We provide two methodologies in the area of computation theory to solve optimal strategies for games such as Watermelon chess and Go. From experimental results, we find relevance to graph theory, group representation, and mathematical consciousness. We prove that the decision strategy of movement for Watermelon chess and Chinese checker games belongs to a matrix that is a noncommutative ring or an abelian group over set Y={-1,0,1}. Additionally, the movement for any chess game with two players belongs to a noncommutative ring or an abelian group from Occam’s razor principle. We derive the closed form of the transition matrix for any chess game with two players and discover that the element of the transition matrix belongs to a rational number. We propose a different methodology based on abstract algebra to analyze the complexity of chess games in their entirety, instead of being limited solely to endgame results. It is probable that similar decision processes of people may also belong to a noncommutative ring or an abelian group.