I have played three dimensional tic-tac-toe with three players before, and we had no problem visualizing it. We drew three layers on a sheet of paper and just remembered all the different ways you could win. My question is, how do we go about playing four dimensional tic-tac-toe with four players? I don't know how to visualize it, but I'm pretty sure it's possible.
Could someone please show me what the board would look like and all the ways a player could win?
Perhaps this will help
Each vertical column represents one game of 3D-tic-tac-toe, so that we have 3 such games adjacent to each other. Each number represents a different win.
The win with $1$ maintains its 3D coordinates and traverses in the 4th. The $2$ maintains its 2D coordinates but traverses diagonally in its 3rd and 4th coordinates. The $3$ maintains its row but traverses in its column (2nd) coordinates along with its 3rd and 4th. The $4$ traverses diagonally in all coordinates and so must pass through the middle.
Basically, any win can be constructed by traversing with respect to at least one coordinate and maintaining in the others.