The Sprague-Grundy function for some nearly disjunctive sums of Nim and Silver Dollar games

We introduce and analyse an extension of the disjunctive sum operation on some classical impartial games. Whereas the disjunctive sum describes positions formed from independent subpositions, our operation combines positions that are not completely independent but interact only in a very restricted way. We extend the games Nim and Silver Dollar, played by moving counters along one-dimensional strips of cells, by joining several strips at their initial cell. We prove that, in certain cases, computing the Sprague-Grundy function can be simplified to that of a simpler game with at most two tokens in each strip. We give an algorithm that, for each Sprague-Grundy value g, computes the positions of two-token Star Nim whose Sprague-Grundy values are g. We establish that the sequence of differences of entries of these positions is ultimately additively periodic.