HMMT 二月 2013 · COMB 赛 · 第 8 题

英文解析

8. It is known that exactly one of the three (distinguishable) musketeers stole the truffles. Each musketeermakes one statement, in which he either claims that one of the three is guilty, or claims that one of thethree is innocent. It is possible for two or more of the musketeers to make the same statement. Afterhearing their claims, and knowing that exactly one musketeer lied, the inspector is able to deduce whostole the truffles. How many ordered triplets of statements could have been made?
   Answer: 99 We divide into cases, based on the number of distinct people that statements aremade about.
   • The statements are made about 3 distinct people. Then, since exactly one person is guilty, andbecause exactly one of the three lied, there are either zero statements of guilt or two statements ofguilt possible; in either case, it is impossible by symmetry to determine who is guilty or innocent.
   • The statements are made about 2 distinct people or 1 distinct person. Then, either at least twoof the statements are the same, or all are different.
   – If two statements are the same, then those two statements must be true because only onemusketeer lied. Consequently, the lone statement must be false. If all the statements areabout the same person, there there must be 2 guilty claims and 1 innocence claim (otherwisewe would not know which of the other two people was guilty). Then, there are 3 choices forwho the statement is about and 3 choices for who makes the innocence claim, for a 3 · 3 = 9
   triplets of statements. Meanwhile, if the statements are about two different people, then thisis doable unless both of the distinct statements imply guilt for the person concerned (i.e.
   where there are two guilty accusations against one person and one claim of innocence againstanother). Consequently, there are 3 sets of statements that can be made, 3 · 2 = 6 ways todetermine who they are made about, and 3 ways to determine who makes which statement,
   for a total of 3 · 6 · 3 = 54 triplets in this case.
   – If all the statements are different, then they must be about two different people. Here, theremust be one person, who we will call A, who has both a claim of innocence and an accusationof guilt against him. The last statement must concern another person, B. If the statementaccuses B of being guilty, then we can deduce that he is the guilty one. On the other hand, ifthe statement claims that B is innocent, either of the other two musketeers could be guilty.
   Consequently, there are 3 · 2 = 6 ways to choose A and B, and 3! = 6 ways to choose whomakes which statement, for a total of 6 · 6 = 36 triplets of statements.
   In total, we have 9 + 54 + 36 = 99 possible triplets of statements.