Problem-solving through problems
This is surely one of the best books on problem solving to appear in many years. As the author states in the preface, it is both an anthology of over 700 problems, more than one-third of which are worked out in detail, and a manual of instruction. It is tempting to describe it as a textbook on problem solving -- it would surely be an excellent choice as the primary text for a course or seminar on the subject. At the same time it is well suited to independent study, and problem solvers from the level of high school students to professional mathematicians will find their time spent in studying this book amply rewarded. The first two chapters use problems to illustrate numerous problem-solving principles and techniques applicable in nearly all areas of mathematics. Illustrative section headings: Search for a pattern; Draw a figure; Formulate an equivalent problem; Exploit symmetry; Divide into cases; Pursue parity; Consider extreme cases; Induction (several sections); Pigeonhole principle. Each section contains several solved problems illustrating the principle, followed by a set of problems for the reader. The remaining six chapters are organized by mathematical subject: Arithmetic, Algebra, Summation of Series, Intermediate Real Analysis, Inequalities, Geometry. In these chapters most of the fundamental facts to be used as problem solving tools are developed themselves as solved problems, or proofs are sketched for the reader to complete as problems. Among the few exceptions: Abel's limit theorem for power series, L'Hôpital's rule and the fundamental theorem of calculus are stated and used without any discussion of proofs. The problems themselves vary in the level of difficulty, from a few which can be solved by obviously mimicking an illustratively solved problem to those which will challenge the best among readers. Some problems appear several times in different parts of the book, with different solutions illustrating different ideas. There are excellent cross references. The book is visually appealing, with good quality paper and print. Misprints are rare and most are not likely to be confusing. Only an occasional dangling participle mars an otherwise well written book.
scientific article; zbMATH DE number 3829822