If I am interpreting the problem correctly, it is equivalent to Chomp, a very well-known game. It is famous for having an ingenious proof (strategy-stealing) of the existence of a winning strategy for the first player, yet no general strategy has been constructed. Please let me know if I have misinterpreted the problem. In case my interpretation is correct, this belongs in the Combinatorics, as it deals with a mathematical game.

As written the problem is incomprehensible: what does "to break into a single square" mean? What are the restrictions on the moves? As written, the first player could simply break along all edges between all squares, which presumably satisfies the condition of "breaking into a single square". Hopefully this was not the official wording . (It seems unlikely to me that, once everything is clarified, this will have anything to do with Chomp.) Edit: okay, revised version now makes sense (and, indeed, has nothing to do with Chomp).

Sorry about the wording. I had to translate it from Spanish into English. I'll take a look into the original problem again and try to fix it. EDIT: Ok hopefully made it better.