Breakthrough is a well-known abstract board game played regularly at the Computer Games Olympiad. Despite the straightforward gameplay mechanics of the game, determining optimal play strategies poses significant challenges. We have (weakly) solved a variant of the game played on the 6 × 6 board, representing the largest board instance of the game solved to date. The game is a win for the first player. We apply a hybrid approach of computing endgame tablebases, defining game-specific race patterns, using a family of parallelizable proof-number-based solvers, and training a neural-network-based heuristic job scheduler to construct a solution tree and proof of the game’s game-theoretic value. This approach allowed us to not only solve the 6 × 6 game variant, but also lays the foundation for solving the game’s larger board-size variants.
2048 is a stochastic single-player game involving 16 cells on a 4 & times; 4 grid, where a player chooses a direction among up, down, left, and right to obtain a score by merging two tiles with the same number located in neighboring cells along the chosen direction. This paper presents that a variant 2048(4 & times;3) with 12 cells on a 4 & times; 3 board, one row smaller than the original, has been strongly solved. In this variant, the expected score achieved by an optimal strategy is about 50, 724.26 for the most common initial states: ones with two tiles of number 2. The number of reachable states and afterstates are identified to be 1, 152, 817, 492, 752 and 739, 648, 886, 170, respectively. The key technique is to partition the state space by the sum of tile numbers on a board, which we call the age of a state. An age is invariant between a state and its successive afterstate after any valid action and is increased by two or four by a stochastic response from the environment. Therefore, we can partition the state space by ages and enumerate all (after)states of an age depending only on states with the recent ages. Similarly, we can identify (after)state values by going along with ages in decreasing order.
