In a $2$-dimensional checkerboard which is $2\times2$, the player going first necessarily achieves $2$ in a row. In a $3$-dimensional board which is $3\times3\times3$, a player going first using a good strategy can always achieve $3$ in a row.
Conjecture: In an $n$ dimensional board, with all sides $n$, in which the aim is to get $n$ in a row, the first player can always win if they play perfectly.
Any thoughts on whether this is likely to be true, and if there is any way to prove or disprove it?
Useful link: https://en.wikipedia.org/wiki/M,n,k-game