The game of \(n\)-times nim
type: Item
instance of: scholarly article
title: The game of \(n\)-times nim
zbMATH Open document ID: 1050.91022
DOI: 10.1016/S0012-365X(02)00667-2
author: Norihide Tokushige
author: Peter Frankl
published in: Discrete Mathematics
publication date: 2003-03-16T00:00:00Z
review text: In the two-player game of \(n\)-times nim there is a single pile of stones from which the players alternate drawing. The first player may initally remove any number of stones but not all the stones. After that each player can remove at most \(n\) times as many stones as the previous player did, where \(n\) is a real number, \(n\geq1\). The first player unable to move loses, and the other wins. The paper deals with the increasing sequence of numbers \(f_k\) such that the second player has a winning strategy when the first player has to move from a pile of size \(f_k\). \textit{M. J. Whinihan} [Fibonacci nim, Fibonacci Q. 1, 9 (1963)] showed that in 2-times nim the sequence satisfies \(f_{k+1}=f_k+f_{k-1}\) and is the Fibonacci sequence. \textit{A. J. Schwenk} [Take-away games, Fibonacci Q. 8, 225--234, 241 (1970; Zbl 0213.46402)] showed that in \(n\)-times nim there are constants \(c=c(n)\) and \(k_0=k_0(n)\) such that \(f_{k+1}=f_k+f_{k-c}\) for all \(k>k_0\), and asked about the behavior of \(c(n)\). The present paper gives a new proof of Schwenk's result, and shows that \(c(n)\) is asymptotic to \(n\log n\) as \(n\to\infty\).
reviewed by: Gerald A. Heuer
Mathematics Subject Classification ID: 91A46
Mathematics Subject Classification ID: 11B39
zbMATH DE Number: 1882193
zbMATH Keywords: \(n\)-times nim
zbMATH Keywords: Fibonacci nim
zbMATH Keywords: take-away games
MaRDI profile type: Publication
full work available at URL: s0012-365x(02)00667-2
OpenAlex ID: W2111741711
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description: scientific article; zbMATH DE number 1882193