7. [4] My friend and I are playing a game with the following rules: If one of us says an integer $n$, the opponent then says an integer of their choice between $2 n$ and $3 n$, inclusive. Whoever first says 2007 or greater loses the game, and their opponent wins. I must begin the game by saying a positive integer less than 10 . With how many of them can I guarantee a win?
Answer: 6. We assume optimal play and begin working backward. I win if I say any number between 1004 and 2006. Thus, by saying such a number, my friend can force a win for himself if I ever say a number between 335 and 1003. Then I win if I say any number between 168 and 334 , because my friend must then say one of the losing numbers just considered. Similarly, I lose by saying 56 through 167 , win by saying 28 through 55 , lose with 10 through 17 , win with 5 through 9 , lose with 2 through 4 , and win with 1.