Across the Board

by Watkins, John J.

• ISBN13: 9780691115030
• ISBN10: 0691115036
• Format: Hardcover
• Copyright: 2004-03-01
• Publisher: Princeton Univ Pr

Summary

Across the Boardis 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 Boardwill captivate students and instructors, mathematicians, chess enthusiasts, and puzzle devotees.

Author Biography

John J. Watkins is professor of mathematics at Colorado College.

Table of Contents

Preface	ix
CHAPTER ONE Introduction	1	(24)
CHAPTER TWO Knight's Tours	25	(14)
CHAPTER THREE The Knight's Tour Problem	39	(14)
CHAPTER FOUR Magic Squares	53	(12)
CHAPTER FIVE The Torus and the Cylinder	65	(14)
CHAPTER SIX Klein Bottle and Other Variations	79	(16)
CHAPTER SEVEN Domination	95	(18)
CHAPTER EIGHT Queens Domination	113	(26)
CHAPTER NINE Domination on Other Surfaces	139	(24)
CHAPTER TEN Independence	163	(28)
CHAPTER ELEVEN Other Surfaces, Other Variations	191	(22)
CHAPTER TWELVE Eulerian Squares	213	(10)
CHAPTER THIRTEEN Polyominoes	223	(24)
References	247	(4)
Index	251