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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 $k$ pilesand remove exactly one stone from each. The player who has to move but cannotis the loser. Cases $k=1$ and $k = n$ are trivial. For $k=2$ the game wassolved for $n \leq 6$. For $n \leq 4$ the Sprague-Grundy function wasefficiently computed (for both the normal and mis\`ere versions). For $n = 5,6$a polynomial algorithm computing P-positions was obtained. Here we consider thecase $2 \leq k = n-1$ and compute Smith's remoteness function, whose evenvalues define the P-positions. In fact, an optimal move is always defined bythe following simple rule: if all piles are odd, keep a largest one and reduceall other; if there exist even piles, keep a smallest one of them and reduceall other. Such strategy is optimal for both players, moreover, it allows towin as fast as possible from an N-position and to resist as long as possiblefrom a P-position.

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