New Results for Domineering from Combinatorial Game Theory Endgame Databases

TL;DR

This paper constructs comprehensive endgame databases for Domineering, introduces new theorems for CGT value computation, and explores the existence of various CGT values including dyadic rationals, infinitesimals, and nimbers, supporting conjectures about game temperature.

Contribution

It extends Conway's Bridge Splitting Theorem with a new Bridge Destroying Theorem, enabling precise CGT value calculations and the construction of positions with specific values.

Findings

Proved the existence of Domineering positions for all dyadic rationals.

Constructed positions with various infinitesimal and nimber values, including *2 and *3.

Supported Berlekamp's conjecture that the maximum temperature in Domineering is 2.

Abstract

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.