The game of \(n\)-times nim (Q1861258)

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 \(n\to\infty\).