In the land of Symutra, there was a murder last night. Witnesses saw one person shooting the guy, but no one could see him well enough to identify him. In addition, Alice, Bob, Cindy, Derek, and Evelyn were all seen near the place of the murder shortly after and before the murder. The police brought them in for questioning and they all gave one sentence.

Alice: I didn't do it if and only if Derek didn't do it.

Bob: Alice did do it if I didn't do it.

Cindy: If I did it and Alice didn't do it, Derek didn't do it.

Derek: I didn't do it or Bob did do it, or Cindy did it.

Evelyn: I did it.

It is known for a fact that ONLY one of them did do it. If only one person told the truth, who did it? And who told the truth?

If Cindy lied, i.e., gave a false statement, then the logical expression of her material conditional statement as p --> q must be FALSE, p ("[Cindy] did it and Alice didn't do it") is TRUE and q ("Derek didn't do it") is FALSE. Given the logical conjunction p ("[Cindy] did it") AND q ("Alice didn't do it") as TRUE, then Cindy did it.

If Alice lied, then her logical biconditional statement results in either [Alice] didn't do it and Derek did it, OR Alice did it and Derek didn't do it.

Thus either Cindy or Alice must have told the truth.

If Alice had told the truth, then either Alice and Derek would both be guilty (a given impossibility), or both innocent -- and hence, Cindy must be guilty. Yet, Cindy's guilt and lie and Alice's veracity do not concur with the Derek's statement. In order for Derek statement to be FALSE, all three parts of the logical disjunctive propositions of Derek's statement would need be FALSE. As "[Derek] didn't do it" would be TRUE, Derek would have told the truth. This contradicts the given that only one of the five told the truth, thus Alice did not tell the truth.

As Derek must have lied, with the deduction that Cindy told the truth, all three logical disjunctive propositions of Derek's statement need be FALSE. This only occurs if Derek had committed the crime.

This concurs with both Evelyn's statement ("[Evelyn] did it" = FALSE), and Bob's converse implication {("Alice did do it") FALSE, ("[Bob] didn't do it") TRUE = FALSE} being falsehoods.

Thus, Derek committed the crime, and Cindy told the truth.

Posted by Dej Mar on 2012-04-01 09:22:55