Unitile 2026

Unitile is played on a 5×5 board of cells. Every cell but one holds a tile. Each tile is red or blue. The board has twelve red tiles and twelve blue tiles. The one unoccupied cell is called the bubble. In the starting position, the bubble is the center cell. The 24 tiles fill the other cells in reading order, from left to right within each row, and row by row from the top. The colors of the tiles alternate in that order. The first tile, in the top left corner, is blue.

The players alternate sliding tiles, one slide per turn, starting with Red. The player to move chooses a tile of either color that is orthogonally adjacent to the bubble and slides it into the bubble. The cell that the tile has left becomes the new bubble.

A group is a maximal set of orthogonally interconnected, same-colored tiles. A color is united when all of its tiles form a single group.

The game ends as soon as a move unites a color. The player of that color wins. A move can unite the opponent’s color, and a player who makes such a move loses. No move unites both colors at once, because a move changes the cell of only one tile.

These rules alone do not make the game end. For example, after Red slides a tile into the bubble, Blue may slide the same tile back, and the position is exactly the same as before. One further rule decides which moves are legal.

The Unitile ruleLet P be a position. Red’s value r(P) is the average number of random slides, starting from P, until the red tiles are united. Blue’s value b(P) is the average number of random slides, starting from P, until the blue tiles are united. When the red tiles are united in P, r(P) = 0. When the blue tiles are united in P, b(P) = 0.

The player to move must make a move that either unites a color or lowers the total r(P) + b(P).

In every position in which neither color is united, at least one move lowers the total r(P) + b(P) by 2 or more. Thus, under the Unitile rule, no position is a deadlock, and every game ends in a win for one of the players.