Troy has a collection of gems.特洛伊收藏了一些宝石。
He ranks his gems from his overall favorite to his least favorite.他将自己收藏的宝石从最喜爱的到最不喜爱的依次排列。
Sarah knows which gems are in Troy's collection, but she does not know how he has ranked them.萨拉知道特洛伊收藏的宝石都有哪些，但她不知道他是如何给这些宝石排序的。
Sarah has a plan to find out which gem is Troy's overall favorite:萨拉有个计划，要弄清楚特洛伊最喜欢哪种宝石：
- Sarah chooses four of Troy's gems and asks Troy: "Out of this group of four, which gem is your favorite?"萨拉从特洛伊的宝石中挑了四颗，然后问特洛伊：“在这四颗宝石中，哪一颗是你最喜欢的？”
- Sarah chooses a new set of four gems and asks her question again.萨拉挑选了一套新的四颗宝石，再次提出了她的问题。
- Then she chooses a third set of four gems and asks her question for the last time.然后她挑选出第三组四颗宝石，并最后一次提出她的问题。
Note: When Sarah chooses her second and third set of four gems, she may sometimes include gems she has chosen before.注意：当萨拉选择她的第二组和第三组四颗宝石时，有时可能会包含之前选过的宝石。
Question:问题:
If Sarah is to successfully find Troy's overall favorite gem, what is the largest possible number of gems in Troy's collection?如果萨拉要成功找到特洛伊最喜欢的宝石，那么特洛伊收藏的宝石最多可能有多少颗？
A. 8
B. 10
C. 11
D. 12
Explanation 解释
Answer: 答案:
10
Explanation: 解释:
With 10 gems, Sarah can ask Troy about eight different gems with the first two requests. Troy's answer to each of these requests is a candidate to be his favourite of all his gems, but the other three cannot be. Therefore, with Sarah's third and final request, she can include these two candidates and the two gems that have not yet been part of a request. Troy's answer to this third request must be his favourite gem.萨拉有 10 颗宝石，她可以用前两次询问来询问其中的 8 颗。特洛伊对这两次询问的回答可能是他所有宝石中最喜欢的那颗，但另外三颗则不可能。因此，在萨拉的第三次也是最后一次询问中，她可以把这两个可能的答案以及之前未被询问过的两颗宝石都包括进去。特洛伊对这次询问的回答一定是他最喜欢的那颗宝石。
We have shown that there is a strategy of Sarah's that works if there are 10 gems. (Note that there are other correct strategies that work for 10 gems.)我们已经证明，如果宝石有 10 颗，那么萨拉有一种策略是可行的。（请注意，对于 10 颗宝石，还有其他正确的策略也是可行的。）
If Troy has at least 11 gems, we consider Sarah's first two requests.如果特洛伊至少有 11 颗宝石，我们就考虑萨拉的前两个请求。
If at least one gem is part of both requests, then at least four gems will be unconsidered after the first two requests. In this case, Sarah must ask about these four unconsidered gems, because if she doesn't, an unconsidered gem could be Troy's favourite. On the other hand, she won't have any information about Troy's favourite among the last four unconsidered gems compared to the other seven gems. In this case, her strategy cannot determine which gem is Troy's most favourite.如果至少有一颗宝石同时属于两个请求，那么在前两个请求之后，至少会有四颗宝石未被考虑。在这种情况下，萨拉必须询问这四颗未被考虑的宝石，因为如果不这样做，未被考虑的宝石中可能就有特洛伊最喜欢的那颗。另一方面，与另外七颗宝石相比，她对最后四颗未被考虑的宝石中哪一颗是特洛伊最喜欢的毫无头绪。在这种情况下，她的策略无法确定哪颗宝石是特洛伊最喜欢的。
If there is no gem that is part of Sarah's both first two requests, then candidates for Troy's favourite include the two answers to these requests and the three remaining unconsidered gems. This is a total of five possibilities for Troy's most favourite gem, and Sarah does not have any information about how they rank, so her strategy cannot work.如果莎拉最初的两个请求中没有共同包含的宝石，那么特洛伊最喜欢的宝石候选者就包括这两个请求的答案以及剩下的三颗未被考虑的宝石。这样特洛伊最喜欢的宝石就有五种可能性，而莎拉对它们的排名毫无头绪，所以她的策略就行不通了。
We have shown that there is not a strategy of Sarah's that works, if there are more than 10 gems.我们已经证明，如果宝石数量超过 10 颗，那么萨拉的任何策略都无法奏效。
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