The goal of this game is to be the first player to get four in a row of their own cube colour!
The four in a row can be horizontal, vertical, or diagonal in any orientation.
Players take turns choosing an uncolored cube to make their own colour.
You'll play as Red, and go first!
The board can be rotated using the arrow keys, or by using the WASD keys.
If it is your turn, you can click on a cube to turn it your colour!
A combinatorial line is given by a root. A root is a coordinate where at least one of the dimensions is replaced by a free character, in this case the character *. To generate a line, the character * is replaced with the possible values of a coordinate; in this case, iterating through [1, 2, 3]. This creates a line of coordinates, which is also always what would be a valid Tic-Tac-Toe win. However, not every valid Tic-Tac-Toe win is a combinatorial line. The Hales-Jewett Theorem implies that for a high enough dimensional grid, a combinatorial line must exist. This then implies that on a Tic-Tac-Toe board of that dimension, a tie-game is not possible.
By matching a colouring of a grid to a colouring of the natural numbers, a monochromatic combinatorial line will provide a monochromatic arithmetic progression. Multiple colouring schemes can work, but summing the coordinates of each point in the grid and assigning corresponding colour is a simple method. Since the Hales-Jewett theorem guarantees that a high enough dimensional grid must contain a monochromatic combinatorial line, this shows that the Hales-Jewett theorem implies van der Waerden's theorem, guaranteeing us a monochromatic arithmetic progression.