The Mathematics of Various Entertaining Subjects, Volume 2

This is the second in what I hope will become a long continuing series on The Mathematics of Various Entertaining Subjects. The editors once again have brought together an extraordinary list of authors to produce nineteen engaging papers, split into five groups: puzzles and brainteasers, geometry and topology, graph theory, games of chance, and computational complexity. It is an unfortunate choice by the publisher to put this book into the recreational mathematics category. I found this book anything but relaxing and recreational (and even Ron Graham in his Forward makes a point of defining recreation as something we do to relax or have fun). In fact, it is often deeply challenging mathematically and, as a result, all the more fun. Each reader will find chapters that appeal to them. I will describe some of my favorites.

One surprisingly interesting chapter by Tanya Khovanova is about the behavior of six four-armed dragons sitting on the faces of a cube trying to steal kasha (a very healthy buckwheat porridge) from their neighbors. Writing with far more humor than mathematicians are usually known for, the author manages to draw us into connections between group theory, linear algebra, and representation theory.

In a wonderful paper about logic puzzles Jason Rosenhouse pays due homage to Raymond Smullyan (who clearly is the winner of the of the Best Book Title Ever for his 1978 book What Is the Name of This Book?). He includes the following unusual knight/knave puzzle by Smullyan (a knight always tells the truth, a knave always lies):

While visiting a knight/knave island, a native says to you, “This is not the first time I have said what I am now saying.” What can you conclude about the native?

Richard Guy’s chapter “A Triangle Has Eight Vertices But Only One Center” was for me perhaps the most eye-opening. I rather foolishly have always thought of plane geometry as somewhat of a dead end, but Guy’s beautifully illustrated chapter showed me just how wrong I was.

In his chapter “Trees, Trees, So Many Trees’ Allen Schwenk begins brilliantly by showing four graphs (including the Petersen graph) and asking the simple question: how many spanning trees does each graph have? He then proceeds to develop the mathematics needed to knock off one of these four graphs at a time (this has the feel of a well-crafted murder mystery where the crime is gradually solved one piece at a time). Needless to say, he comes to a deeply satisfying conclusion.

I greatly enjoyed Geoffrey Dietz’s analysis of the game Pop-O-Matic Trouble and only wish I had a 5-year-old to play it with. I do plan to buy one immediately and find colleagues to play it with. His analysis of tactics and strategy was very appealing. In the final section I found two chapters on the computational complexity of two games I am completely unfamiliar with — Multinational War and Clickomania — far less appealing. On the other hand, the final chapter by Erik Demaine and William Moses, “Computational Complexity of Arranging Music,” is the very model of how mathematics can be applied in unexpected places, in this case to the question of the computational complexity of arranging music written for one set of instruments to, say, a viola or a piano.

See also our review of the first volume in the series.

John J. Watkins is Professor Emeritus of Mathematics at Colorado College.