  Given $n$ piles of tokens and a positive integer $k \leq n$, we study the
following two impartial combinatorial games Nim$^1_{n, \leq k}$ and Nim$^1_{n,
=k}$. In the first (resp. second) game, a player, by one move, chooses at least
$1$ and at most (resp. exactly) $k$ non-empty piles and removes one token from
each of these piles. For the normal and mis\`ere version of each game we
compute the Sprague-Grundy function for the cases $n = k = 2$ and $n = k+1 =
3$. For game Nim$^1_{n, \leq k}$ we also characterize its P-positions for the
cases $n \leq k+2$ and $n = k+3 \leq 6$.
