On the Sprague-Grundy values of games with a pass

In this paper, we consider two-player impartial games with a pass-move. A disjunctive compound of games is a position in which, on each turn, the current player chooses one of the components and makes a legal move in it. For disjunctive compounds, it is known that the time to determine which player has a winning strategy is bounded by the time to compute the SG-values of the components plus the time for their XOR. However, if we allow a pass-move during the play, the analysis of such games becomes much more difficult. A pass-move allows each player to skip exactly one turn in non-terminal positions during the game, after which neither player may use a pass-move again. We establish a homomorphism on the SG-values of games with a pass-move. That is, if every component satisfies a condition called one-move game, the SG-value of the disjunctive compound of the components with a pass-move is the same as the SG-value of nim with a pass-move where the size of every pile is the same as the SG-value of every component of the compound. This guarantees that the time to determine which player has a winning strategy in a disjunctive compound with a pass can be bounded by the sum of the time to determine SG-values of all components without a pass and a position in nim with a pass. We also show how the homomorphism is used for determining SG-values of some chocolate games.