Row Impartial Terminus

Abstract: We introduce Row Impartial Terminus (RIT), an impartial combinatorial game played on integer partitions. We show that any position in RIT can be uniquely decomposed into a core and a remnant. Our central result is that the Conway pair of any RIT position-which determines the outcome under both normal and mis`ere play-is identical to the Conway pair of a corresponding position in the game of Nim defined by the remnant. This finding provides a complete winning strategy for both variants of RIT, reducing its analysis to the well-understood framework of Nim. As a consequence, we classify RIT within the Conway-Gurvich-Ho hierarchy, showing it to be forced and miserable but not pet.