The game begins with empty $n\times n$ chessboard and a fixed number $m\in\{1,2,\dots,n\}$.
Two players are making moves alternately, each move is placing a coin on one empty square, each row and column can contain at most $m$ coins, the guy who cannot put a coin when he is to play, loses.
Who has the winning strategy?
In the original problem there was $n=2011$ and $m=1005$.
My solution:
The first guy wins. First move: a coin in the centre, then symetrical reflections of opponent's moves.
After solving the problem, I generalised it.
My above solution works for all $n,m$ both odd.
If $n$ is even, then the second guy wins by symetrical reflections.
What about remaining cases?