Fair and Square

Andy and Brian are playing a game on a square grid. They take turns placing their initial ('A' or 'B') into an empty cell on the grid, and win if they are the first to claim all four corners of any square larger than the trivial $1$ x $1$. Note that this square must be axis aligned (for instance, $(0, 1), (1, 0), (1, 2), (2, 1)$ does not form a winning square).

This game can be played either on a $36$ by $36$ board, or an infinitely large board. For both versions, either find a winning strategy for one player, or prove that the game will end in a draw after optimal play.

Up to $7$ marks will be awarded for proving the correct result of one version, and up to $10$ marks will be awarded for proving the correct result in both versions.

For all the tasks, including this one, you can submit as many times as you like with no penalty! Only your best submission will be counted.

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