Problem 550

Problem 550

Divisor game

Two players are playing a game. There are k piles of stones. When it is his turn a player has to choose a pile and replace it by two piles of stones under the following two conditions:

  • Both new piles must have a number of stones more than one and less than the number of stones of the original pile.
  • The number of stones of each of the new piles must be a divisor of the number of stones of the original pile.

The first player unable to make a valid move loses.
Let f(n,k) be the number of winning positions for the first player, assuming perfect play, when the game is played with k piles each having between 2 and n stones (inclusively).f(10,5)=40085.

Find f(107,1012).Give your answer modulo 987654321.



  • 新的两堆石子的数目必须大于1且小于原来那一堆石子的数目;
  • 新的两堆石子的数目必须是原来那一堆石子的数目的约数。