# A Mathematical Orchard:

## Problems and Solutions

### by M. I. Krusemeyer, G. T. Gilbert, L. C. Larson

The book is an expanded version of "The Wohascum County Problem Book" published by the MAA in 1993. While the title has changed, some problems - old and new - still draw from the (un)reality of that locality. There are now 208 problems vs 130 in the older edition. This is unabashedly a problem collection in which the Solutions part takes about 85% of the book. The book is very thoughtfully organized. Each problem is followed by a solution page number; there are also three indices (a thorough term index, prerequisites by problem number, and problem numbers by subject); that and the fact that about 25% of the problems come with more than one solution, attest to the effort made by the authors and the sincerity of their desire to accommodate diverse skills and interests. On the whole, the book is very easy to work with. The metaphorical title A Mathematical Orchard is supposed to "evoke images of such good things as vigorous growth, thoughtful care, and delectable fruit..."

The problems are loosely ordered by the difficulty. Earlier problems are generally more difficult than the later ones, but this is not necessarily true for nearly adjacent problems. Solutions are lovely, very detailed and structured: some even contain lemmas. Several are preceded with an Ideas paragraph or two, others are followed by Comments for broader perspective. As the authors write, "In an ideal world, may be no one would look at a solution before trying seriously to solve the problem, but if you're feeling curious and are pressed for time, you can still appreciate the problem by reading the solution and the underlying ideas behind it." Solutions in the book are truly designed to be read not just to navigate the reader to an answer.

The problems in the book are all but routine and all are original with the authors. For example, problem #29 classified as Elementary geometry requires to prove that any polygon can be tiled by convex pentagons. Similarly, problem #31 asks to construct the midpoint of a segment with a straightedge and a trisector tool (a device that divides any segment into three equal parts.) Problem #172 (Analytic geometry / Conic sections) calls to describe all points P that lie on an ellipse that passes through the vertices of a fixed parallelogram. In Problem #132 (Arithmetic / Rational / Irrational numbers), one needs to find all rational points (a, b) from which all the distances to any rational point on y = 13x are rational. Problem #11 asks to prove that the determinant of a 4×4 matrix whose entries are either 2 or -1 is divisible by 27.

The problems have been tried on undergraduates at Carleton and St. Olaf Colleges and participants at Canada/USA Mathcamp. If you are not as lucky as those kids, you may still enjoy solving the book problems or benefit from reading the solutions. The book does not require proficiency beyond beginning calculus and linear algebra.

A Mathematical Orchard: Problems and Solutions, MAA. Softcover, 400 pp, \$49.95. ISBN 978-0883858332.