Across the Board: The Mathematics of Chessboard Problems

Synopsis

Across the Board is the definitive work on chessboard problems. It is not simply about chess but the chessboard itself--that simple grid of squares so common to games around the world. And, more importantly, the fascinating mathematics behind it. From the Knight's Tour Problem and Queens Domination to their many variations, John Watkins surveys all the well-known problems in this surprisingly fertile area of recreational mathematics. Can a knight follow a path that covers every square once, ending on the starting square? How many queens are needed so that every square is targeted or occupied by one of the queens?

Each main topic is treated in depth from its historical conception through to its status today. Many beautiful solutions have emerged for basic chessboard problems since mathematicians first began working on them in earnest over three centuries ago, but such problems, including those involving polyominoes, have now been extended to three-dimensional chessboards and even chessboards on unusual surfaces such as toruses (the equivalent of playing chess on a doughnut) and cylinders. Using the highly visual language of graph theory, Watkins gently guides the reader to the forefront of current research in mathematics. By solving some of the many exercises sprinkled throughout, the reader can share fully in the excitement of discovery.

Showing that chess puzzles are the starting point for important mathematical ideas that have resonated for centuries, Across the Board will captivate students and instructors, mathematicians, chess enthusiasts, and puzzle devotees.

Excerpt

In play there are two pleasures for your choosing—
The one is winning, and the other losing.

Lord Byron

Games of all kinds were an important part of my childhood. There is a comfortable pleasure, not unlike that of sitting down with a really good book, in retreating from the real world into a simpler realm where the rules are crystal clear and the objectives well defined. I was taught to play chess one summer by my older brother, Reed, when I was very young. Even then, chess had a special, slightly mysterious, attraction that appealed to me and, that summer, I forced all the kids in my neighborhood to join my newly formed chess club. Gradually, though, my enthusiasm for chess waned and by high school my interests had turned elsewhere.

Meanwhile, when I was about 13 or 14, my brother came home from college and showed me a copy of Scientific American that contained a column by Martin Gardner called Mathematical Games. Reed knew I liked mathematics and thought I would find his monthly column interesting. Little did he know! Just as for many mathematicians of that era, Gardner’s articles have been for me a continuous source of inspiration. Just as I can vividly remember details of the chemistry classroom in which I first heard that President Kennedy had been shot, I still remember the exact place where I was sitting when, as a graduate student, I first read Gardner’s article on rsa codes. One of the things Gardner teaches us is that mathematics is everywhere— even on the chessboard, a topic to which he returned again and again.

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