New Results for Domineering from Combinatorial Game Theory Endgame Databases
Theoretical Computer Science(2015)
摘要
We have constructed endgame databases for all single-component positions up to 15 squares for Domineering. These databases have been filled with exact Combinatorial Game Theory (CGT) values in canonical form. We give an overview of several interesting types and frequencies of CGT values occurring in the databases. 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 single-component positions as the sum of the values of smaller fragments, but also to compose larger single-component positions with specified values from smaller fragments. Using the theorems, we then prove that for any dyadic rational number there exist single-component 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, in particular that for any multiple up n ¿ ¿ or down n ¿ ¿ there are Domineering positions of size 1 + 5 n with these values. 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. We have constructed Domineering endgame databases for all positions up to 15 squares.For any dyadic rational number there exist Domineering positions with that value.For any multiple up or down there exist Domineering positions with that value.We show Domineering positions with values *2 and *3, reachable in standard play.We found one Domineering position with temperature 2 and no hotter positions.
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关键词
Domineering,Combinatorial game theory,Numbers,Infinitesimals,Ups and downs,Tinies and minies,Nimbers including*2 and*3,Temperatures
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