Question 122
Let $n \ge 3$ be an odd positive integer. We are given an $n \times n$ board which is divided into white unit squares by lines drawn parallel to the sides of the board. Let $N$ denote the number of ways of placing 2 identical red checkers and 2 identical blue checkers on the board such that
(a) in any such placement, no two checkers occupy the same square; and
(b) none of the placements can be obtained from any other by a rotation of the board.
Prove that
$$
N = \left(\frac{n^2-1}{2}\right)^4 - \left(\frac{n^2-1}{2}\right)^3
$$
(a) in any such placement, no two checkers occupy the same square; and
(b) none of the placements can be obtained from any other by a rotation of the board.
Prove that
$$
N = \left(\frac{n^2-1}{2}\right)^4 - \left(\frac{n^2-1}{2}\right)^3
$$