# A modified divisor game

Consider a two player game. There are N balls marked 1 to N. A move consists of removing a ball n and all other balls which are divisors of n (including 1). The players alternate the moves. The one who takes the last ball wins the game.
Eg. [1 ,2 ,3 , 4, 5]--[Player 1 (4, 2, 1)]-->[3, 5]--[Player 2 (3)]-->[5]--[Player 1 (5)]-->[]. Player 1 wins. The () contains the balls a player removes.
I tried solving the problem for N < 10 manually and was able to observe that it was always possible to force victory for player who starts the game. I also know that this is always the case for all N from here. Can someone share the proof of this result and the playing strategy?
I tried to use the strategy stealing argument from the game of chomp but I am not sure if it is applicable here.

• Well, now I understand that the crux of the strategy stealing is finding a "zero" move which is a subset of any other legal move. Though we can figure out the winning moves with the aid of computer, is it possible to find a general strategy? Is there a mathematical formula as in the case Nims game or its proven that it can`t be found? – forcebrute Apr 7 '12 at 6:12