In how many ways can two queens be placed on a chessboard such that they are mutually attacking each other?
Show how to solve this problem quickly with only pencil and paper (not even a calculator).
Without considering rotations, interchange of colors, etc., there are 1456 ways:
For each of 64 squares, each attacks 7 others in the same row, and 7 others in the same column (64 * 14 = 896).
Considering only diagonals, depending on ring:
Each of the 28 squares forming the outermost perimeter attacks 7 other squares (28 * 7 = 196)
Each of the 20 squares one removed from the outer ring attacks 9 other squares (20 * 9 = 180)
Each of the 12 squares two removed from the outer ring atacks 11 other squares (12 * 11 = 132)
Each of the 4 central squares attacks 13 other squares (4 * 13 = 52)
Adding all these gives 1456. If you consider an exact interchange of white/black as the "same" way, this would divide by 2 (giving 728). If you considered the geometric relation of the two squares the "same" with rotations of the board, this would divide again by 4 giving 182. Other topological equivalences could be considered. Without a strict definition of a "way" any of these are possible answers.
Posted by badger
on 2007-10-11 14:53:40