HMMT 二月 2007 · GEN2 赛 · 第 7 题

HMMT February 2007 — GEN2 Round — Problem 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?

解析

[ 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.