Is there a dimension of tic-tac-toe where there can never be a draw game?

Question

Tic-tac-toe is a game traditionally played in a 2-dimensional 3 by 3 grid, where each cell of the grid can either be an "X", an "O", or it can be empty, and the number of cells empty is the number of moves played subtracted from 9 (so a game can never exceed 9 moves). A game where at least 5 moves has been played can have a winner, but it also can continue going, if in no row, column, or diagonal has 3 of the same symbol in it. 3 empty cells in a row, column, or diagonal does not count, as those have no symbol and thus no player attached to them. Incidentally, due to the alternating turns of the two players, the symmetry of the game board, and it's 3-by-3-ness, tie games are possible, where, after 9 turns, there are no rows, columns, or diagonals where there are 3 matching symbols in them. Is it possible, in higher dimensions, to construct a board with rows, columns, and their higher dimensional counterparts of length 3, such that the game always has a winner?

Answer 1

For the 3 by 3 board in 3 dimensions, the first player will always win with optimal play. Generalizations of $n$ by $n$ boards in $d$ dimensions is explored in this very nice video by PBS Infinite Series.