"This is unashamedly a book for beginners." So begins this wonderful book by A. Gardiner. The goal of this book is to introduce students to the world of problem solving, and it does so marvelously. The preface indicates that the book is aimed at students aged 15 or 16 and above, but this should not prevent older students from picking up this book and learning how to tackle Olympiad type problems. Indeed, problem lovers of all ages and ability levels will find much in this book to entertain and educate.
The book consists of three main sections. Part I contains over thirty pages which set out to teach and/or review much of the mathematical background needed to tackle the problems in Part II. Part II contains the problems themselves. All of the problems from the British Mathematical Olympiad Round One papers dating from 1965 through 1996 are included. Part III contains hints and outline solutions to the problems in Part II.
While it is impossible to fit a review of all necessary mathematics into 34 pages, Gardiner does a remarkable job of covering the basics. Topics in Part I include Numbers, Algebra, Proof, Elementary Number Theory, Geometry, and Trigonometric Formulae. Some of the topics within these sections are explained in some detail so a student who has little or no familiarity with the topic will be able to grasp the material. Other topics consist mostly of the basic facts (sometimes requiring the reader to supply the proofs or derivations of formulae -- good preparation for the problems in Part II). Gardiner has made very good decisions on what to include in the limited space allotted for this material. Also, he includes an additional 10 pages devoted to a categorized list of books on mathematics and/or problem solving which will certainly help satisfy an interested reader's curiosity.
There is not much to say about Part II of the book other than to say that it contains an entertaining and challenging set of problems.
Part III is where the book truly shines. It does NOT contain solutions to the problems from Part II. Instead, it contains hints and outlines of solutions. Thus, a reader finding herself stuck on a problem can turn to the hints to get help, but will not have the fun entirely spoiled by seeing the fully worked out solution. The hints and outlines are complete enough that a bright student (who is willing to do some work) will be able to fill in the gaps and solve the problem herself. Additionally (and importantly), the solutions are not slick proofs which leave students thinking "I would never have thought of that." Instead, the solutions are at a level appropriate for the intended audience, and they make use of the material contained in Part I of the book. This will likely increase the confidence of the reader and make the book that much more exciting to work through.
As an illustration of the structure of the book, I include the following simple example from the 21st British Mathematical Olympiad, 1985:
Given any three numbers a, b, and c between 0 and 1, prove that not all of the expressions a(1-b), b(1-c), and c(1-a) can be greater than 1/4.
The first hint encourages the reader to experiment a little bit ... to note that for various choices of a, b, and c, it is always true that at least one of the expressions is less than or equal to 1/4.
The second hint asks the reader to prove that if a is between 0 and 1, then a(1-a) < ________. The reader needs to fill in this blank and complete the proof. It is pointed out that the AM-GM inequality (discussed in Part I) can help here.
The third hint tells the reader to consider the **o*u** of the three expressions a(1-b) * b(1-c) * c(1-a) and to apply the result from the second hint (after rearranging the terms) to complete the proof.
The hints in the book are somewhat wordier than this. They give the reader the chance to discover what she needs without seeing too much too soon. While not (quite) completely giving it away, the hints are designed to carefully lead the reader to discover the proof for herself.
I would highly recommend this book for anyone interested in this sort of problem solving. It would be a particularly valuable resource for those who participate in mathematics competitions at the high school or college level.
Carl D. Mueller (firstname.lastname@example.org)is Associate Professor of mathematics at Georgia Southwestern State University.