It has been known since the 1850's (or even much earlier) that 5 queens could be placed on an 88 chessboard so that every square on the board lies in the same row, column, or diagonal as at least one of the queens. It was also "known" that this could not be done with 4 queens. But I have not been able to obtain or track down any rigorous mathematical proof of this that could be (or could have been) carried out in a reasonable time by a human being with pencil and paper. There are altogether 635376 ways of placing 4 queens on an 88 chessboard. Does anyone know of a combinatorial algorithm, exploiting the symmetries of the chessboard, which would reduce the number of cases to be considered to the hundreds (or, at most, the low thousands.)? This is, of course, a trivial problem for modern computers, which have many times since verified that 5 queens are needed.