I think everyone knows the game of Cram (see my post from yesterday) I tried to find for what boards the first player has a strategy when each piece is made out of 3 pieces, however I was unable to find a strategy that cover that many cases.
If $m$ and $n$ are both odd then the first player can place a "line" piece that has it's center piece in the center square of the board. After that the first player can just use the "symmetry" strategy.
If $m$ and $n$ are both even, the second player can mirror the first one. – user3257842
I've also studied the case with $m=1$. I didn't manage to find some sort of periodicity. I tried using the Guy-Smith Periodicity Theorem for Octal Games.
Any ideas or thoughts on this?