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Across the Board:
The Mathematics of Chessboard Problems
John J. Watkins

Paperback | 2012 | $18.95 | £14.95 | ISBN: 9780691154985
272 pp. | 5 1/2 x 8 1/2 | 204 line illus.
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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.


"This book is extremely well written and is, no doubt, the best exposition of the connection between the chessboard problems and recreational mathematics. The author surveys all the well-known problems about chess and the chessboard. . . . The problems are treated in depth from their beginnings through to their status today."--Mohammed Aassila, MAA Review

"Torus-shaped boards, three-dimensional boards, a shape called the Klein bottle--the simple checkerboard pattern proves to be creatively malleable when Watkins puts his mind to his hobbylike subject. Watkins' invitational tone ensures attention from the finite but enthusiastic audience for mathematical recreation."--Booklist

"Watkins offers an excellent invitation to serious mathematics."--Choice

"I would be happy to recommend this book to you. . . . The book is an easy and entertaining read that shows numerous paths into various branches of discrete mathematics and graph theory."--Paul J. Campbell, Mathematics Magazine

"This is not just about chess, but also the three centuries of 'recreational mathematics' that the game has inspired. From simple questions, such as whether it is possible for a knight to land on each square of the board on its path, Watkins wades into graph theory, the mathematics of three-dimensional chess and even chess on a torus."--Nature Physics

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Table of Contents:

Preface ix
Chapter One
Introduction 1
Chapter Two
Knight's Tours 25
Chapter Three
The Knight's Tour Problem 39
Chapter Four
Magic Squares 53
Chapter Five
The Torus and the Cylinder 65
Chapter Six
The Klein Bottle and Other Variations 79
Chapter Seven
Domination 95
Chapter Eight
Queens Domination 113
Chapter Nine
Domination on Other Surfaces 139
Chapter Ten
Independence 163
Chapter Eleven
Other Surfaces, Other Variations 191
Chapter Twelve
Eulerian Squares 213
Chapter Thirteen
Polyominoes 223
References 247
Index 251

This book has been translated into:

  • Dutch

Other Princeton books authored or coauthored by John J. Watkins:


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