*Competitions for Young Mathematicians* is a collection of 14 essays loosely organized into eight broad categories. The opening essay by the editor makes the case for essay-style mathematics competitions (often referred to as Olympiads) versus multiple-choice competitions such as the AMC 8, 10, and 12 competitions. Other essays describe how Olympiad competitions are organized and/or provide sample problems as well as some hints about developing and solving Olympiad problems. In some cases, a chapter provides an historical narrative about a particular competition. For example Chapter 9, “The International Mathematical Tournament of Towns,” describes this Tournament in great detail including a table of rating of the various towns. I am not sure that function that list serves. It concludes with a list of past problems. They are quite good.

Soifer is the founder and still director of the Colorado Mathematical Olympiad, which is now in its third decade. He argues, correctly I believe, that for the very best students essay-style competitions provide a more realistic view of how mathematics is actually done and can provide an entry point into independent research. He argues further that such competitions should be an important part of mathematics instruction in the schools. In my view, the MAA competitions provide a tiered approach (from multiple choice to single answer to proofs) that seems to fit well with Soifer’s view. It would be wonderful if a Research 1 institution in each state would step up to the plate, as Soifer and Colorado have done, but I suspect most faculty do not feel they have the time and/or expertise to create and administer Olympiad-style competitions. Soifer’s book seeks to aid such efforts, but I am not convinced it provides sufficient information to ensure a successful competition. In short, I think the main reason to read this book is for the wonderful problems (and solutions) it contains.

Soifer argues that competitions such as the AMC 8, 10, and 12 and MathCounts encourage clever guessing and working fast rather than thoughtfully. While I agree with him in part, there is no avoiding the fact that these competitions attract many smart kids. I think that they attract thousands of students of less than genius status, at least in part, because of the fun of learning how to solve lots of interesting, if less challenging, problems. In Illinois, the NCTM affiliate group has sponsored a team competition each year for over 25 years. The questions are not nearly as difficult as the AMC questions, but the students seem to enjoy the competitions a great deal. Using course-based questions, contests like this allow all students to take part. Olympiads serve only the very best students. Team competitions allow many more students to enjoy doing math. They may not grow up to be mathematicians, but maybe they will vote in favor of school funding bills!

Many subsequent chapters consist mainly of problems and the techniques for solving them. The book is thus a solid source of good problems and, in many cases, solutions. Here are brief descriptions of three of my favorites.

Chapter 6, “The Rainbow of Mathematics,” contains a wonderful defense of the joys of pure mathematics as well as some solid advice about how to get students (and others) to see how much fun pure math can be. Sudoku is offered as a good entry point since lots of people play this game without realizing there is lots of math lurking behind the surface. Sudoku can quickly lead to very interesting questions: What is the minimal number of filled squares required to ensure a unique solution? What is the maximal number of square that can be filled without forcing a unique solution?

Chapter 4, “Arrangements and Transformations of Numbers on a Circle,” is my favorite chapter. Each problem consists in an arrangement of numbers on a circle along with an operation that can be performed on them. The problem is to determine which other arrangements can be reached using the permissible operation a finite number of times. What is nice about these problems is that there is little to no background material assumed, yet they can be quite challenging. These problems could lead nicely into a discussion of Turing Machines and other finite state automata. Chapters 6 and 4 could form the nucleus of a great problem-solving seminar

Chapter 3, “Techniques for Solving Problems in Plane Geometry,” provides several nice examples of the power of analytic geometry as a tool to solve problems that are difficult to solve using only Euclidean techniques. Like several other chapters, it requires more mathematical sophistication and knowledge than chapters 6 and 4. Examples include problems where assigning coordinates to the relevant points allows one to compute slopes, and lengths as well as check whether two lines are perpendicular using the product of their slopes. Other problems use the quadratic equation, trigonometry and linear algebra. As with several other chapters the difficulty level of the problems ranges from (as I assess things) reasonably hard to very hard.

I would recommend this book for persons with an interest in problem solving and as a great source of problems for advanced students.

Richard Wilders is Professor of Mathematics at North Central College in Naperville, IL. He directs the College’s STEM outreach program to pre-college students including offering all of the AMC exams and advising pre-college students enrolled in North Central courses.