The n by n case is easier than other cases because it is symmetric: [the first player picks the 2nd-from-the-left-and-2nd-from-the-bottom box, resulting in a (symmetric) L shape. Then the 1st player can win no matter what the 2nd player do. (how?) He just imitates what the 2nd player pick on the other side of the L shape.]

Unfortunately, this only works for n by n. If we have an n by n+1 L shape, then the 2nd player can use his turn to make it a symmetric L shape and he wins. [This does give insights in finding strategies to win in the asymmetric cases nonetheless]

This method works for special cases of other games:
Nim (play against computer here)
Start with several stacks of stones. 1st player choose a stack and remove stones from that stack. Alternate until everything is removed and the last to remove wins.

Activity 2
In which cases does symmetry help?

[One of the cases is 2 stacks (1st player wins if and only if the stacks are uneven). A good tactic is to get pairs of stacks of even size.]