Solving Dots-And-Boxes
نویسندگان
چکیده
Dots-And-Boxes is a well-known and widely-played combinatorial game. While the rules of play are very simple, the state space for even very small games is extremely large, and finding the outcome under optimal play is correspondingly hard. In this paper we introduce a Dots-And-Boxes solver which is significantly faster than the current state-of-the-art: over an order-of-magnitude faster on several large problems. Our approach uses Alpha-Beta search and applies a number of techniques—both problem-specific and general—that reduce the search space to a manageable size. Using these techniques, we have determined for the first time that Dots-AndBoxes on a board of 4 × 5 boxes is a tie given optimal play; this is the largest game solved to date.
منابع مشابه
Solving 4x5 Dots-And-Boxes
Dots-And-Boxes is a well-known and widely-played combinatorial game. While the rules of play are very simple, the state space for even small games is extremely large, and finding the outcome under optimal play is correspondingly hard. In this paper we introduce a Dots-And-Boxes solver which is significantly faster than the current state-of-the-art: over an order-of-magnitude faster on several l...
متن کاملResearch and Implementation of Dots-and-Boxes Game System
This paper has studied the game rules, victory or defeat rules, and the key techniques of Dots-and-Boxes, and has designed and realized one 6×6 Dots-and-Boxes based on the representing method of Strings and Coins. By corresponding Dots-and-Boxes chessboard to strings and coins chessboard, the proposed representing method is not only intuitive and convenient in chessboard representation, but als...
متن کاملVariations on Narrow Dots-and-Boxes and Dots-and-Triangles
We verify a conjecture of Nowakowski and Ottaway that closed 1×n Dots-and-Triangles is a first-player win when n 6= 2 [2]. We also prove that in both the open and closed 1× n Dots-and-Boxes games where n is even, the first player can guarantee a tie.
متن کاملNarrow Misère Dots-and-Boxes
We study misère Dots-and-Boxes, where the goal is to minimize score, for narrow boards. In particular, we characterize the winner for 1× n boards with an explicit winning strategy for the first player with a score of b(n − 1)/3c. We also give preliminary results for 2 × n and for Swedish 1× n (where the boundary is initially drawn).
متن کاملEvolution of Neural Networks to Play the Game of Dots-and-Boxes
Dots-and-Boxes is a child’s game which remains analytically unsolved. We implement and evolve arti cial neural networks to play this game, evaluating them against simple heuristic players. Our networks do not evaluate or predict the nal outcome of the game, but rather recommend moves at each stage. Superior generalisation of play by co-evolved populations is found, and a comparison made with ne...
متن کامل