Adventures in Mathematical Reasoning

Do you know someone who is curious about mathematics and yet struggled with poor grades while in school? Do you know a mathematically talented student in grades 6–12 who always wants to know “what’s next in mathematics?” Stein’s book may be a good fit for either audience. This book may also make a good supplement for various liberal arts mathematics courses or first-year seminars as the problems and writing style are engaging but do not presume any specific background or mathematical sophistication.

Stein’s book covers eight mathematical problems in mostly independent chapters. The emphasis in this book is on discrete mathematics and problems that can be expressed in terms of properties of strings of characters or counting elements of a set. Each problem is introduced, often with a practical application, re-expressed in terms of abstract symbols (often strings), and then solved.

The first chapter, on the so-called “Buffon needle problem,” is a great opener that will hopefully hook the reader with the unexpected result that the probability that a needle dropped on a lined surface hits one of the lines involves the value of \(\pi\) and is essentially independent of the shape of the needle. Chapters Two and Four are motivated by sports (games that must be won by two points or winning/losing streaks for teams). Chapter Six discusses an issue related to whether a candidate on a ballot will consistently stay ahead in the counting while Chapter Five is motivated by an engineering problem: trying to transmit strings of data efficiently and without error. Chapters Three, Seven, and Eight deal with labeled triangles, Cantor’s theory of infinite sets, and repeated blocks of characters in strings. These three chapters are probably the most abstract and thus most likely to appeal to a more mathematically talented or inquisitive audience.

This book is best read in short, well-spaced sessions when the reader has time to think while reading. Attempting to “binge read” every chapter in a day is not recommended. I would also advise any future reader of this book who is lacking a degree in mathematics (particularly pre-college readers) that not every problem in higher mathematics can be reduced to a problem involving strings of characters as is done for the discrete problems in this book. While I think this book may successfully interest a reader enough to pursue further reading in mathematics, I would not want it to turn away someone who may learn to enjoy another branch of mathematics and yet not find interest in problems related to strings of characters. Overall, the book is very well written and many different audiences should find various parts of the book interesting and easy to read.

Geoffrey Dietz is an Associate Professor at Gannon University in Erie, PA. He is married and has six children.

Editor's Note: This book was originally published in 2001 as How the Other Half Thinks. See our review of the original edition.