Perfectly Solving Domineering Boards

Abstract

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 $10 \times 10$ (omitting the trivial $1 \times n$ and $m \times 1$ 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.

Finally, we prove some more general theorems: (1) all $m \times 3$ boards ($m > 1$) are a win for Vertical; (2) all $2k \times n$ boards with $n = 3, 5, 7, 9,$ and $11$ are a win for Vertical; (3) all $3 \times n$ boards ($n > 3$) are a win for Horizontal; and (4) all $m \times 2k$ boards for $m = 5$ and $9$, all $m \times 2k$ boards with $k>1$ for $m = 3$ and $7$, and all $11 \times 4k$ boards are a win for Horizontal.

Appendix

In this appendix we provide fuller details on the results for all $m \times n$ boards for $m$ and $n \ge 2$ and ${\le }~30$. We first summarize the boards solved per level, distinguishing them into wins for Vertical and wins for Horizontal.

1.1 Level 3

Wins for Vertical: $2 \times 3$, $6 \times 3$, $10 \times 3$, $14 \times 3$, $18 \times 3$, $22 \times 3$, $26 \times 3$, $30 \times 3$; 8 in total.

Wins for Horizontal: $3 \times 4$, $3 \times 8$, $3 \times 12$, $3 \times 16$, $3 \times 20$, $3 \times 24$, $3 \times 28$; 7 in total.

1.2 Level 4

Wins for Vertical: $2 \times 2$, $6 \times 2$, $10 \times 2$, $3 \times 3$, $4 \times 3$, $7 \times 3$, $8 \times 3$, $9 \times 3$, $11 \times 3$, $12 \times 3$, $13 \times 3$, $15 \times 3$, $16 \times 3$, $17 \times 3$, $19 \times 3$, $20 \times 3$, $21 \times 3$, $23 \times 3$, $24 \times 3$, $25 \times 3$, $27 \times 3$, $28 \times 3$, $29 \times 3$, $2 \times 5$, $4 \times 5$, $6 \times 5$, $8 \times 5$, $10 \times 5$, $12 \times 5$, $14 \times 5$, $15 \times 5$, $16 \times 5$, $17 \times 5$, $18 \times 5$, $19 \times 5$, $20 \times 5$, $21 \times 5$, $22 \times 5$, $23 \times 5$, $24 \times 5$, $25 \times 5$, $26 \times 5$, $27 \times 5$, $28 \times 5$, $29 \times 5$, $30 \times 5$, $2 \times 7$, $4 \times 7$, $6 \times 7$, $8 \times 7$, $10 \times 7$, $12 \times 7$, $14 \times 7$, $16 \times 7$, $18 \times 7$, $20 \times 7$, $22 \times 7$, $24 \times 7$, $26 \times 7$, $27 \times 7$, $28 \times 7$, $30 \times 7$, $2 \times 11$, $6 \times 11$, $10 \times 11$, $2 \times 15$, $2 \times 19$; 67 in total.

Wins for Horizontal: $5 \times 2$, $9 \times 2$, $5 \times 4$, $7 \times 4$, $9 \times 4$, $3 \times 5$, $3 \times 6$, $5 \times 6$, $9 \times 6$, $5 \times 8$, $7 \times 8$, $9 \times 8$, $3 \times 9$, $3 \times 10$, $5 \times 10$, $9 \times 10$, $3 \times 11$, $5 \times 12$, $7 \times 12$, $9 \times 12$, $3 \times 13$, $5 \times 13$, $3 \times 14$, $5 \times 14$, $7 \times 14$, $9 \times 14$, $3 \times 15$, $5 \times 15$, $5 \times 16$, $7 \times 16$, $9 \times 16$, $3 \times 17$, $5 \times 17$, $3 \times 18$, $5 \times 18$, $7 \times 18$, $9 \times 18$, $3 \times 19$, $5 \times 19$, $5 \times 20$, $7 \times 20$, $9 \times 20$, $3 \times 21$, $5 \times 21$, $3 \times 22$, $5 \times 22$, $7 \times 22$, $9 \times 22$, $3 \times 23$, $5 \times 23$, $5 \times 24$, $7 \times 24$, $9 \times 24$, $3 \times 25$, $5 \times 25$, $3 \times 26$, $5 \times 26$, $7 \times 26$, $9 \times 26$, $3 \times 27$, $5 \times 27$, $5 \times 28$, $7 \times 28$, $9 \times 28$, $3 \times 29$, $5 \times 29$, $3 \times 30$, $5 \times 30$, $7 \times 30$, $9 \times 30$; 70 in total.

