11 x 11 Domineering is Solved: The first player wins

We have developed a program called MUDoS (Maastricht University Domineering Solver) that solves Domineering positions in a very efficient way. This enables the solution of known positions so far (up to the 10 x 10 board) much quicker (measured in number of investigated nodes). More importantly, it enables the solution of the 11 x 11 Domineering board, a board up till now far out of reach of previous Domineering solvers. The solution needed the investigation of 259,689,994,008 nodes, using almost half a year of computation time on a single simple desktop computer. The results show that under optimal play the first player wins the 11 x 11 Domineering game, irrespective if Vertical or Horizontal starts the game. In addition, several other boards hitherto unsolved were solved. Using the convention that Vertical starts, the 8 x 15, 11 x 9, 12 x 8, 12 x 15, 14 x 8, and 17 x 6 boards are all won by Vertical, whereas the 6 x 17, 8 x 12, 9 x 11, and 11 x 10 boards are all won by Horizontal

An update on domineering on rectangular boards

Domineering is a combinatorial game played on a subset of a rectangular grid between two players. Each board position can be put into one of four outcome classes based on who the winner will be if both players play optimally. In this note, we review previous work, establish the outcome classes for several dimensions of rectangular board, and restrict the outcome class in several more.Comment: 9 pages. References fixe

New Results for Domineering from Combinatorial Game Theory Endgame Databases

We have constructed endgame databases for all single-component positions up to 15 squares for Domineering, filled with exact Combinatorial Game Theory (CGT) values in canonical form. The most important findings are as follows. First, as an extension of Conway's [8] famous Bridge Splitting Theorem for Domineering, we state and prove another theorem, dubbed the Bridge Destroying Theorem for Domineering. Together these two theorems prove very powerful in determining the CGT values of large positions as the sum of the values of smaller fragments, but also to compose larger positions with specified values from smaller fragments. Using the theorems, we then prove that for any dyadic rational number there exist Domineering positions with that value. Second, we investigate Domineering positions with infinitesimal CGT values, in particular ups and downs, tinies and minies, and nimbers. In the databases we find many positions with single or double up and down values, but no ups and downs with higher multitudes. However, we prove that such single-component ups and downs easily can be constructed. Further, we find Domineering positions with 11 different tinies and minies values. For each we give an example. Next, for nimbers we find many Domineering positions with values up to *3. This is surprising, since Drummond-Cole [10] suspected that no *2 and *3 positions in standard Domineering would exist. We show and characterize many *2 and *3 positions. Finally, we give some Domineering positions with values interesting for other reasons. Third, we have investigated the temperature of all positions in our databases. There appears to be exactly one position with temperature 2 (as already found before) and no positions with temperature larger than 2. This supports Berlekamp's conjecture that 2 is the highest possible temperature in Domineering

Computer Chess: From Idea to DeepMind

Computer chess has stimulated human imagination over some two hundred and fifty years. In 1769 Baron Wolfgang von Kempelen promised Empress Maria Theresia in public: “I will invent a machine for a more compelling spectacle [than the magnetism tricks by Pelletier] within half a year.” The idea of an intelligent chess machine was born. In 1770 the first demonstration was given.The real development of artificial intelligence (AI) began in 1950 and contains many well-known names, such as Turing and Shannon. One of the first AI research areas was chess. In 1997, a high point was to be reported: world champion Gary Kasparov had been defeated by Deep Blue. The techniques used included searching, knowledge representation, parallelism, and distributed systems. Adaptivity, machine learning and the recently developed deep learning mechanism were only later on added to the computer chess research techniques.The major breakthrough for games in general (including chess) took place in 2017 when (1) the AlphaGo Zero program defeated the world championship program AlphaGo by 100-0 and (2) the technique of deep learning also proved applicable to chess. In the autumn of 2017, the Stockfish program was beaten by AlphaZero by 28-0 (with 72 draws, resulting in a 64-36 victory). However, the end of the disruptive advance is not yet in reach. In fact, we have just started. The next milestone will be to determine the theoretical game value of chess (won, draw, or lost). This achievement will certainly be followed by other surprising developments.Algorithms and the Foundations of Software technolog