The maximum number of bishops that can be placed on an N by N chessboard such that no two bishops threaten each other is N. This is because bishops can only attack pieces that are on squares of the same color.

In order to place N bishops on an N by N chessboard without them threatening each other, they must be placed on alternating colors. For example, if the board is 8 by 8, then bishops can be placed on squares of one color and the other bishops can be placed on squares of the other color. In this case, you would be able to place 4 bishops of one color and 4 bishops of the other color, for a total of 8 bishops.

Therefore, the reasoning behind this arrangement is that by placing bishops on alternating colors, they are unable to attack each other.
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