2004 IMO Shortlist S17

$A$ and $B$ play a game, given an integer $N$ , $A$ writes down $1$ first, then every player sees the last number written and if it is $n$ then in his turn he writes $n+1$ or $2n$ , but his number cannot be bigger than $N$ . The player who writes $N$ wins. For which values of $N$ does $B$ win?

*Proposed by A. Slinko & S. Marshall, New Zealand*

$A$ 和 $B$ 玩一个游戏，给定一个整数 $N$ ， $A$ 首先写下 $1$ ，然后每个玩家都会看到最后写下的数字，如果是 $n$ 则依次写下 $n+1$ 或 $2n$ ，但他的数字不能大于 $N$ 。写下$N$的玩家获胜。对于 $N$ 的哪些值，$B$ 获胜？

*由新西兰 A. Slinko 和 S. Marshall 提议*