Extended Complementary Nim

In the standard {\sc Nim} with $n$ heaps, a player by one move can reduce (by a positive amount) exactly one heap of his choice. In this paper we consider the game of {\em complementary {\sc Nim}} ({\sc Co-Nim}), in which a player by one move can reduce at least one and at most $n-1$ heaps, of his choice. An explicit formula for the Sprague-Grundy (SG) function of {\sc Co-Nim} was obtained by Jenkyns and Mayberry in 1980. We consider a further generalization, called {\em extended complementary {\sc Nim}} ({\sc Exco-Nim}). In this game there is one extra heap and a player by one move can reduce at least one and at most $n-1$ of the first $n$ heaps, as in \textsc{Co-Nim}, and (s)he can also reduce the extra heap, whenever it is not empty. The $\P$-positions of {\sc Exco-Nim} are easily characterized for any $n$. For $n \geq 3$ the SG function of {\sc Exco-Nim} is a simple generalization of the SG function of {\sc Co-Nim}. Somewhat surprisingly, for $n = 2$ the SG function of {\sc Exco-Nim} looks much more complicated and "behaves in a chaotic way". For this case we provide only some partial results and some conjectures.