Across the Board: The Mathematics of Chessboard Problems

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Princeton University Press, Jun 4, 2012 - Games - 272 pages
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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.

 

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Contents

Introduction
1
Knights Tours
25
The Knights Tour Problem
39
Magic Squares
53
The Torus and the Cylinder
65
The Klein Bottle and Other Variations
79
Domination
95
Queens Domination
113
Domination on Other Surfaces
139
Independence
163
Other Surfaces Other Variations
191
Eulerian Squares
213
Polyominoes
223
References
247
Index
251
Copyright

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About the author (2012)

John J. Watkins is professor emeritus of mathematics at Colorado College. An award-winning teacher, he is the author of "Topics in Commutative Ring Theory" (Princeton) and coauthor of "Graphs: An Introductory Approach".

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