The Proof and the Pudding
Jim Henle is a lover of mathematical puzzles and an admirer of Martin Gardner. His interest clearly shows in this book that he has written with another of his avatars: the man who loves to cook. From the title of the book, one could expect some recreational mathematics to model the cooking process or some optimization problem to compose a recipe that satisfies certain criteria, or a recipe to bake π. However, to my surprise this is not what this book is about. The subtitle describes the content more precisely: What mathematicians, cooks, and you have in common. This is indeed the essence of the message he wants to transmit. We find real recipes for baking bread, pizza, pancakes, apple pie, etc. on the one hand, and there are several mathematical puzzles or games with their analysis, possible complications, solutions, winning strategies, etc. as well. And then there is of course what they have in common. You need to experiment, learn by experience, combine different techniques, learn from mistakes, be not afraid, and most of all enjoy the fun of it. Everybody can learn to cook, just like everybody can learn to do mathematics. Everybody has preferences for certain tastes, yes, and that is just like mathematicians have preferences for analysis, algebra, etc. A cook can use prepared ingredients, i.e., ingredients with ingredients in it, and a mathematical proof can rely on proofs of other theorems. Sometimes, the parallel is rather thin, and it may just be a word like simplicity, complexity, exotic, weird,... Anyway, there are many parallels to be drawn if one is willing to.
One advise he gives for budding cooks and mathematicians: when the solution is not immediately clear, just mess around and try out something. If it does not work, try something else, if it does work, find your mistake. Just an erratic random walk, hoping to end up somewhere. This may raise a shivery reaction of unbelief at first, but he relaxes this algorithm a bit later when he says that this will not always lead to a solution. However, this tinkering should never be a burden. It should always be fun, end even if you did not find the solution, you always did learn something and you gained insight and understanding of the problem and that is a comforting feeling.
Although he does not mention it so explicitly, there is one thing Henle does extremely well: he transfers an attitude that is a proper one to become a good professional mathematician or a proper chef, or that is apt to any challenging profession for that matter. When a simple puzzle is solved, you should start modifying the boundary conditions, or the rules, and try to answer the question 'what if this or that is added as an extra condition'. Although this is presented at a recreational level, this is indeed what professional mathematical research is about: inquisitiveness, questioning results, and the 'what if' attitude.
On the mathematical side, there are a few simple proofs, some ideas on constructivism and whether such proofs are acceptable, some number theory, the construction of finite algebras, and even a generalization of the Euclidean algorithm. There is a discussion of Cheney's card trick, and there are several games like phutball, nimrod (a variant of the nim game). As puzzles, we find e.g. variants of Sudoku puzzles, but a recurrent idea throughout the book is a puzzle that starts in its simplest form as a rectangular pool table where you shoot a ball along the bisector of a corner and see if and when it will end in the same or another corner. One may start an analysis of the influence of the aspect ratio of the table, or if length and breadth are integers, and the table is covered by unit square tiles, will the ball pass through all these tiles? What if the same problem has to be solved in a cuboid, or if the ball is following a three-dimensional path on the surface of a cuboid. This kind of problems and many variants show up in several chapters.
The cookery is fun, but since, as Henle already mentioned, there are different tastes, and although I love cooking myself, the recipes are a bit too American style for my taste, which is rather different from European cuisine. But they are original, as far as I can judge, and it is certainly fun to try out a particular combination of ingredients that was still unknown to you.
The puzzles and games are really fun. The spectrum may be less broad, but they differ from what can be found in the many other books with similar collections. They certainly are accessible to anyone with no or very elementary mathematical knowledge. And yet, they bring the proper message of an attitude that is needed to do mathematics professionally.
The cartoon illustrations and the imaginative typesetting cheer up the whole book. An ideal present for a family with young children that are confronted with mathematics at school. It might give ideas to spend some quality time together as a family while cooking and/or gaming.