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Article

On the Sprague–Grundy function of extensions of proper NIM

Boros E., Gurvich V., Bao Ho N., Makino K.

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−1n−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≥3n≥3. Surprisingly, the case of n=2n=2 is much more complicated. For this case we obtain several partial results and propose some conjectures.