On the Sprague–Grundy function of extensions of proper Nim

Endre Boros (), Vladimir Gurvich (), Nhan Bao Ho () and Kazuhisa Makino ()
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Endre Boros: Rutgers University
Vladimir Gurvich: National Research University Higher School of Economics
Nhan Bao Ho: La Trobe University
Kazuhisa Makino: Kyoto University

International Journal of Game Theory, 2021, vol. 50, issue 3, No 4, 635-654

Abstract: Abstract 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 $$n-1$$ 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 $$n\ge 3$$ n ≥ 3 . Surprisingly, the case of $$n=2$$ n = 2 is much more complicated. For this case we obtain several partial results and propose some conjectures.

Keywords: Nim; Proper Nim; Moore’s Nim; Sprague–Grundy function (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s00182-020-00707-3

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