Originally published in 1969, *Excursions in Mathematics* has since then become one of the most popular mathematics books. Three decades later, we the expanded and revised *Millennium Edition.* In addition to the original material it also discusses new research and solutions to problems that have been solved since then, such as the Four Color Theorem and Fermat’s Last Theorem. Thus, it is a valuable record of mathematical developments that have taken place these three decades.

The authors, Anatole Beck, Michael N. Bleicher, and Donald W. Crowe, give the reader real insight into the intriguing world of mathematics. As the title of the book suggests, they take the reader on short “excursions” into mathematical subjects. The six chapters are independent and range over important elementary concepts from number theory, Euclidean and non-Euclidean geometries, and game theory.

There is a very well written explanation of why the binary analysis of the remarkable game of Nim works and a complete analysis of the game of Hex. There is a detailed discussion on matrix games (such as Matching Pennies) and an interesting proof on the minimax theorem for 2×2 games. Another appealing chapter, by Michael N. Bleicher, comprehensively discusses Farey numbers, Egyptian fractions, the Euclidean algorithm, and continued fractions. Chapter 4 on “Some Exotic Geometries” covers spherical and hyperbolic geometries, finite affine and projective planes, and gives Poincaré’s famous model of the hyperbolic plane. Lastly, there is a chapter on Euclidean and non-Euclidean perfect numbers, prime numbers and factorization techniques, and amicable numbers.

The text in the book is interspersed with rich mathematical background and history, amazing illustrations and drawings, a play excerpt (*Sapientia* by Roswitha), and applications of the mathematics. The authors show how to use matrix games to compute the expectation of finding and destroying German submarines in the Second World War and how to use cooperative games to study production and other real economic problems.

All of the mathematical concepts in the book are introduced in such a “smooth”, elementary, and friendly manner as to be easily understood by the general reader. The book contains diverse exercises and extensive lists of references for further reading, a few elegant and ingenious proofs, and numerous answered and unanswered problems.

*Excursions into Mathematics* is a very exciting and readable book, which aims “to acquaint the general reader with some of the flavor of mathematics” and to make it accessible to anyone. It is truly an excellent way to enrich any mathematical education. I would recommend it to any reader with a genuine interest in mathematics.

A native of Macedonia, Ana Momidic-Reyna has an M.S. in Mathematics and has also worked for the high energy physicists at Fermilab. While waiting for the opportunity to work on her Ph.D. in mathematics, she keeps up with the field by reading as many mathematics books as she can.