On Remoteness Functions of Exact Slow kk-NIM with k+1k+1 Piles

Given integer $n$ and $k$ such that $0 < k \leq n$ and $n$ piles of stones, two player alternate turns. By one move it is allowed to choose any $k$ piles and remove exactly one stone from each. The player who has to move but cannot is the loser. Cases $k=1$ and $k = n$ are trivial. For $k=2$ the game was solved for $n \leq 6$. For $n \leq 4$ the Sprague-Grundy function was efficiently computed (for both the normal and mis\`ere versions). For $n = 5,6$ a polynomial algorithm computing P-positions was obtained. Here we consider the case $2 \leq k = n-1$ and compute Smith's remoteness function, whose even values define the P-positions. In fact, an optimal move is always defined by the following simple rule: if all piles are odd, keep a largest one and reduce all other; if there exist even piles, keep a smallest one of them and reduce all other. Such strategy is optimal for both players, moreover, it allows to win as fast as possible from an N-position and to resist as long as possible from a P-position.Comment: 20 page

Sprague-Grundy function of matroids and related hypergraphs

We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph \cH on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge H \in \cH and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. In this paper we give an explicit formula that describes the Sprague-Grundy function of hypergraph $NIM$ for several classes of hypergraphs. In particular we characterize all $2$-uniform hypergraphs (that is graphs) and all matroids for which the formula works. We show that all self-dual matroids are included in this class