What is the maximum number of rooks that can be placed on an $N \times N$ chessboard so that they do not attack each other?
A chessboard has $2 N-1$ diagonals that run from left to right upwards. On each of these, we can place at most one rook so that they do not attack each other. However, the two extreme diagonals each consist of only one square, and these lie on a diagonal running from left to right downwards. Consequently, at most $2 N-2$ rooks can be placed. We will show that this is indeed possible. Place $N-2$ rooks in the first row of the chessboard so that there are no pieces on the first and last squares, and place $N$ rooks in the last row. It is easy to verify that this placement indeed meets the requirements of the problem.

Note. We considered solutions to be incorrect if they only examined a few specific numerical examples, or did not justify their results at all.