# Across the Board: The Mathematics of Chessboard Problems

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

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