Table 7. Game-theoretic values of many $m \times n$ Domineering boards, for $m, n \le 30$, Vertical moving first. V or v indicates a Vertical win, H or h a Horizontal win. An uppercase character (V or H) is used for boards solved by search or combinatorial game theory, a lowercase character (v or h) is used when the game is solved using the translational symmetry rules. An overline on any character means that our program perfectly solves the game.

1.3 Level 5

Wins for Vertical: $3 \times 2$, $7 \times 2$, $11 \times 2$, $14 \times 2$, $15 \times 2$, $18 \times 2$, $19 \times 2$, $22 \times 2$, $23 \times 2$, $26 \times 2$, $27 \times 2$, $30 \times 2$, $5 \times 3$, $11 \times 5$, $13 \times 5$, $11 \times 7$, $15 \times 7$, $17 \times 7$, $19 \times 7$, $21 \times 7$, $23 \times 7$, $25 \times 7$, $29 \times 7$, $2 \times 9$, $4 \times 9$, $6 \times 9$, $8 \times 9$, $10 \times 9$, $12 \times 9$, $14 \times 9$, $16 \times 9$, $18 \times 9$, $20 \times 9$, $22 \times 9$, $24 \times 9$, $26 \times 9$, $28 \times 9$, $30 \times 9$, $4 \times 11$, $8 \times 11$, $12 \times 11$, $14 \times 11$, $16 \times 11$, $18 \times 11$, $20 \times 11$, $22 \times 11$, $24 \times 11$, $26 \times 11$, $28 \times 11$, $30 \times 11$; 50 in total.

Wins for Horizontal: $2 \times 4$, $11 \times 4$, $7 \times 6$, $2 \times 8$, $11 \times 8$, $5 \times 9$, $7 \times 10$, $5 \times 11$, $2 \times 12$, $11 \times 12$, $7 \times 13$, $2 \times 16$, $11 \times 16$, $7 \times 17$, $7 \times 19$, $2 \times 20$, $11 \times 20$, $7 \times 21$, $7 \times 23$, $2 \times 24$, $11 \times 24$, $7 \times 25$, $7 \times 27$, $2 \times 28$, $11 \times 28$, $7 \times 29$; 26 in total.

1.4 Level 6

Wins for Vertical: $4 \times 2$, $8 \times 2$, $12 \times 2$, $16 \times 2$, $20 \times 2$, $24 \times 2$, $28 \times 2$, $4 \times 4$, $6 \times 4$, $8 \times 4$, $10 \times 4$, $12 \times 4$, $14 \times 4$, $16 \times 4$, $18 \times 4$, $20 \times 4$, $22 \times 4$, $24 \times 4$, $26 \times 4$, $28 \times 4$, $30 \times 4$, $2 \times 6$, $4 \times 6$, $6 \times 6$, $8 \times 6$, $10 \times 6$, $12 \times 6$, $14 \times 6$, $16 \times 6$, $18 \times 6$, $20 \times 6$, $22 \times 6$, $24 \times 6$, $26 \times 6$, $28 \times 6$, $30 \times 6$, $2 \times 10$; 37 in total.

Wins for Horizontal: $4 \times 8$, $6 \times 8$, $4 \times 10$, $4 \times 12$, $6 \times 12$, $4 \times 14$, $4 \times 16$, $6 \times 16$, $4 \times 18$, $4 \times 20$, $6 \times 20$, $4 \times 22$, $4 \times 24$, $6 \times 24$, $4 \times 26$, $4 \times 28$, $6 \times 28$, $4 \times 30$; 18 in total.

We next provide in Table 7 an up-to-date overview of all known game-theoretic values of Domineering boards with sizes up to 30, including information on whether they are perfectly solved, solved by search or combinatorial game theory, or by using translational symmetry rules.