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What is the maximum number of bishops that can be placed on a chessboard such that at most three bishops lie on any diagonal?
The answer is 38.
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1 solution
If the chessboard is colored black and white as usual, then any diagonal is a solid color, so we may consider bishops on black and white squares separately. In one direction, the lengths of the black diagonals are 2 , 4 , 6 , 8 , 6 , 4 and 2 . . Each of these can have at most three bishops, except the first and last which can have at most two, giving a total of at most 2 + 3 + 3 + 3 + 3 + 3 + 2 = 1 9 bishops on black squares. Likewise there can be at most 1 9 bishops on white squares for a total of at most 3 8 bishops.