme.) In Mathematics of Plausible Reasoning Volume II, he discusses more general forms of inductive logic that can be used to roughly determine to what degree a conjecture (in particular, a mathematical conjecture) is plausible.
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