NB. I was sent this book as a review copy.

From Princeton University Press

I tell my first year students that whether or not they will use their first year maths directly in the future, taking a course in mathematics is like going to a gym for your brain. Unless you are doing some good mental sweating, you are not benefiting from the study. It should be a subject in which you grow by gently (or not) applying more and more intellectual pressure to your thought patterns, and over time you will find that you can understand more complicated, or more abstract concepts than you ever thought that you could before. This translates into solving problems which may not have anything to do with maths, but require a similar pattern of logical juggling.

This book (The Mathematics of Various Entertaining Subjects, Volume 2) feels like Crossfit for the mathematics world. It’s a book filled with strength, endurance, flexibility and power exercises, each of which will stretch you in different ways. I found myself chuckling regularly at the amazing ways my brain was forced to work while trying to understand the puzzles discussed here. I should note that it is not an exercise book at all, but a book in which puzzles are presented and their solutions explained. However, simply following allow with the solutions takes in such a depth and breadth of mathematical topics, that just about anyone will find something to stretch them.

It is beautiful in that just about every problem could be explained to anybody with almost no mathematics background at all, but the methods of solving them take you deeply into many complex areas of mathematics.

The book gathers together problems which pop up through what one might consider ‘silly’ or ‘frivolous’ questions, but which lead to new ways of thinking and have applications in enormously wide-ranging areas of mathematics.

The book is split into sections covering different areas of mathematics, including Geometry, Topology, Graph Theory, Games of Chance, and Computational complexity, together with a section of miscellaneous puzzles and brain teasers.

The first chapter, is a puzzle about prisoners creating a system to distribute information with very little communication allowed. This, of course, has implications for the world of distributed computing, and the solutions offered are written in the language of group theory. Group theory in fact turns up in a lot of the problems here, and I would say that many of the chapters here would be ideal to get advanced students thinking in an undergraduate course on any of the above topics. I am certainly planning on sharing the chapters on Graph Theory to the graph theorist in my department, and I hope that he may share some of the ideas with his Honours class.

Some of my favourite questions in the book were about simple, linguistic logic puzzles. From the logical syllogism “all men are mortal and all Greeks are men” it is possible to conclude that all Greeks are mortal. This is simple to understand and one can picture the flow of logic. However, the logical consequence of “No nice cakes are unwholesome, and some new cakes are unwholesome” is (at least for me) a lot harder to juggle in my head. A diagrammatic method for analysing such statements developed by Lewis Carroll is explained in lovely detail. In the same chapter (by Jason Rosenhouse), the classic puzzles of finding which person always lies and which person always tells the truth when you come across one of each and are able to ask one question is presented…and then extended to absurd and wonderful lengths, to islands where fuzzy and three valued logic hold amongst the populace and one has to determine the nature of a group of islanders. Again, such frivolous games have clear parallels in the real world where one may want to most efficiently determine the nature of a statistical system using as little communication as possible.

Questions about the logical extension to the duel (where multiple gunslingers have known probabilities of hitting their target) are encountered, the tangling and untangling of tangled toys are explored, numerically balanced dice are tackled using magic squares and Integer Programs and generating functions show up across many different graph theoretic questions. I think that perhaps another interesting exercise for students using this book could be to find real world applications of the techniques expounded on here. It is possible, on reading each chapter, to see a huge number of different applications and extensions and this is precisely the sort of gym for the brain an enthusiastic maths student should be exposed to!

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