In 3D Tic-Tac-Toe, there are 3 2D planes, each 3x3 in size, stacked on top of each other. A winning line can go through each of the layers (in addition to the standard winning lines of normal Tic-Tac-Toe). If two players are playing with the standard rules, how could they achieve a tie?
It is not possible.
There are only 3 essentially different ways have a filled a layer without three in a row:
X O X O X X O X O X X O X O O X O X O X O X O X X O XEach of these can of course be rotated/reflected, or have O and X interchanged. It is not possible to have 6 pieces of one player and 3 of the other without a winning line.
Count how many rows/column in these patterns alternate (i.e. are
OXO). All three filling patterns have exactly two such rows/columns on the outside, and only the third pattern has an additional alternating row/column in the middle.
If you fill the middle layer of the cube, the middle row of two of the faces on the side of the cube are alternating. This forces those two side faces of the cube to use the third filling pattern. Those two faces can be opposite or adjacent, but either way it leads to problems. In both cases it makes two edges of the top and bottom layers alternating, which means the other two edges in each of those layers are not alternating. The other two side faces of the cube then have at least 3 or 4 edges that are not alternating, and so cannot be filled with one of the patterns.
Therefore it is impossible to fill the cube without having 3 in a row.