# A Mathematical Orchard: Problems and Solutions

### By Mark I. Krusemeyer, George T. Gilbert, and Loren Larson

Catalog Code: MOR
Print ISBN: 978-0-88385-833-2
Electronic ISBN: 978-1-61444-403-9
409 pp., Paperbound, 2012
List Price: $49.95 Member Price:$39.95
Series: Problem Books

A Mathematical Orchard: Problems and Solutions, by Krusemeyer, Gilbert, and Larson, is a re-publication and expansion of the very popular MAA volume Wohascom County Problem Book, published in 1993. The original 130 problems have been retained and supplemented by an additional 78 problems. This collection of problems provides something for problem solvers working at many different levels, and the volume includes an appendix identifying the prerequisite knowledge for each problem. A second appendix organizes the problems by subject matter so that readers can focus their attention on particular types of problems if they desire. As with the original volume, the primary purpose of the volume is simply to provide an engaging set of problems for readers to work through..

Preface
The Problems
The Solutions
Appendix 1: Prerequisites by Problem Number
Appendix 2: Problem Numbers by Subject
Index

### Excerpt: The Problems (p. 20)

98. The proprietor of the Wohascum Puzzle, Game and Computer Den, a small and struggling but interesting enterprise in Wohascum Center, recently was trying to design a novel set of dice. An ordinary die, of course, is cubical, with each face showing one of the numbers 1, 2, 3, 4, 5, 6. Since each face borders on four other faces, each number is “surrounded” by four of the other numbers. The proprietor’s plan was to have each die in the shape of a regular dodecahedron (with twelve pentagonal faces). Each of the numbers 1, 2, 3, 4, 5, 6 would occur on two different faces and be “surrounded” both times by all five other numbers. Is this possible? If so, in how many essentially different ways can it be done? (Two ways are considered essentially the same if one can be obtained from the other by rotating the dodecahedron).

Mark Krusemeyer has been teaching at Carleton College since 1984 and, during the summers, at Canada/USA Mathcamp since 1997. He has served two three-year terms as a problem setter for the William Lowell Putnam Mathematical Competition, and he has recently succeeded Loren Larson as an Associate Director of that competition. Over the years, Mark’s research interests have varied from commutative algebra to combinatorics; he has also written a textbook on differential equations. He has received an award for distinguished teaching from the North Central Section of the MAA. Mark plays and teaches recorder, and his other activities include mountain scrambling, duplicate bridge, and table tennis.

George Gilbert grew up in Arlington, Virginia, and went on to earn degrees in mathematics from Washington University and Harvard. His interest in problem solving, begun as a student contestant, has led to his posing problems for the Putnam, for national high school contests, and for the Konhauser Problemfest and the Iowa Collegiate Mathematics Competition. He was Problems Editor of Mathematics Magazine from 1996–2000. He has been a professor in the Department of Mathematics at TCU for the past 20+ years. His two grown-up sons and he each claim to be the best poker player in the family.

Loren Larson is Professor Emeritus at St. Olaf College in Northfield, Minnesota. His interest in problem-solving dates to his early days of teaching when he engaged students in seminars that featured problems from the Putnam and the American Mathematical Monthly. Solutions were submitted under the byline St. Olaf College Problem Solving Group. Twenty years of such experience culminated in a book Problem-Solving Through Problems, which isolated and illustrated the most common problem-solving techniques encountered in undergraduate mathematics. Loren has served the MAA as a governor, a reviewer of books and articles, two five-year terms as Problem Editor for Mathematics Magazine and twenty-six years as Associate Director of the William Lowell Putnam Exam. He presently enjoys woodworking and specializes in crafting artistic mathematical puzzles.

### MAA Review

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…” Continued...

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