What is 3D Tic-Tac-Toe 4×4×4?
3D Tic-Tac-Toe 4×4×4 is played in a cube made of four 4×4 layers stacked on top of each other, 64 squares in all. X moves first, and the players take turns marking any empty square on any layer; there is no gravity, so an upper square can be taken before the one below it. The first player to complete four of their marks in a straight line wins. A full cube without such a line is a draw.
A line can lie flat in one layer, as a row, a column or a diagonal of that layer, or it can pass through all four layers: straight up a pillar, diagonally up a side of the cube or a slice through it, or along one of the four great diagonals from a corner of the bottom layer to the opposite corner of the top layer. That makes 76 lines in all, and spotting the ones that cross the layers is most of the skill.
The squares are not equal. The 8 corners of the cube and the 8 squares at its centre, the middle four of each of the two inner layers, lie on 7 lines each, while every other square lies on only 4. Those 16 squares are the strongest, and both players usually fight for them first. The way to win is a double threat: two lines that each need one more mark, so that your opponent can block only one of them.
Rule changes
| Rule | Standard Tic-Tac-Toe | 3D Tic-Tac-Toe 4×4×4 |
|---|---|---|
| Board | Standard Tic-Tac-ToeOne flat 3×3 board of 9 squares. The table setup also offers 4×4 and 5×5 boards. | 3D Tic-Tac-Toe 4×4×4A cube of four stacked 4×4 layers, 64 squares. You may mark any empty square on any layer. |
| How to win | Standard Tic-Tac-ToeComplete a line of three of your marks, across, down or diagonally. | 3D Tic-Tac-Toe 4×4×4Complete a straight line of four of your marks, within one layer or through all four layers. |
| Winning lines | Standard Tic-Tac-Toe8 lines: three rows, three columns and two diagonals. | 3D Tic-Tac-Toe 4×4×476 lines: 48 rows, columns and pillars, 24 diagonals of the flat and upright 4×4 slices of the cube, and 4 diagonals from corner to opposite corner through the whole cube. |
| Draws | Standard Tic-Tac-ToeA full board without a line is a draw. | 3D Tic-Tac-Toe 4×4×4The same: a full cube without a line of four is a draw. |
| Perfect play | Standard Tic-Tac-ToeA draw. | 3D Tic-Tac-Toe 4×4×4A win for the first player, proved with the help of computer searches. |
| Board and options | Standard Tic-Tac-ToeA 3×3 board. The table setup also offers 4×4 and 5×5 boards and the Misère, Wild and gravity rules. | 3D Tic-Tac-Toe 4×4×4Always the 4×4×4 cube. Board size, Misère, Wild and gravity are not available with it. |
History
Tic-tac-toe extends naturally to more dimensions, and the cube of side four is the size most often played. The smaller 3×3×3 cube makes a poor game: the first player wins easily by taking the centre, and a full 3×3×3 cube without a line is impossible, so it can never end in a draw. The 4×4×4 cube has neither problem, and sets of four clear stacked boards have been sold as a game since the 1960s.
With best play the 4×4×4 game is not a draw. In 1980 Oren Patashnik published a computer-assisted proof that the first player can always force a win. Victor Allis and a co-author confirmed the result in 1992 with proof-number search, a method Allis developed for solving games.