Some results on split sum of impartial games

Abstract

As it is stated in "Unsolved problems in combinatorial games" as problem A13, Nim with pass is a difficult problem. Nim was one of the first combinatorial games to be solved but when we introduce a pass the game becomes far more complex. In this article we extend the use of the split sum introduced in "Investigations of Impartial Games With a Pass" and we find cases where the split sum act the same as an classical disjunctive sum. We also introduce generalization of Grundy value for the split sum and find that games of same Grundy values can give very different properties with a split sum.