  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.
