It is also possible to achieve the 512 tile and the 1024 tile, but somewhat ironically, not the 2048 tile. The stated goal of the 3x3 variant app is to achieve the 256 tile. In contrast, the 3x3 variant, with its far more restrictive board, requires just minutes to achieve relatively high scores (Figure 2).įigure 1: Highest score and board position achieved by author in the 4x4 original 2048 version (without computer assistance)įigure 2: Highest score and board position achieved by author in the 3x3 2048 variant app (with computer assistance)
While scoring this high in the 4x4 version is fun for the player, this does require thousands of moves and several hours to achieve, and so can become somewhat tedious. The author was motivated to seek out the 3x3 variant of 2048 (, ) after achieving some very high scores in the original 4x4 version (Figure 1). The main goal of the original 2048 game was to achieve the 2048 tile, but shortly after the game creation, the game was modified to allow the player to continue to strive for higher tiles and higher scores. No points are added to the player's score for randomly appearing tiles. After all tile merges are completed, all remaining tiles on the board slide as far as possible in the direction of the player move.Įach time a pair of tiles combines, it adds the total of those tiles (the value of the new tile) to the player's score. In the case of four identical adjacent tiles (impossible in the 3x3 variant), both pairs of tiles combine into higher tiles. In the case where three identical tiles are adjacent, the two tiles closest to the edge of the board towards the direction of movement are combined, with the third tile remaining uncombined. When there are adjacent identical tiles in the direction of movement, they merge into a tile with the sum of the two tiles. If none of the four directions are possible (no empty board spaces, and no adjacent identical tiles), the game ends. On each turn, the player may move in any of four directions (left, up, right, or down), as long as at least one tile may slide in that direction (having an available blank space toward the direction of movement), or as long as two identical tiles are adjacent in the direction of movement. Each randomly appearing tile is a 2 with 90% probability or a 4 with 10% probability, and appears in any blank space with equal probability. At the start of the game, two tiles randomly appear within an empty grid, and after each player move, one tile randomly appears. The rules of the original game are simple:
The original 4x4 version of 2048 is a single-player game created in March 2014 by Gabriele Cirulli. For each of these four objectives, the expected value (win probability or expected additional score) has been calculated for each of the four allowable moves from any achievable board position. The 3x3 variant of 2048 has been strongly solved for four objectives: Achieving a 256 tile, achieving a 512 tile, achieving a 1024 tile, and maximizing expected game score. Solving the 3x3 Variant of 2048 SOLVING THE 3x3 VARIANT OF 2048