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A mathematical orchard. Problems and solutions. (English)
MAA Problem Books Series. Washington, DC: The Mathematical Association of America (MAA) (ISBN 978-0-88385-833-2/pbk; 978-1-61444-403-9/ebook). xi, 397~p. (2012).
Problem books is a well-established series of the Mathematical Association of America consisting of collections of problems and solutions from various sources, including diverse annual mathematical competitions, compilations of problems specific to particular branches of mathematics, discourses on the art and practice of problem solving in mathematics instruction, and other (inter)national year-books of this kind. Within this series of meanwhile twenty-three volumes, the book under review is actually an extended version of an older book. In 1993, the MAA published “The Wohascum County problem book” by the same three authors [The Dolciani Mathematical Expositions. 14. Washington, DC: Mathematical Association of America. ix, 233 p. (1993; Zbl 0997.00521)] with an original list of 130 problems and their solutions. Having been asked to consider reissuing that book with a less rustic and more descriptive title, the authors added many more problems and published the current version “A mathematical orchard. Problems and solutions”. The present book is now a collection of 208 challenging, highly interesting, and quite original problems, all of which come with carefully worked, detailed, instructive, and well-explained solutions. Moreover, for many problems, the authors present multiple solutions, together with the crucial ideas underlying them. As the authors point out in the preface, the main purposes of this particular collection of problems can be summarized as follows: (1) mathematical entertainment; (2) instruction with regard to creative problem solving; and (3) invitation to the composition of further, related problems. Generally, the problems presented here are intended for undergraduates, and many of them should be even accessible to high school students. In fact, most of the problems require nothing beyond elementary calculus, elementary school geometry, or elementary discrete mathematics. For a few problems, some knowledge of linear or abstract algebra is needed, but also in these rare cases the elementary level is barely exceeded. However, most of the problems should be far from being routine for average students, with quite a number of them even requiring considerable mathematical maturity. In this regard, there is a wide-range of challenge and difficulty represented by the problems in this book. For the convenience of the reader, there are two appendices at the end of the book. Appendix 1 lists the mathematical prerequisites for each of the 208 individual problems, whereas Appendix 2 arranges the problem numbers by topic. In addition, the carefully compiled index refers to problems with a specific theme, as well, and provides a useful overview of the mathematical concepts occurring in the numerous problems and solutions. As over 80 percent of the text is devoted to discussing lucid, enlightening, detailed, and often multiple solutions to the original problems, the didactic value of this true “mathematical orchard” is absolutely outstanding, especially with a view toward the art of teaching problem solving in class, including solution strategies, pattern finding, experimentation, and the use of suitable mathematical techniques. No doubt, this book is a perfect companion for students who like mathematical problem solving, in general, and a highly useful source for problem solving classes or contest preparation teams in particular. Besides, the problems and solutions collected in this volume provide a very rich source of instructive and illustrating examples within various elementary mathematics courses, and that in both college and high school likewise.
Reviewer: Werner Kleinert (Berlin)
Classification: U40 K20 I10 G10
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