Across the Board: The Mathematics of Chessboard Problems [NOOK Book]

Across the Board

Princeton University Press

Chapter One

INTRODUCTION

We are no other than a moving row
Of Magic Shadow-shapes that come and go
Round with the Sun-illumined Lantern held
In Midnight by the Master of the Show;
But helpless Pieces of the Game He plays
Upon this Chequer-board of Nights and Days; Hither and thither moves, and checks, and slays,
And one by one back in the Closet lays.
        -Omar Khayyám, The Rubáiyát

This book is about the chessboard. No, not about chess, 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 Scrabble; 18 × 18 boards for Go; and even non-square sizes such as 4 × 8 and 2 × 6 boards for Bau and Owari, two games that are widely played in various forms and under several different names in Africa.

In some board games there are no special colors given to individual squares of the board, all squares are the same, but in other games the color of individual squares can be very important. The familiar checkered black and white or black and red alternating coloring of the squares in chess or checkers are the best examples of this. Alfred Butts spent years during the 1930s and 1940s tinkering with the coloring of the board for the game that eventually became Scrabble, deliberating on exactly where the double and triple valued squares should go, what their colors should be, and how many of each type he wanted. We, too, will discover that ingenious colorings of the squares on an otherwise ordinary chessboard can pay surprising dividends.

Board games, like other of our games, are usually in some form a metaphor for life itself. Chess, of course, is about war, conquering your enemy and protecting your king; Go is about the territorial imperative; Snakes and Ladders a morality play for children about how to achieve Nirvana. Omar Khayyám, in the two quatrains I quoted for you from The Rubáiyát, saw in such games a reflection of our lives as mere 'pawns' in a game run by the 'Master of the Show', a theme picked up by Shakespeare 300 years later in As You Like It, the play with which he possibly opened the Globe Theatre in 1599:

       All the world's a stage,
And all the men and women merely players;
They have their exits and their entrances,
And one man in his time plays many parts,
His acts being seven ages.

Even now, in an age, and especially in a society, such as ours, in which the problematic notion of free will is simply taken for granted, this is still an idea that carries a chilling weight.

However, this is not a book about such games, and still less is this book concerned with the cultural significance of the games we humans play. Instead, this book is about the game board itself, the simple grid of squares that forms such a common feature of games played around the world, and, more importantly, about the mathematics that arises from such an apparently simple structure. Let us begin with a puzzle.

Guarini's Problem

The earliest chessboard puzzle that I know of dates from 1512, almost 500 years ago. This puzzle is known as Guarini's Problem and involves four knights, two white and two black, at the four corners of a small 3 × 3 chessboard. The white knights and the black knights wish to exchange places. Their situation is shown in Figure 1.1. A knight can move on a chessboard by going two squares in any horizontal or vertical direction, and then turning either left or right one more square. Since, in this problem, each knight will have only two moves available from any position, this is a very simple puzzle to solve, even by trial and error. Still, it is somewhat harder than it might look at first glance, so I urge you to try to do it for yourself before reading on.

If you solved this problem you undoubtedly observed its underlying basic structure. Note that this rather simple structure comes from two things: the geometry of the board itself along with the particular way in which knights are allowed to move. In Figure 1.2, the structure of the possible knight moves is first exhibited explicitly, on the left, by lines that are drawn to connect any two squares of the board between which a knight can move. This diagram, or graph, can then be 'unfolded', as shown on the right in Figure 1.2, and the underlying structure of the graph immediately emerges. The graph, which looked somewhat complicated to us on the left, turns out to consist of only a single cycle, and so the solution to our puzzle is now completely clear. In order to exchange places, the knights have no choice at all. They must march around this cycle, all in the same direction, either clockwise or counterclockwise, until their positions are exactly reversed. This graphical solution, of course, still has to be translated back to the original 3 × 3 chessboard, but this final step is quite straightforward.

I like Guarini's Problem in part because it is so very old, but also because it is such a nice illustration of the way in which mathematical abstraction can clear away the messy details of a problem and lead us gently toward a solution. In the case of Guarini's Problem, the first level of abstraction avoids for us the need to keep worrying about the awkward business of how a knight moves on a chessboard by the simple device of drawing lines on the board to represent these moves. This allowed us to forget about chess completely. The next level of abstraction is to forget also about the actual board and focus instead entirely on the diagram. Then, the final level of abstraction is to eliminate the clutter inherent in the diagram simply by unfolding it. This general process of turning a problem into a diagram is so useful and so natural that an entire area of mathematics, now called graph theory, has evolved that is dedicated to studying the properties and uses of such diagrams, called graphs. This book, then, is really a book about graphs in disguise. Usually, explicit graphs such as the one drawn in Figure 1.2 will be kept offstage during our drama, but rest assured, they are always there and ready to appear at a moment's notice should we ever have need of them.

Problem 1.1 In Figure 1.3 we now have six knights, three white and three black, on opposite sides of a 4 × 3 chessboard. Find the minimum number of moves required for these knights to exchange places. This variation of Guarini's Problem appeared in Scientific American in the December 1979 issue. A solution was given the following month. Remember though that you can, if you like, find solutions to these problems at the end of each chapter.

The Knight's Tour Problem

Since we have just studied a problem involving knights, and we now know how they move, let's continue with them for the moment. Note that in Guarini's Problem the center square of the 3 × 3 chessboard is inaccessible to any of the four knights, but that otherwise a single knight could visit each square of the board exactly once and return to its original position merely by making a complete circuit of the cycle graph in Figure 1.2. What about on a larger board, such as a 4 ×4 board or an 8 ×8 board, might a knight be able to do a tour of the entire board, that is, visit every square exactly once and return to the start? The answer is: well, sometimes yes, and sometimes no.

