*Mathematics Galore!* is a compilation of puzzles that appeared in a series of newsletters or weekly emails from the St. Marks Institute of Mathematics. It is filled with several interesting problems and puzzles that can help inspire mathematical curiosity in students of all ages. It is difficult to read through the book without pausing to put pen to paper and try to solve the problems. The chapters are laid out so that several problems are presented; followed by hints, solutions, open problems and references.

The beauty of these problems is that they are often tactile. Even a student with minimal mathematics background can begin working on the problems to see what is going on. One example of this was the puzzle that dealt with “Folding.” It asks if the reader can draw a line on a page and make two crease marks that divide the angle made by the line and the bottom of the page into three angles of the same size. One approach would be to get out a piece of paper and start folding. As the chapter continues it links this concept to decimal expansions of fractions in base two which gives the reader an algebraic way to study this problem. It is easy to envision a high school or college student making progress on this project.

One question that came to mind while reading this book was “how could I use these problems?” I think they can be used in several settings. In the section about factor trees, for example, the author makes a slight change to the definition of a prime number by restricting the factors to a subset of the integers. If, for instance, we restrict allowable factors to the set of even numbers, then the number 10 is now prime, because it cannot be decomposed into two smaller even numbers. When reading this section as an algebraist I thought this would be an excellent activity to introduce unique factorization. One can see through the examples in this section that unique factorization is lost by making this change. It is often difficult for students in abstract algebra to understand when unique factorization can fail; since it does not fail for the integers, the ring with which they are most familiar.

These problems and projects can also be used to create supplemental projects for a class in discrete mathematics or as an after-school activity for high school students. These problems show students what it means to do research in mathematics: they can explore the concepts and come up with their own answers. Students can make many observations by working through examples to conjecture what might happen in general. Several problems included have no known answers; they lead to open questions included in the text and possible generalizations that students could conjecture on their own.

I think the best use of these problems, however, would be for a faculty member who is interested in getting undergraduates involved in research but does not know where to start. Often it can be difficult to find a real research problem that might be accessible to an undergraduate student. These problems would be great for this purpose, because students can begin working right away, experiencing the joy of discovering math for themselves, and then later increasing their knowledge base to extend their work. Since each chapter also contains a list of references, students can continue their work beyond the introductory problems once they have a handle of what is going on.

I think this book is a unique resource that would be a great reference for any college faculty or high school math club advisor. I certainly enjoyed reading it!

Ellen Ziliak is an Assistant Professor of mathematics at Benedictine University in Lisle IL. Her training is in computational group theory. More recently she has become interested in ways to introduce undergraduate students to research in abstract algebra through applications.