Complexity, appeal and challenges of combinatorial games
Theoretical Computer Science - Algorithmic combinatorial game theory
Theoretical Computer Science
Invariant and dual subtraction games resolving the Duchêne-Rigo conjecture
Theoretical Computer Science
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An invariant subtraction game is a 2-player impartial game defined by a set of invariant moves (k-tuples of non-negative integers) M. Given a position (another k-tuple) x=(x"1,...,x"k), each option is of the form (x"1-m"1,...,x"k-m"k), where m=(m"1,...,m"k)@?M, and where x"i-m"i=0, for all i. Two players alternate in moving and the player who moves last wins. The set of non-zero P-positions of the game M defines the moves in the dual game M^@?. For example, in the game of (2-pile Nim)^@? a move consists in removing the same positive number of tokens from both piles. Our main results concern a double application of @?, the operation M-(M^@?)^@?. We establish a fundamental 'convergence' result for this operation. Then, we give necessary and sufficient conditions for the relation M=(M^@?)^@? to hold, as is the case for example with M=k-pile Nim.