Across the Board

Across the Board: The Mathematics of Chessboard Problems

John J. Watkins
Copyright Date: 2004
Pages: 272
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    Across the Board
    Book Description:

    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.

    eISBN: 978-1-4008-4092-2
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Preface
    (pp. ix-xii)
  4. CHAPTER ONE Introduction
    (pp. 1-24)

    This book is about the chessboard. No, not aboutchess, but about the board itself. The chessboard provides the field of play for any number of games, both ancient and modern: chess and its many variants around the world, checkers or draughts, Go, Snakes and Ladders, and even the word game Scrabble. Boards for these games come in many sizes: 8 × 8 boards for chess; 8 × 8 and 10 × 10 boards for checkers, depending on what part of the world you are in; 10 × 10 boards for Snakes and Ladders; 15 × 15 boards for these...

  5. CHAPTER TWO Knight’s Tours
    (pp. 25-38)

    There I was, sitting in my seat at theSecond Stage Theatre, a seat I had paid $80 for mind you, watching renowned sleight-of-hand artist Ricky Jay do a knight’s tour on stage in front of a sold-out crowd of several hundred theater goers each of whom had also shelled out roughly $80 for the privilege. I could hardly believe my eyes, it was the summer of 2002, roughly 1000 years after the idea of a knight’s tour first emerged somewhere in India, and here I was watching the world’s most famous conman—you’ve probably seen him in such wonderful...

  6. CHAPTER THREE The Knight’s Tour Problem
    (pp. 39-52)

    In Chapter 1 we saw that a knight’s tour is impossible on the 4 × 4 chessboard and also on any board both of whose dimensions are odd. Knight’s tours are also impossible for a number of other chessboards. For example, if a board has only one or two rows, there is not enough room for a tour; with one row, a knight can’t even move, and with two rows, a knight can only move in one direction but is then stuck at that far end of the board. By the way, I should mention that, in general, we will...

  7. CHAPTER FOUR Magic Squares
    (pp. 53-64)

    At least a generation before Euler was doing his own work on knight’s tours in Switzerland, there was a Fulani mathematician and astronomer living in the city of Katsina in a region of West Africa that is now the northernmost part of Nigeria, but which at that time was at the southern end of major trade routes that crossed the Sahara to northern Africa. In 1732, this African mathematician, Muhammad ibn Muhammad, wrote a manuscript, in Arabic, concerning the construction of magic squares. Amazingly, this manuscript has survived.

    Claudia Zaslavsky described Muhammad ibn Muhammad’s work on magic squares in her...

  8. CHAPTER FIVE The Torus and the Cylinder
    (pp. 65-78)

    In this chapter we will look at a number of the many variations on the theme of knight’s tours that have been explored by considering chessboards on two surfaces: the torus and the cylinder. We begin with the most natural variation of all, one which we have discussed several times already, and which involves simply wrapping both the columns and the rows of the chessboard into closed loops.

    Atorusis the name we give to a surface that is shaped like a doughnut. We can think of creating a torus by starting with a rectangle, say a piece of...

  9. CHAPTER SIX The Klein Bottle and Other Variations
    (pp. 79-94)

    Up to this point we have looked at rectangular chessboards in three different ways. The first way is as the perfectly ordinary chessboard with standard edges that fully contain the chess pieces within the chessboard. The second way is to identify the top and the bottom edges of the board and also to identify the left and the right edges, that is, the chessboard becomes a flat torus and the chess pieces gain considerable freedom of movement since the edges, in effect, disappear. The third way to look at rectangular chessboards is to identify just one pair of opposite edges,...

  10. CHAPTER SEVEN Domination
    (pp. 95-112)

    The concept of domination is one of the central ideas in graph theory, and is especially important in the application of graph theory to the real world. Imagine a network of some kind, it could be a communication network such as a cellular phone system or perhaps a network of roads in your local community. Such systems often require vital transmission stations to make them work effectively. A cellular phone company must provide an adequate number of communication links suitably spaced so that customers always have a strong signal for their cell phones. Similarly, your local community needs to provide...

  11. CHAPTER EIGHT Queens Domination
    (pp. 113-138)

    Of all the chessboard-domination problems, it is that of the queen that continues to hold the most interest among mathematicians. This seems to be a remarkably difficult problem and one that is far from solved even today, although there is much that is known. For example, as we noted in Chapter 1, five queens are required to dominate the 8 × 8 chessboard, and it has been observed by Yaglom and Yaglom that there are exactly 4860 different ways that these five queens can be arranged so as to dominate the board. Several of these arrangements are shown in Figure...

  12. CHAPTER NINE Domination on Other Surfaces
    (pp. 139-162)

    We closed the last chapter with the observation that among chessboards with more than four rows, the 5 × 12 chessboard and the 6 × 10 chessboard are the largest boards that can be dominated by four queens. In particular, the four queens in Figure 1.12 in Chapter 1 failed to cover the 8 × 8 chessboard shown there. However, as we noted earlier, they only missed two squares on that board, and if we were to think of this chessboard as being on a torus, then those two squareswouldbe covered. In fact, it would even be sufficient...

  13. CHAPTER TEN Independence
    (pp. 163-190)

    The concept of independence is closely related to that of domination, and is, in its own right, one of the central ideas in graph theory. We call two vertices in a graphindependentif they are not adjacent, that is, if there is no edge joining them. This quite naturally gives rise to the following definitions.

    Definition 10.1 A setSof vertices in a graphGis said to be anindependent setif no two vertices inSare adjacent. Theindependence numberof a graphGis, then, the maximum size of an independent set in the...

  14. CHAPTER ELEVEN Other Surfaces, Other Variations
    (pp. 191-212)

    In this chapter, we will look to see what happens to the independence number for each of the chess pieces on a variety of other surfaces, such as the torus and the Klein bottle, and we will also come to see that, in addition to the two numbers on which we have been focusing our attention—the domination number and the independence number—there are several other interesting numbers related to chessboard problems that are natural to investigate as well. Let’s start by looking at the independence problem for each of the chess pieces on the 2 × 2 ×...

  15. CHAPTER TWELVE Eulerian Squares
    (pp. 213-222)

    W. W. Rouse Ball concluded his essay on chessboard recreations in [2] with a brief discussion of Euler’s contributions to the study of Latin squares. ALatin squareis ann×narray of the integers 0, 1, 2, …,n−1—or, equivalently, if you prefer, a labeling of the squares of ann×nchessboard with these integers—such that each integer appears once and only once in each row, and once and only once in each column. There is nothing special about using integers in Latin squares, except that on occasion their arithmetical properties may be...

  16. CHAPTER THIRTEEN Polyominoes
    (pp. 223-246)

    There is a long history of geometric dissection problems in recreational mathematics and these problems often involve a chessboard in one way or another. Here are two problems that nicely illustrate the genre taken from a marvelous collection of puzzles,Mathematical Puzzles of Sam Loyd[12], America’s foremost puzzle expert of the late nineteenth century. I urge you to look at the originals in the book. All of the charm of the original drawings and of Loyd’s prose has been lost in the following versions, although the mathematics remains.

    Problem 13.1 In this problem, a square tract of land containing...

  17. References
    (pp. 247-250)
  18. Index
    (pp. 251-257)