On the Sprague–Grundy function of extensions of proper Nim

We consider the game of proper Nim, in which two players alternately move by taking stones from n piles. In one move a player chooses a proper subset (at least one and at most n- 1) of the piles and takes some positive number of stones from each pile of the subset. The player who cannot move is the loser. Jenkyns and Mayberry (Int J Game Theory 9(1):51–63, 1980) described the Sprague–Grundy function of these games. In this paper we consider the so-called selective compound of proper Nim games with certain other games, and obtain a closed formula for the Sprague–Grundy functions of the compound games, when n≥ 3. Surprisingly, the case of n= 2 is much more complicated. For this case we obtain several partial results and propose some conjectures.