Joseph Kirtland's recent addition to the MAA's **Classroom Resource Materials** provides a thorough collection of examples of check digit schemes for identification numbers organized for a liberal arts mathematics course. Simple coding and cryptography are featured in other liberal arts mathematics texts because of their applicability and mathematical simplicity, but this book expands on this in a way that takes students beyond identification numbers and into the realm of permutations and finite groups.

The book is short, consisting of 174 pages organized into the following five chapters:

- Identification Numbers and Check Digit Schemes
- Number Theory, Check Digit Schemes, and Cryptography
- Functions, Permutations, and Their Applications
- Symmetry and Rigid Motions
- Group Theory and the Verhoeff Check Digit Scheme

The logical organization of these topics is appropriate for introductory students, but the execution may seem dry to the type of student in a "general studies" course. For example, in The Heart of Mathematics (Burger and Starbird, Key College Press, 2000), many of the same ideas are treated (in a single section) in a vastly more engaging style. Of the dozen "liberal arts" math textbooks on my shelf, the only other one that even mentions check digit schemes is the Seventh Edition of *Topics in Contemporary Mathematics* (Bello and Britton, Houghton Mifflin, 2001), and there the topic is relegated to the exercises in a section on modular arithmetic. Kirtland's book goes into much greater depth than either of these other books.

The pedagogy involved in the structure of the sections (preliminary activity, followed by text and examples, followed by exercises, writing assignments and group activities) is interesting. The structure of each section is very sound, but the quantity and quality of the components vary. The "preliminary activities" occasionally seem too vague to be successful, and the "group activities" often simply tell the students to do one of the regular section exercises, but "as a group." Another difficulty, and perhaps the most serious problem with this book as a text, comes from the shortage of exercises. Liberal arts mathematics texts typically have dozens of exercises for each concept of varying degrees of difficulty. Kirtman's book averages about six exercises per section. Overall the author has developed a good plan for a textbook, but in many places, the execution of this plan is lacking.

The main strength of this book is in the comprehensiveness of the treatment of this topic. No other single source contains as many different types of check digit schemes for identification numbers. Every mathematical construction required for understanding the check digit schemes is introduced in advance, and the progression through the material is well planned.

Unfortunately the book also has some weak moments, particularly when the author strays from his main focus. These lapses are particularly disappointing because the Preface bills these excursions as a feature of the book:

Not only are all mathematical concepts developed within the context of studying check digit schemes, but as each mathematical topic is studied, other applications are discussed. This will lead to a study of not only check digit schemes, but also 'public key' cryptography systems, graphing data, presenting data and symmetry.

- The section entitled Graphs of Functions is a pure non sequitur, consisting of six pages tacked on the end of a chapter, containing exactly two exercises, and lacking any explicit ties to the concepts before or after. This is apparently the section in which we study both "graphing data" and "presenting data."
- The section that treats "public key cryptography systems" is rather poorly motivated for such a popular topic. For example, the author suggests that a computer can test 27! permutations "in a matter of minutes" to break a substitution cipher. Not only is this nonsense, it misleads the students on how substitution ciphers are actually broken. The RSA system is introduced, appropriately with manageable numbers, but it is then used to encrypt single characters with the author never pointing out that this is equivalent to a substitution cipher! In fairness, the other example given for RSA does encrypt pairs of characters, but the author never explains how the RSA system is really used or why it is a more secure encryption scheme than a simple substitution cipher.

On the positive side, the excursion into symmetry is terrific. Chapters 4 and 5 are quite well done, presumably as the material gets closer to the author's main interest of finite group theory. The exercises and activities on symmetry in Chapter 4 are well-conceived and well-connected to the notion of a permutation. Just enough of these ideas carry over to Chapter 5 that the students will feel that their investment in Chapter 4 has paid off. Chapter 5 also revisits the modular arithmetic material from Chapter 2 in a way that should effectively tie things together.

The author proclaims this book to be an "ideal textbook for a liberal arts math course," but its dry exposition, shortage of quantity and variety in its problem sets, and questionable activities leave one wondering if it has been used successfully in this way before. It is difficult to decide if this would best be used as a supplement or as a text to be supplemented. It seems unlikely that it can be used alone as a text for a semester-long course.

This book would be most successful as an informal, application-driven introduction to finite groups for a general audience. The book will be most satisfying for those who wish to pursue this route with their liberal arts mathematics class. Those interested in devoting only a portion of their semester to check digit schemes without the finite group ideas will probably find better material available.

Doug Ensley ([email protected]) is associate professor of mathematics and computer science at Shippensburg University in Pennsylvania. His current interests are in the use of recreational mathematics to motivate student exploration and research and in the use of technology for teaching and learning. He is the first editor of the MAA's Digital Classroom Resources, a piece of the Mathematical Sciences Digital Library.