Perfectly Solving Domineering Boards

In this paper we describe the perfect solving of rectangular empty Domineering boards. Perfect solving is defined as solving without any search. This is done solely based on the number of various move types in the initial position. For this purpose we first characterize several such move types. Next we define 12 knowledge rules, of increasing complexity. Of these rules, 6 can be used to show that the starting player (assumed to be Vertical) can win a game against any opposition, while 6 can be used to prove a definite loss (a win for the second player, Horizontal). Applying this knowledge-based method to all 81 rectangular boards up to 10×10 (omitting the trivial 1×n and m×1 boards), 67 could be solved perfectly. This is in sharp contrast with previous publications reporting the solution of Domineering boards, where only a few tiny boards were solved perfectly, the remainder requiring up to large amounts of search. Applying this method to larger boards with one or both sizes up to 30 solves 216 more boards, mainly with one dimension odd. All results fully agree with previously reported game-theoretic values. Moreover, among the larger boards solved 4 were not solved before, i.e., 10× 11, 14× 11, 18× 6, and 18× 11, all being a win for Vertical. Finally, we prove some more general theorems: 1) all m×3 boards (m > 1) are a win for Vertical; 2) all 2k × n boards with n = 3, 5, 7, 9, and 11 are a win for Vertical; 3) all 3×n boards (n > 3) are a win for Horizontal; and 4) all m × 2k boards for m = 5 and 9, and all m × 4k boards for m = 3, 7, and 11 are a win for Horizontal.