DeepAI
Log In Sign Up

New Results for Domineering from Combinatorial Game Theory Endgame Databases

06/12/2015
by   Jos Uiterwijk, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

06/03/2021

Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

We settle two long-standing complexity-theoretical questions-open since ...
04/18/2018

Faster Evaluation of Subtraction Games

Subtraction games are played with one or more heaps of tokens, with play...
07/20/2017

Pictures of Combinatorial Cubes

We prove that the 8-point algorithm always fails to reconstruct a unique...
05/13/2022

Discrete density comonads and graph parameters

Game comonads have brought forth a new approach to studying finite model...
01/03/2022

Mechanization of Incidence Projective Geometry in Higher Dimensions, a Combinatorial Approach

Several tools have been developed to enhance automation of theorem provi...
03/06/2022

Intertwining of Complementary Thue-Morse Factors

We consider the positions of occurrences of a factor x and its binary co...
04/27/2020

A Phase Transition in Arrow's Theorem

Arrow's Theorem concerns a fundamental problem in social choice theory: ...