On the Sprague-Grundy function of compound games

The classical game of {\sc Nim} can be naturally extended and played on an arbitrary hypergraph $\text{cH}\subseteq 2^V\setminus \{\mathrm{\varnothing }\}$ whose vertices $V=\{1,\dots ,n\}$ correspond to piles of stones. By one move a player chooses an edge $H$ of $\text{cH}$ and reduces arbitrarily all piles $i\in H$. In 1901 Bouton solved the classical {\sc Nim} for which $\cH = {{1}, \ldots, {n}}$\mathrm{.}In1910Mooreintroducedandsolvedamoregeneralgame$k$-{\sc Nim}, for which $\text{cH}=\{H\subseteq V\mid \mathrm{\mid }H\mathrm{\mid }\le k\}$, where $1 \leq k < n$. In 1980 Jenkyns and Mayberry obtained an explicit formula for the Sprague-Grundy function of Moore’s {\sc Nim} for the case $k+1=n$. Recently it was shown that the same formula works for a large class of hypergraphs. In this paper we study combinatorial properties of these hypergraphs and obtain explicit formulas for the Sprague-Grundy functions of the conjunctive and selective compounds of the corresponding hypergraph {\sc Nim} games.