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Problem 640

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Shut the Box

Bob plays a single-player game of chance using two standard $6$-sided dice and twelve cards numbered $1$ to $12$. When the game starts, all cards are placed face up on a table.

Each turn, Bob rolls both dice, getting numbers $x$ and $y$ respectively, each in the range $1,\ldots,6$. He must choose amongst three options: turn over card $x$, card $y$, or card $x+y$. (If the chosen card is already face down, it is turned to face up, and vice versa.)

If Bob manages to have all twelve cards face down at the same time, he wins.

Alice plays a similar game, except that instead of dice she uses two fair coins, counting heads as $2$ and tails as $1$, and that she uses four cards instead of twelve. Alice finds that, with the optimal strategy for her game, the expected number of turns taken until she wins is approximately $5.673651$.

Assuming that Bob plays with an optimal strategy, what is the expected number of turns taken until he wins? Give your answer rounded to $6$ places after the decimal point.

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关上盒子

鲍勃在玩一种带有运气成分的单人游戏，这种游戏需要用到两个六面骰子和十二张标记有$1$至$12$的卡片。在游戏开始时，把所有的卡片正面朝上放在桌上。

每一回合，鲍勃抛掷两枚骰子，得到的数字分别是$x$和$y$，显然这两个数字都在$1,\ldots,6$的范围内。他必须在以下三种操作中选择一种：翻转卡片$x$、翻转卡片$y$、翻转卡片$x+y$。（如果被翻转的卡片已经正面朝下，则翻转后变回正面朝上；反之亦然。）

如果鲍勃成功地将所有十二张卡片都翻转至正面朝下，则获得胜利。

爱丽丝则在玩一种类似的游戏，只是她用的不是骰子而是两枚标准硬币，并将正面视为$2$，反面视为$1$，此外她只用四张卡片而非十二张。爱丽斯发现，当她采用最优策略时，获得胜利所需回合数的期望约为$5.673651$。

假设鲍勃采用最优策略，他获胜所需回合数的期望是多少？将你的答案保留小数点后$6$位小数。

（注：标题指一种流行于英国的酒吧骰子游戏，本题题面对游戏规则有所修改。）

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