Question 8: What is the answer if R = C = N and all the squares along…

You have been given a R × C chessboard, with a list of squares cut out. This problem will involve placing rooks on a chessboard, so that they cannot attack each other. For a rook to attack a target piece, it must share the same row or column as the target.
Find the maximum number of rooks that can be placed on the chessboard, such that no pair of rooks can attack each other.
Rooks cannot be placed on cut out squares. The cut out squares do not affect where the rooks can attack.
Based on the above data, answer the given subquestions.

What is the answer if $R = C = N$ and all the squares along a diagonal are cut out? In particular, assume the square at the intersection of the $i$-th row (from the top) and the $i$-th column (from the left) is a cut out square, for all $1 \leqslant i \leqslant N$. For this question, assume that $N = 2K$, i.e, that $N$ is even.