XT Domineering: A new combinatorial game

摘要

This paper introduces a new combinatorial game, named XT Domineering, together with its mathematical analysis. XT Domineering is modified from the Domineering game in which 1 × 2 or 2 × 1 dominos are allowed to be placed on empty squares in an m × n board. This new game allows a player to place a 1 × 1 domino on an empty square s while unable to place a 1 × 2 or 2 × 1 domino in the connected group of empty squares that includes s. After modifying the rule, each position in the game becomes an infinitesimal. This paper calculates the game values of all sub-graphs of 3 × 3 squares and shows that each sub-graph of 3 × 3 squares is a linear combination of 8 elementary infinitesimals. These pre-stored game values can be viewed as a knowledge base for playing XT Domineering. Instead of searching the whole game trees, a simple rule for determining the optimal outcome of any sum of these positions is presented.