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To get the most out of this entry, try the following puzzle. This is a moderately difficult puzzle of its type, so don't get discouraged too easily. After you've tried the puzzle, check out my comments below.
Puzzle: Too Many Suspects
Instructions: Victor Timm was not a popular man; no one mourned him when he died. Timm made his living as a blackmailer and his only "friends", as he called them, were those he blackmailed. However, Timm did have a sense of humor, however mordant. On the evening of his demise, which happened to be his birthday, he invited all five of his current "friends" to dine with him. Each of the five took him up on the invitation only because they feared not to. Timm's practical joke was the biggest, and also the last, mistake of his life. At the end of the evening, he was dead, stabbed multiple times by the very knife he had used to carve the roast he had served.
Detective David Davidson was baffled. All five of Timm's dining companions had means, motive, and opportunity. Means: the carving knife; motive: blackmail; opportunity: presence in the house when and where Timm died. Any one of the five could have killed Timm, and perhaps all had done so, or some combination of two or more. How was he supposed to prove whodunnit?
Of course, Davidson interrogated each of the suspects separately, but none had confessed or implicated any of the other four. However, he did determine six facts about the crime. To protect the innocent, I will refer to the five suspects as: Mr. A, Mrs. B, Mr. C, Ms. D, and Mr. E.
Can you help Detective Davidson solve the mystery? Which of the suspects stabbed Timm?
Remember that more than one suspect may be guilty.
Reason by cases:
The second clue says that either Mrs. B or Mr. C are guilty, but not both. So, there are two cases: Mrs. B is guilty but Mr. C is not, or Mr. C is guilty and Mrs. B is not. First, assume that Mrs. B is guilty but Mr. C is not and see what happens, then do the same for the other case.
Mr. C and Ms. D stabbed Timm.
Explanation: There are many ways to solve this puzzle, but here's one using reasoning by cases: By clue 2, either Mrs. B or Mr. C stabbed Timm, but not both, so there are two cases: Mrs. B stabbed him but not Mr. C, or Mr. C did stab him and not Mrs. B.
Case 1: Assume that Mrs. B stabbed Timm but Mr. C did not. By clue 5, Ms. D is guilty only if Mr. C is too, but we've assumed that Mr. C is innocent. Thus, Ms. D is also innocent. By clue 6, if Ms. D is innocent then so is Mr. E, which means that Mr. E is not guilty. However, by clue 1, either Ms. D or Mr. E is guilty, but we've just concluded that neither is. Therefore, our assumption must be false since it leads to a contradiction, which means that the second case is true.
Case 2: Mrs. B is innocent but Mr. C is guilty. By clue 5, Ms. D is guilty if Mr. C is, so Ms. D is also guilty. By clue 3, if Mr. A is guilty then so is Mrs. B, but Mrs. B is not guilty, which means that Mr. A is innocent. Finally, by clue 4, Mr. E is guilty only if Mr. A is, but Mr. A is not guilty. Thus, Mr. E is also not guilty. Therefore, of the five suspects, only Mr. C and Ms. D are guilty of stabbing Timm.
Comments: In the previous entry in this series on problem-solving*, I warned against making false assumptions. However, I also explained that it was sometimes useful to make assumptions, and reasoning by cases is an example. Reasoning by cases is a way of dividing and conquering―see entry six in this series―since it divides a problem into two or more sub-problems, each of which is simpler and easier to solve than the problem as a whole. Moreover, in reasoning by cases, we make a different assumption for each case, giving us additional information for solving it.
Reasoning by cases is also a way of solving a problem by elimination―see entry three in this series―because it breaks a problem down into two or more cases, and then eliminates those cases that cannot be true. So, in the above puzzle solution, the first case was eliminated because it led to a contradiction, which meant that the second case had to be the truth.
To reason by cases, examine the problem to be solved to see if it can be broken down into cases. Specifically, look for disjunctions among the clues, such as the first two clues in the puzzle above. There are two types of disjunction to be on the lookout for: a weak or inclusive and a strong or exclusive disjunction. A weak/inclusive disjunction is one that says at least one of the cases is true, but more than one may be true; in contrast, a strong/exclusive disjunction says that exactly one of the cases is true. The first clue, above, is a weak disjunction and the second is a strong one.
When reasoning by cases, strong disjunctions are better than weak ones, so if you have a choice between using a strong or a weak disjunction to establish cases, choose the strong one. For instance, in the above puzzle, the first clue is a weak disjunction and the second a strong one. The puzzle can be solved using the first clue to establish the cases, but the cases are a bit harder to solve than those arising from the second clue.
It's not just explicit disjunctions that can be used to break a problem down into cases; in fact, each one of the six clues in the above puzzle can be so used, but this is an advanced topic for another time.
* Previous entries in the "How to Solve a Problem" series: