As George Polya writes in the preface to his book How to Solve It, the experiences of tackling challenging mathematical problems "at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime." The stimulation of intriguing problems is undoubtedly something that draws many people to mathematics, so it's always worthwhile to check out a new problem book when it becomes available. Two such books have hit the shelves recently.
The first of these books is The Contest Problem Book V by George Berzsenyi and Stephen B. Maurer. This book collects together the problems and solutions for the American High School Mathematics Exam (AHSME) and the American Invitational Mathematics Exam (AIME) from 1983 to 1988.
A bit of background for the uninitiated: the AHSME and the AIME are two high school mathematics competitions sponsored by American Mathematics Competitions (AMC), an organization which coordinates numerous high school and junior high school examinations and is supported by nine different mathematics organizations (including the MAA). The oldest of the exams sponsored by AMC, the AHSME, is a thirty-question, ninety-minute multiple choice exam covering mathematics up to the level of pre-calculus. It is given annually to more than 300,00 high school students. Students who score higher than 100 points out of the possible 150 points on the exam are invited to take the AIME, an honor typically obtained by fewer than two percent of the students taking the AHSME. The students who do make it to this exam are treated to a 15-question, three-hour examination markedly more difficult than the AHSME. The AIME also has an interesting twist: Though the questions suppose the same pre-calculus breadth assumed by the AHSME, every solution is an integer between 0 and 999. Students who do well on the AIME are invited to participate in the United States Mathematical Olympiad (USAMO). In 1997, only 184 students made it to this point.
The Contest Problem Book V is the fifth in a series of AHSME problem books published in the New Mathematical Library Series of the MAA. It is also the first book in this series to include problems and solutions from the AIME, which began in 1983 as an intermediary step between the AHSME and the USAMO. Other volumes in the New Mathematical Library Series cover problems from the USAMO and the International Mathematical Olympiad.
The book, ideally suited for students and teachers preparing for the AHSME and the AIME, has more than just the exam questions and detailed solutions to those questions. Also included are problems and solutions which were dropped from the examinations and a section with frequency tables detailing how AHSME "Honor Roll Students" (i.e., students eligible to continue on to the AIME) responded to individual questions on each AHSME.
There is also a lot about this book to appeal to anyone with a general interest in solving challenging mathematical problems. For instance, there is a 14-page annotated bibliography of problem solving resources, including books on problem solving, collections of mathematical problems, and journals with problem sections. The authors have also included a topical index to the problems in the book, and numerous bits of problem-posing wisdom are scattered throughout the book.
Incidentally, information on all of the junior high and high school exams administered by the American Mathematics Competitions--including registration and eligibility information as well as sample questions from recent exams--can be found on the AMC Web Site.
The second book, Critical Thinking Puzzles by Michael DiSpezio, is a problem book targetted at a much younger audience. The book consists of some seventy brainteasers (along with solutions), each less than a page in length and most accompanied by a descriptive diagram or cartoon. The questions have very few mathematical prerequisites and the author has endeavoured to phrase the questions in concrete, easy-to-imagine terms that, say, middle school students may find appealing. In fact, most of the problems can be solved simply by visualizing the situation, and the author encourages his readers to experiment with household objects such as toothpicks and construction paper in solving these problems.
Most of the problems will seem at least vaguely familiar to older readers, but the author usually prefaces the problems with a short lead-in of his own. Sometimes he ties together several problems using a unifying scenario. For example, the idea of going to the movie leads to some graph theory (counting the number of routes through a graph from home to the movie theatre) and combinatorics (counting the number of ways to arrange people in the row of a movie theatre) and several other questions. At other times he embellishes the problem with a physical or historical piece of trivia. Did you know that the word "cup" is derived from the Sanskrit word "kupa", which means "water well"? This comes up in a question asking how to measure one cup of water given one container which holds three cups and another container which holds five cups.
The book is not limited to discussing mathematics--many of the problems require a bit of physical intuition or could be classified as general logical puzzles--but it touches on a surprisingly wide variety of mathematical ideas for a book geared at such a young audience. For example, Hamilton paths make an appearance in the plight of an ant trying to traverse the girders of a bridge, passing through each girder once and only once. Equitable distribution is presented when two friends try to divide the last slice of a pizza. The author also describes how to construct a Mobius strip, explains some of its uses, and has the reader cut it down the middle to see what shape results. The most complicated computation involved in any of the puzzles is solving a system of two equations in two variables.
The book's greatest shortcoming is that it sometimes lacks the precise language expected from good mathematical writing. In one example, there are three girls and two boys sitting in a row with five seats. The author asks, "How many different ways can the two boys and three girls be seated in this row?" If you answered 5!, you are not correct. In another question, he asks his reader to make a perfect square from four rectangles of varying sizes: one rectangle of 4x1 units, one rectangle of 2x1 units, and two rectangles of 3x1 units. After a while and maybe a little frustration, the young reader may begin to realize that the author wants a hole in the middle of the square. On still another question, the author is just flat-out wrong: he asks his readers to assemble a triangle from three identical trapezoids which, based on the accompanying picture, could only be described as isosceles trapezoids with the bottom twice the length of the top. (I took out my ruler and measured it just to be sure.) But the solution, given simply by the picture
DiSpezio has undertaken the difficult task of writing a problem book to entertain and intrigue young people with a small mathematical vocabulary. Such books are vital if students "at a susceptible age" are to be infected with a love of mathematics. But Critical Thinking Puzzles should be approached with a bit of caution. Though the book brings together many interesting problems and is written in a way which may appeal to younger people, the ambiguities in the book may cause more than a little unnecessary frustration.
----Andrew Leahy
Andrew Leahy is an Assistant Professor of Mathematics at Knox College.
Critical Thinking Puzzles, by Michael A. DiSpezio. Sterling Publishing Company, 1996, ISBN 08006994304.
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