This problem is about that multiple players place rooks on a chess board.
We consider the most natural situation. There are two players and each of the two players place $k$ non-attacking rooks on a $n\times n$ chess board. Rooks don't attack rooks of the same player. In other words, rooks in the same row or column attack each other only when the rooks belong to different players.
$a(n)$ is the maximal number of rooks that each player can place on a $n \times n$ board such that rooks are non-attacking. Each player places the same number of rooks. (It's a fair game!)
More generally, suppose there are $m$ players. $A(n,m)$ is the maximal number of rooks that each player can place on a $n \times n$ board such that rooks are non-attacking. Each player places the same number of rooks.
Clearly, $A(n,m)\le \frac{n^2}{m}$. $A(n,1)=n^2$, $A(n,2)=a(n)$ and $A(n,n)=1$.
Find $A(n,m)$.