Domineering
A two-player game, also called crosscram, in which player 
has horizontal dominoes and player 
has vertical dominoes. The two players alternately place a domino on a board until the other cannot move, in which case the player having made the last move wins (Gardner 1974, Lachmann et al. 2000). Depending on the dimensions of the board, the winner will be 
, 
, 1 (the player making the first move), or 2 (the player making the second move). For example, the 
board is a win for the first player.
Berlekamp (1988) solved the general problem for 
board for odd 
. Solutions for the 
board are summarized in the following table, with 
a win for 
for 
.
 |
win |
 |
win |
 |
win |
| 0 |
2 |
10 |
1 |
20 |
H |
| 1 |
V |
11 |
1 |
21 |
H |
| 2 |
1 |
12 |
H |
22 |
H |
| 3 |
1 |
13 |
2 |
23 |
1 |
| 4 |
H |
14 |
1 |
24 |
H |
| 5 |
V |
15 |
1 |
25 |
H |
| 6 |
1 |
16 |
H |
26 |
H |
| 7 |
1 |
17 |
H |
27 |
1 |
| 8 |
H |
18 |
1 |
28 |
H |
| 9 |
V |
19 |
1 |
29 |
H |
Lachmann et al. (2000) have solved the game 
for widths of 
, 3, 4, 5, 7, 9, and 11, obtaining the results summarized in the following table for 
, 1, ....
 |
winner |
| 3 |
2, V, 1, 1, H, H, ... |
| 4 |
H for even  and all  |
| 5 |
2, V, H, V, H, 2, H, H, ... |
| 7 |
H for  |
| 9 |
H for  |
| 11 |
H for  |
Bullock created a program called Obsequi that solved the additional cases 
, 
, 
, 
, and 
.
REFERENCES:
Berlekamp, E. R. "Blockbuster and Domineering." J. Combin. Th. Ser. A 49, 67-116, 1988.
Berlekamp, E. R.; Conway, J. H.; and Guy, R. K. Winning Ways for Your Mathematical Plays, Vol. 2: Games in Particular. London: Academic Press, 1982.
Breuker, D. M.; Uiterwijk, J. W. H. M.; van den Herik, H. J. "Solving 
Domineering." Theor. Comput. Sci. 230, 195-206, 2000.
Bullock, N. "Obsequi's Domineering Page." http://www.cs.ualberta.ca/~games/domineering/.
Conway, J. H. On Numbers and Games, 2nd ed. Wellesley, MA: A K Peters, 2000.
Gardner, M. "Mathematical Games: Cram, Crosscram and Quadraphage: New Games having Elusive Winning Strategies." Sci. Amer. 230, 106-108, Feb. 1974.
Lachmann, M.; Moore, C.; and Rapaport, I. "Who Wins Domineering on Rectangular Boards?" 8 Jun 2000. http://arxiv.org/abs/math.CO/0006066.
Uiterwijk, J. W. H. M. and van den Herik, H. J. "The Advantage of the Initiative." Info. Sci. 122, 43-58, 2000.
Uiterwijk, J. W. H. M. "Domineering Results." http://www.cs.rulimburg.nl/~uiterwyk/Domineering_results.html.
Wolfe, D. "The Gamesman's Toolkit." In Games of No Chance, Proc. MSRI Workshop on Combinatorial Games, July, 1994 (Ed. R. J. Nowakowski). Cambridge, England: Cambridge University Press, 1998.
