How many ways can you mark 8 squares of an $$8 \times 8$$ chessboard so that no two marked squares are in the same row or column, and none of the four corner squares is marked? (Rotations and reflections are considered different.)

Marking 8 squares with no two in the same row or column means exactly one marked square in every row and every column. The first row must avoid its two corner squares, so it has 6 choices (columns 2 to 7). The eighth row must avoid columns 1 and 8 and the column already used by the first row, leaving 5 choices. The six middle rows contain no corners and fill the remaining 6 columns in $$6! = 720$$ ways. Hence the total is $$6 \times 5 \times 720 = 21600$$.