Chess is an extremely complex game, with intricate strategies and mathematical patterns embedded in the rules themselves. There are a wide variety of pieces that move across the board in different ways, the most mobile of which is the queen.
The queen can move up, down, left, right, or diagonally, as far as the edge of the board. For this reason, the queen can attack many different pieces in various locations. One may try to think of how many queens can be placed on the board with no queen attacking any other. Since a chessboard is an 8x8 grid, the maximum number of queens is 8. However, actually placing the 8 queens on the board with the aforementioned rule is an entirely different problem - the 8-queens problem.
The 8-Queens Problem, where 8 queens are placed on a chessboard so that none can attack the other, has 96 solutions. However, if rotating or reflecting the board is considered the same solution, that narrows it down to 12. Here they are:
The queen can move up, down, left, right, or diagonally, as far as the edge of the board. For this reason, the queen can attack many different pieces in various locations. One may try to think of how many queens can be placed on the board with no queen attacking any other. Since a chessboard is an 8x8 grid, the maximum number of queens is 8. However, actually placing the 8 queens on the board with the aforementioned rule is an entirely different problem - the 8-queens problem.
The 8-Queens Problem, where 8 queens are placed on a chessboard so that none can attack the other, has 96 solutions. However, if rotating or reflecting the board is considered the same solution, that narrows it down to 12. Here they are:
However, there is a certain asymmetry to the way that a queen moves - the distance that it can travel along diagonals depends on its position. For example, a queen near the corner of the board cannot move far diagonally (except toward the center), while a queen in the middle of the board can move far in all directions. Ideally, the queen should be able to go out of the left side of the board and reappear on the right, and the same should be true for the top and bottom. If the left and right sides of the board are one and the same, they can be glued together, so that the flat board turns into a cylinder. If the top and bottom of the cylinder are glued in the same way, it becomes a donut shape, or torus. Because of this, this new version of the 8-Queens Problem, where queens can "wrap around" from one side of the board to another, is called the Toroidal 8-Queens Problem.
The Toroidal 8-Queens Problem has no solutions on any board whose side is a multiple of 3 or 2, meaning that it cannot be solved on a traditional 8x8 chessboard. 5 is the smallest number that is not a multiple of 3 or 2 (discounting 1), and on that board, there are 10 different solutions to the Toroidal 8-Queens Problem. Discounting rotation and reflection, there are only 2 solutions, shown below.
The Toroidal 8-Queens Problem has no solutions on any board whose side is a multiple of 3 or 2, meaning that it cannot be solved on a traditional 8x8 chessboard. 5 is the smallest number that is not a multiple of 3 or 2 (discounting 1), and on that board, there are 10 different solutions to the Toroidal 8-Queens Problem. Discounting rotation and reflection, there are only 2 solutions, shown below.
However, even then, if one takes the first solution, moves every queen 2 squares to the left (making sure that queens that go off the edge wrap around to the other side), and then reflects the whole thing, the second solution is formed.
There are also solutions for the 7-by-7 board, the 11-by-11, 13-by-13, etc.
For more information, see Wolfram Mathworld and Wikipedia.
There are also solutions for the 7-by-7 board, the 11-by-11, 13-by-13, etc.
For more information, see Wolfram Mathworld and Wikipedia.