XT Domineering: A new combinatorial game

Authors:
Kuo-Yuan Kao;I-Chen Wu;Yi-Chang Shan
Affiliations:
Department of Information Management, National Penghu University, Penghu, Taiwan;Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan;Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan
Venue:
Knowledge-Based Systems
Year:
2012

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Abstract

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 1x2 or 2x1 dominos are allowed to be placed on empty squares in an mxn board. This new game allows a player to place a 1x1 domino on an empty square s while unable to place a 1x2 or 2x1 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 3x3 squares and shows that each sub-graph of 3x3 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.