一道关于博弈的笔试题

hehanz

2377

收藏 2016-12-25

悬赏 40 个论坛币未解决

Consider 2016 coins, all of which initially showing heads, lying in a straight line on a long table. Two players, Alex and Ben, standing by the same side of the table, play the following game with alternating moves: each "good move" consists of choosing a block of 15 consecutive coins, the leftmost of which showing head, and turning them all over. Alex starts first, and the last player who can make a "good move" wins the game.
a) Will this game always end?
b) Does Alex have a winning strategy, and why?

这道题是英文的，说说我的理解。一共有2016个硬币，甲和乙分别翻硬币。2016个硬币开始默认都是head朝上。甲先翻硬币，规则是：必须连续翻15个硬币（如果是head就会变成tail，如果是tail就会变成head），而且这15个硬币中最左边一定得是head才可以翻。最后一个可以连续翻15个硬币的人赢。
第一问，这个游戏一定会结束吗？
第二问，乙有没有可能赢？策略是什么？
这道题我知道答案，第一问是游戏一定会结束，第二问是乙没有可能赢，因为无论怎么翻，甲都会赢，但是我不会严格的证明。请大家提供证明过程，谢谢。