Problem

There are three suspects for a murder: Adams, Brown, and Clark. Adams says “I didn’t do it. The victim was an old acquaintance of Brown’s. But Clark hated him.” Brown states “I didn’t do it. I didn’t even know the guy. Besides I was out of town all that week.” Clark says “I didn’t do it. I saw both Adams and Brown downtown with the victim that day; one of them must have done it.” Assume that the two innocent men are telling the truth, but that the guilty man might not be. Who did it?

Solution

Suppose that Adams and Brown were telling the truth. Then since Adams claims the victim was friends with Brown, but Brown didn’t know the victim, their statement contradict each other. Thus, one of Adams or Brown must be lying.

On the other hand, suppose that Brown and Clark were telling the truth. Brown claims he was out of town that week, but Clark claims that he saw Brown with the victim that day. Again, the statements contradict each other, so one of Clark and Brown must be lying.

Since we know that one of Adams or Brown is a liar and one of Brown or Clark is liar, and since there is only one liar, we know that the liar must be Brown. Thus, Brown is guilty of the murder.