The game of n-times nim

The following game is considered. The 0rst player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player 0rst unable to move loses and his opponent wins. Let f1; f2; : : : be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size fi. It is proved that there exist constants c= c(n) and k0 = k0(n) such that fk+1 =fk +fk−c for all k ¿k0, and limn→∞ c(n)=(nlogn)=1. c © 2002 Elsevier Science B.V. All rights reserved.