When students today solve mathematical “problems”, they are not actually solving problems. Rather, they are doing exercises. In The Art and Craft of Problem Solving, Paul Zeitz establishes the distinction and introduces readers into the culture of problem solving. Zeitz’s book trains readers to be mathematical problem solvers while demonstrating the beauty and intricacy of mathematics.
The first four chapters prepare the readers by filling their problem-solving toolkit with important “strategies” and “tactics”. Zeitz then tackles the four main categories of mathematical problem: algebra, combinatorics, number theory, and geometry. There is a chapter devoted to each category. The book concludes with a chapter on calculus. The individual chapters teach important concepts of mathematics and provide sample problems that are then solved. Each chapter is not a systematic lesson, but rather enhances the reader’s mathematical knowledge and understanding.
The heart of a problem solving book is, of course, the problems. Zeitz’s book presents problem sets at the end of every subchapter. The problems come in three types: classic, contest, and exploration. Recreational problems, as the name suggests, are fun puzzles that require creativity. Contest problems, from sources such as the AIME and IMO, are formal problems that can be quite difficult. Open-ended problems that invite exploration are a unique aspect of the book. They do not require an answer, and sometimes do not even have one. They are the most challenging problems in the book, but at the same time are the most rewarding.
What sets Zeitz’s book apart from other books on problem solving its emphasis on the methodology of problem solving. Problem solving books of the same kind often present many problems, but do not explain how a problem is solved. Certainly, other books have sparse solutions for their presented problems, but they neither explain how to explore the problem nor how to arrive at the solution. Zeitz’s book focuses on the exploration process and so teaches the reader how to engage with the problem. The sample problems brought up in each chapter are thoroughly explored and investigated. Limitations and implications of the presented solutions are fully discussed and sometimes built on in later chapters.
One defect of Zeitz’s book is the lack of a solutions section. Though each chapter explains sample problems in great detail, the problem sets presented at the end of each subchapter do not have solutions. Given the difficulty of some of the problems, a solutions section would be a welcome amenity. There is, however, an additional instructor’s manual that can be purchased. The manual provides full solutions to most problems, hints for others, but no solutions for contest problems (readers are expected to consult the wealth of online sources for solutions to the contest problems).
Overall, The Art and Craft of Problem Solving is an excellent gateway to the culture of problem solving. It is challenging and rewarding. Zeitz’s book shines a new light on mathematics and engages readers with its wonderful insights and problems.
Jack Chen (firstname.lastname@example.org) is a high school student who enjoys mathematics. He hopes to pursue mathematics in his post-secondary education. His current mathematical interests are in modular arithmetic and cryptography.