The general question of which chessboards have a knight's tour is known as the Knight's Tour Problem. This famous problem has a long and rich history. The Knight's Tour Problem dates back almost to the very beginnings of chess in the sixth century in India and will in fact be one of the main topics considered in this book.

You might wish to try the following problem before continuing further. The key feature of a knight's tour is that a knight visits each square of the board exactly once. A tour can be closed, meaning the knight returns to its original position, or it can be open, meaning it finishes on a different square than it started. Unless otherwise indicated, the word 'tour' in this book will always mean a closed tour.

Problem 1.2 The smallest chessboards for which knight's tours are possible are the 5 × 6 board and the 3 × 10 board. Find a tour for each of these boards. The smallest board for which an open tour is possible is the 3 × 4 board. Find an open tour for this board.

Leonhard Euler, perhaps the most prolific mathematician of all time, did extensive work on the Knight's Tour Problem. One particularly attractive knight's tour for the 8 × 8 chessboard, done by Euler in 1759, is shown in Figure 1.5. What is especially interesting about this particular tour is that Euler first does an open tour of the lower half of the board, starting at square 1 and ending at square 32. He then repeats exactly this same tour, in a symmetric fashion, for the upper half of the board, starting at square 33 and ending at square 64. Note that Euler has also very carefully positioned both the beginning and the end of these two open half-tours so that they can be joined together into a tour of the entire chessboard.

This tour of Euler's tells us something very useful about the Knight's Tour Problem, at least we now know for sure that the 8 × 8 chessboard has a knight's tour. On the other hand, we already know that not all chessboards have knight's tours. For example, as we have seen, the 3 × 3 board can't have a tour for the simple reason that the center square is inaccessible. A more interesting, and much more surprising, example is the 4 × 4 board. A quick glance at the graph in Figure 1.6 shows why a knight's tour is impossible for this chessboard. The two bold lines, or edges as they are usually called in graph theory, coming from the upper left-hand corner represent the only possible way for a knight to get into or away from that particular corner square. Thus, both of these bold edges must be a part of any knight's tour of the entire board. For exactly the same reason, the two bold edges coming from the lower right-hand corner must also be part of any tour of the entire board. But, as we can easily see, these four bold edges form a small closed cycle visiting just four squares, which means that they cannot simultaneously also be part of a larger tour of the entire board. So, because of this unavoidable conflict, a tour of the entire 4 × 4 board is impossible.

Are there any other chessboards on which a knight's tour is impossible? In order to answer this question, it is going to turn out to be useful for us to color the squares of our chessboards, as we shall see.

Coloring Chessboards

Omar Khayyám used the black and white 'chequer'-board pattern of the chessboard to represent nights and days in the passage I quoted earlier from The Rubáiyát, thereby invoking a disturbing sense of the endless cyclic repetitiveness of life. These days, we see the checkerboard pattern used so often for decorative purposes-on taxicabs, on table cloths in Italian restaurants, for tiling floors, for the checkered flags at auto races-that it will seem quite amazing to us how wonderfully useful this pattern also turns out to be for purely mathematical purposes. This is a theme to which we will return again and again. But, for now, let's use coloring to see how to find infinitely many chessboards that can't possibly have knight's tours!

Up to this point, of course, we have not bothered to color the squares on any of our chessboards. But now, look at the 5 × 5 chessboard in Figure 1.7, where we have colored the squares in the usual way. The main thing to note is that a knight on this board always moves from a square of one color to a square of the other color. You should verify this for yourself. Therefore, a knight's tour necessarily alternates between black squares and white squares; and so, a knight's tour must consist of an equal number of black squares and white squares. But, the 5 × 5 chessboard doesn't have an equal number of black squares and white squares! In fact, our 5 × 5 chessboard has 12 white squares and 13 black squares. And so a tour of the 5 × 5 chessboard is impossible.

The same argument, of course, works for any 'odd' chess-board, such as a 13 × 17 board, where both of the dimensions are odd, simply because in that case the total number of squares would also be odd, and so the number of black squares could not be equal to the number of white squares. Thus, there are infinitely many chessboards for which a knight's tour is impossible.

Playing on Other Surfaces

It may have occurred to you, however, that just because there isn't a closed knight's tour for a 5 × 5 chessboard doesn't mean that there can't be an open knight's tour. After all, couldn't an open tour begin on a black square and then also end on a black square and in that way successfully alternate 13 black squares with 12 white squares? This is indeed possible! One such tour is shown in Figure 1.7.

This particular open tour suggests a natural extension of the Knight's Tour Problem, or at least it suggests something that might seem natural to anyone who has spent a significant amount of time playing video games. What if we change the rules for the knight and now allow the knight to move o3 the bottom of the board and then reappear at the top, exactly as happens in some video games? Or if we allow the knight to go o3 of one side of the board and instantly reappear at the other side? This is not an idea that would have seemed particularly natural to Euler in the middle of the eighteenth century when he was playing with knight's tours, but it certainly seems natural to us now. With this new-found, computer-aided freedom, the poor knight who was previously stuck at square 25 in Figure 1.7 can now happily return to square 1 in a single move, thus closing the tour.

In this particular situation, a mathematician might say we were playing chess on a flat torus. As we shall see later, this is actually the same as playing chess on the surface of a doughnut! Since I intend for us to be investigating knight's tours on a wide variety of strange surfaces like the torus, you should perhaps warm up a bit first on the following problem.

Problem 1.3 Find a knight's tour on the surface of a 2 × 2 × 2 cube. This needs some explaining.

Continues...

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Excerpted from Across the Board by John J. Watkins Excerpted by permission.
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