$4^4$ Tic-Tac-Toe is a first player win I have seen the claim that in $4^4$ Tic-Tac-Toe there is an obvious first player win, and more generally that it is easy to see that $n^n$ Tic-Tac-Toe is a win for the first player. But I can't seem to get a proof to work. It is indeed easy to see that $3^3$ Tic-Tac-Toe is a first player win. (After moving in the center, regardless of your opponents move you can find a $3\times3$ sub-board containing the center where your opponent has not moved and play in it. Then you have a sequence of forcing moves to win.) I suppose there is a higher dimensional generalisation of this idea. I hope it is obvious what rules I am thinking of, I guess the only thing that should be clarified is that in $n^d$ Tic-Tac-Toe we need $n$ in a row to win.

In the paper Qubic: $4 \times 4 \times 4$ Tic-Tac-Toe, Patashnik proves that even the $4^3$ game is a win for the first player with optimal play; therefore the $4^4$ game should also be a first-player win. The proof is a computer-aided solution, however, which might not be satisfying.

A table in this paper shows that the status of the $5^5$ game was still open at the time (in 1980). It's possible that the problem has been resolved by now, but it makes me skeptical that an "obvious win for the first player" exists; it definitely wasn't obvious to anyone at the time!