I took great pleasure in reading Across the Board: The Mathematics of Chessboard Problems
by John J. Watkins. This book is extremely well writen 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: "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?" The problems are treated in depth from their beginnings through to their status today. Using graph theory, the author gently guides the reader to the forefront of current research in this area of mathematics. Exercises are provided to enhance the reader's involvement.
The book is organized in thirteen chapters. The first chapter introduces the main topics with which we will be concerned: knight's tours, domination, independence, coloring, geometric problems, chessboards on other surfaces, and polyominoes. The next two chapters deal with knight's tours, from the earlier workof De Moivre, Euler, Hamilton up to most recent results known today. Chapter four is devoted to magic squares. The beginning of the chapter includes a delightful presentation of the work of Muhammad Ibn Muhammad, an African Mathematician who discovered a very ingenious idea for constructing magic squares of odd order. This construction was later on rediscovered by Bachetin the early 1600s.
Chapters five and six generalize in different ways from the ordinary chessboard. If we identify the left and the right edges, the chessboard becomes a flat torus and the chess pieces gain considerable freedom of movement since the edges, in effect, disappear. If we identify just one pair of opposite edges, we have a cylindrical board. Finally, we can identify the top and bottom edges of the rectangle just as it was done for the torus, and also identify the left and the right edges, but this time with opposite orientation, so the top and bottom are identified inthe normal way, but the sides get a half-twist before they are identified. A rook in the left-most column moving up and going off the top of the board reappears at the bottom of the board in the same column; but a rook in the bottom row going off the right side of theboard will reappear at the upper left in the top row. This makes the board into a the Klein bottle.
In chapters 7, 8 and 9 the central concept is domination. This is one of the central ideas in graph theory, and is especially important in the application of graph theory to the real world. Of all the chessboard-domination problems, it is that of the queen that continues to hold the most interest among mathematicians. It is a remarkably difficult problem and one that is far from solved even today, although there is much that is known. One nice result from chapter nine is that among chessboards with more than four rows, the 5x12 chessboard is the largest board that can be dominated by four queens. I have a strong feeling that Across the Board will reveal the beauty of mathematics to students, teachers and math lovers.
Mohammed Aassila is a mathematics professor whose research area is analysis. He is interested in mathematics competitions and is the author of two books on the subject: 300 Défis Mathématiques and Olympiades Internationales de Mathématiques.