There is no question that mathematics is a creative endeavor. New ideas are constantly developed, examined, polished and then presented. From those ideas, it is possible to create specializations or generalizations, in many cases both. In this book, Beardon takes eleven problems, presents them, gives a solution and then gives a generalization of the problem-solving strategy. Throughout the process, there is an explanation of how mathematical results are derived, sometimes by solving a specific case and generalizing and other times by solving the general case first and then working to the special case.
The problems cover a wide range of topics in geometry, number theory, probability, polyominoes and a set of weights that will allow any amount to be weighed, although the specific nature of the problems is not the major point. The main idea in this book is to illustrate how solutions to problems are derived in a step-by-step manner and then those results used to solve additional problems. Beardon does an excellent job, making this a book that would be an effective text in courses for math majors where the focus is on learning how to do mathematical proofs.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
Preface; Part I. Style and Presentation: 1. Solving problems; 2. Writing mathematics; 3. Giving a presentation; Part II. The Problems: 4. A first look at the problems; Part III. Solutions and More Problems; Part IV. Discussion and Generalisations; Index.