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Mathematics as Problem Solving

Alexander Soifer
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Barbara E. Reynolds
, on

This is a very readable collection of interesting problems of varying levels of difficulty. The author has been involved with mathematics competitions since childhood, and this book could be used to coach a team preparing for such a competition. It is so delightfully written that anyone who simply likes working on challenging problems could read it independently.

There are seven chapters: 1) Language and Some Celebrated Ideas, 2) Numbers, 3) Algebra, 4) Geometry, 5) Combinatorial Problems, 6) Chess 7 × 7, and 7) Farewell to the Reader. In the first chapter, the author includes a brief discussion about interpreting and reasoning with complex sentences, without laboring over the details of formal mathematical language. He introduces some general problem solving principles such as arguing by contradiction, the pigeonhole principle, and mathematical induction, and illustrates these with numerous examples, some of which are solved. The second chapter includes a number of problems about rational and irrational numbers, while the third illustrates problem-solving strategies for situations involving systems of equations and inequalities.

Chapter 4, Geometry, is the longest chapter in the book, including more than 60 problems in sections titled Loci, Symmetry and Other Transformation, Proofs in Geometry, Constructions, Computations in Geometry, and Maximum and Minimum in Geometry. The problems in Chapter 5 include combinatorial problems of sets and problems of existence, as well as ways of using coloring to solve certain kinds of problems. Chapter 6 opens with a problem about scheduling games in a chess tournament so that each player on one team plays each player on the opposing team. Three complete solutions to this problem are worked out in detail: The first solution exploits an algebraic description of convexity, the second solution is presented as a problem on a 7×7 chessboard, and the third solution connects this problem to a representation of the 7-point real projective plane.

More than 150 problems are included in this collection. While these problems here do not require anything beyond high school algebra and geometry, many of them would be challenging to undergraduate mathematics students.

One question that goes through my mind when I review a book like this one is “How would I use this book?” Aside from the fact that I found the problems given here simply engaging and enjoyable, I could imagine using some of these problems to challenge my students in a variety of courses. The combinatorial problems in Chapter 5 could be used to augment discussion in a discrete mathematics course, the loci and construction problems of Chapter 4 would fit nicely into a geometry course, and problems involving systems of inequalities could be used to challenge students in a algebra or precalculus course. The problems are grouped into collections of types of problems — which is helpful for an instructor who is looking for ways of integrating problem solving into a particular course, and might be useful for coaching a mathematics competition team. This little book would be a good resource for an instructor or coach.

I found it a little bewildering at first that this book begins with three Forwards and two Prefaces. Later I realized that these Forwards are one way of showing that it has been endorsed by Branko Grünbaum, Peter D. Johnson, Jr., and Cecil Rousseau, each of who has been involved with coaching students in one or another mathematics competition. Since this is a Second Edition, it seems appropriate that both the original Preface and a Preface to the Second Edition be included here.

Sr. Barbara E. Reynolds, SDS, is Professor of Mathematics & Computer Science at Cardinal Stritch University, Milwaukee, WI.

The table of contents is not available.