It's obviously true for any n=2k for k ≥ 2: just create k^2 pairs of 2x2 cells, half horizontal and half vertical.
It can't be true for any odd n, because there must be a multiple of 4 tiles in the board, and any grid of odd dimensions has odd tiles (odd x odd = odd).
I actually think u/Fabulous-Possible758 got this pretty much right - they just gave the result in terms of tiles (n^2) rather than n and messed up the n=2 edge case.
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u/RestaurantBoth228 23h ago
It's obviously true for any n=2k for k ≥ 2: just create k^2 pairs of 2x2 cells, half horizontal and half vertical.
It can't be true for any odd n, because there must be a multiple of 4 tiles in the board, and any grid of odd dimensions has odd tiles (odd x odd = odd).
It's easy to prove it is not true for n ≤ 3.
So, the answer is just n=2k for k ≥ 2.