I think your (plural) problem is that you're talking about chess as opposed to a chess board.
A chess board is 2D. But, you could technically say it's 4D (3 dimensions + time). You could also technically say it's 1D (a string) if you play by just writing the chess notation.
Just a thought experiment. I was implying that chess is a game, not a board. That game can be encoded/coded without any need for a Z variable, can't it? My Rook doesn't go up and down an ladder, it doesn't need a 3D array to record its position. So that's 2D + Time, or 3D. YRMV.
