April 25, 2013
Mahina Kaholokula
Math 7: Mathematical Puzzles
Peter Winkler
Chess Mobility
I have always had a soft spot for an elegant puzzle. The challenge of coming up with the perfect strategy and the subsequent discovery of an economic (meaning every step has a purpose) and rational solution can be deeply satisfying. Many of the most elegant puzzles that I have come across are in the form of chess problems.
These puzzles require only a chessboard and the chess pieces on it, and the challenge is usually to solve some task using these pieces. One of the things I especially appreciate about these puzzles is that one doesn’t have to be a master chess player to complete them. Although a grandmaster may have an advantage in perspective and experience with seeing connections and interactions between the pieces, usually a general knowledge of how the pieces can move is all the problemsolver needs.
Some of the most famous chess puzzles include the Knight’s Tour (finding the sequence of moves that a knight can take to visit every square exactly one time) and the Eight Queens Puzzle (figuring out where to place eight queens on a chess board so that no two queens can attack each other). Other tasks may include strategizing how to mate in the shortest number of moves given a set-up board, or backtracking

from an endgame to what moves were made and where each piece started. The problem that stood out to me in Gardner’s book is another common one, involving the placing of chess pieces to attack the smallest number of squares and then a placement attacking the largest possible number of squares. Here is the problem as
Gardner states it (Problem 12.6 on pg. 341):
“Many beautiful chess problems do not involve positions of competitive play; they use the pieces and board only for posing a challenging mathematical task.
Here are two classic task problems that surely belong together:
1.
The minimum-attack problem: Place the eight pieces of one color (king, queen, two bishops, two knights, two rooks) on the board so that the smallest possible number of squares are under attack. A piece does not attack the square on which it rests, but of course it may attack squares occupied by other pieces. In Figure [33], 22 squares (gray) are under attack, but this number can be reduced considerably. It is not necessary that the two bishops be placed on opposite colors.

2.
The maximum-attack problem: Place the same eight pieces on the board so that the largest possible number of squares are under attack. Again, a piece does not attack its own square, but it may attack other occupied squares. The two bishops need not be on opposite colors. In Figure [34], 55 squares (gray) are under attack. This is far from the maximum.”
I must admit that part of the reason I enjoy this puzzle is because of its context. I am naturally fond of chess, so the puzzle appealed to that side of me. But more importantly, and more generally, I like the puzzle because it isn’t quite like the majority of riddle-type problems that Gardner introduces in the beginning of his book. In class, we determined that a “good” puzzle has ten requirements, among which are that it includes a paradoxical statement, that it seems to easy to solve, that it has a memorable statement, and that it draws on common experience. But this problem, like most chess puzzles, tends to hold few of these qualities. There is no trick, no plan to fool or lead anyone astray; the puzzle is straightforward and success is based purely upon will and focus and determination. The emphasis on strategy alone makes the problem more respectable and elegant in my opinion.
The solution, as well, is not paradoxical. It is not a surprise, nor does it leave any room for debate. It simply is the truth, easily tested by any party willing to map out the moves of each piece. Here is how the solution goes:
The minimum number of squares that can be attacked is 16, and the setup is as follows:

This can easily be worked out with some logic and strategy. The key to finding this setup is in the realization that one must trap the far-reaching pieces and thus cut off the range of their movement. The queen, two bishops, and two rooks are generally considered to be the far-reaching pieces as they can move along any continuous, unblocked straight line along the board. Thus, these pieces must be given no room for movement, which is most easily achieved by blocking them into a corner with the remaining pieces. Where each specific piece is placed can be found by a simple guess and check method, but it is also important to notice that some “outside” pieces
(king and two knights) will have to block two long-range pieces, and will therefore have to be placed in a square that is the intersection of the paths of two long-range pieces.
The maximum number of squares that can be covered by the 8 chess pieces is actually the entire board, unless the bishops are required to be on separate colors, in which case, all but one square of the board can be covered:

Finding these two set ups can be a little more difficult. I was unable to personally solve the two problems on my own, yet I can appreciate the solution and the range that each piece has because it teaches me how, when used in concert with each other, the pieces of the chess board can be used to their maximum potential to cover every square of the board. The main strategy used in finding these solutions is in the key fact that it is useful to try to overlap attack squares as little as possible. This is especially important in setting up the long-range pieces first, and then placing the short-range pieces in the “clean” (unattacked) spaces.
The particular case of the maximum attack problem with bishops on different colors is interesting because of that one left-out square. As far as anyone has ever found, this is the maximum number of squares that can be covered given this restriction. It would be interesting if there existed some sort of proof to show that this leftover square is inevitable under these conditions. Gardner also points out that the unattacked square may be at any spot on the board, which gives rise to a whole new collection of problems in which the goal is to create a specific pattern with the unattacked square (for example, to have it in the middle of the board).

This problem has multiple solutions, so even after one solution is found, the puzzler may continue on with the puzzle and find another, or solve a variant of the puzzle, like the unattacked pattern described above. Overall, the puzzle is a challenging one, but a rewarding one. When a solution is found, it is obvious, so the puzzle is one you can play with yourself over and over without relying on a third party source to check your work. From personal experience, it is just as satisfying every time. The economy of the problem (how each piece has a completely unique and necessary function without which the puzzle is impossible) only adds to the elegance of the solution. The popularity of this puzzle emphasizes its attraction, and the fact that there are no proofs among these puzzles adds to the excitement of the puzzle. There is always the thought that perhaps there is a positioning that would allow for that last square to be filled, or to minimize the number of attacked squares to below 16, and that is what makes the puzzle so interesting. Maybe we’re all just missing one key positioning or relationship between the pieces, and that is what we must all strive to find.