An inspector knows that exactly one of 3 suspects committed a crime, and interviews them to find out which. Each person lies in one of his statements, and tells the truth in the other. A says: I did not do it. B did it. B says: I did not do it. I know that C did it. C says: I did not do it. B does not know who it was. a) Find out who committed the crime; b) Convert the puzzle into a propositional proof (first create the propositional signature and then construct the premises); c) Prove your conclusion in a) using natural deduction.

Here's an accurate explanation as to why C is the correct answer:

A condition of the puzzle is that each person MUST make one statement that is false, and one that is true.

Start with suspect A: If the statement "B did it" is true, then A's other statement "I did not do it" must also be true. Since that does not meet the one true statement, one false statement condition, the statement "B did it" must be FALSE, meaning that A's statement "I did not do it" is TRUE.

On to suspect B: If the statement "I know that C did it" is true, then B's other statement "I did not do it" must also be true. Since that also does not meet the one true statement, one false statement condition, the statement "I know that C did it" must be FALSE, meaning that B's statement "I did not do it" is TRUE.

So if A and B did not do it, C did.

This perfectly harmonizes with C's statements: "I did not do it" (FALSE), and "B does not know who it was" (TRUE, as we determined that B's statement "I know that C did it" was false).