When I see a book called *Mathematical Puzzling* by someone named Gardiner, I am likely to assume that the “i” is a typo and it is another collection by Martin Gardner, master of all recreational mathematics. A. Gardiner, however, wrote a different kind of puzzle book that is not about the puzzles themselves so much as using puzzles as a mechanism for mathematical learning.

Each of the short 27 chapters includes a collection of related puzzles of varying difficulty, often with open-ended “investigation” sections inviting the reader to explore or generalize the ideas. The chapters close with detailed commentary featuring hints and assorted observations which are themselves sometimes followed by “extension” commentary inviting still more exploration. The presentation often feels disorganized, but there is some method to it.

The structure of the chapters depend on the chapter number modulo three. Chapters congruent to 1 mod 3 are comprised of puzzles that have some loose theme; 2 mod 3 attacks a more specific idea from several angles, and 0 mod 3 presents a puzzle sequence that develops a topic. The chapters are more or less independent, but are clustered with purpose. It doesn’t matter much since it’s a short book, but the strange numbering and unhelpful chapter titles do make it a chore to navigate.

The multi-layered discussions are what make the book different and well-suited to teachers. Each chapter is a ready-made special project, with the variety of structure described above. The puzzles themselves are nothing special, generally including common fare such as magic squares, word-sums (my favorite being to find digits that fit the pattern 1777+1855+CARL=GAUSS), or a discovery of properties of the Fibonacci sequence. The puzzles vary in the topics they explore, including number theory, combinatorics, and geometry. All puzzles require minimal mathematical background — only a few even require much in the way of algebra. Puzzles are arranged in rough order of difficulty, allowing the reader to get the first few by experiment or calculation and later inviting more cleverness and even research.

As a collection of puzzles, this book probably won’t be compelling to mathematicians or puzzle enthusiasts. But thought of more as a collection of short problem-sequences rather than puzzles, it has value as a guide for discovery learning in several rewarding topics. It is worth considering for libraries or resource centers specifically serving educators.

Bill Wood is an Assistant Professor of Mathematics at the University of Northern Iowa. His involvement with puzzles and games, mathematical and otherwise, has probably transitioned from hobby to illness.