Only one of the three is guilty, and only one of each man statements is false. Who killed Billy the Knife ?

Question

Billy "The Knife," the infamous tough guy, was found murdered one night in an alleyway behind the nightclub he used to frequently visit. The police brought in three suspects on the next morning. One of the officers interrogated the three men and noted down the following statements.

Alex:
1. I did not kill Billy.
2. Jimmy is not my friend.
3. I knew Billy.

Jimmy:
1. I did not kill Billy.
2. Alex and Dexter are friends of mine.
3. Alex did not kill Billy.

Dexter:
1. I did not kill Billy.
2. Alex lied when he said that Jimmy was not his friend.
3. I do not know who killed Billy.

Only one of the three is guilty, and only one of each man statements is false. Who killed Billy the Knife ?

Answer 1

All three men claim they did not kill Billy. That means the for the real killer the other two statements he made must be true.
If Alex killed Billy, statements # 2 and 3 must be true. For Jimmy, this would mean statements # 2 and 3 are false. So Alex did not kill Billy.
If Dexter killed Billy, statement #1 is false. But as he is the killer he knows who is the killer and statement # 3 is also false.
Therefore, Jimmy killed Billy and statements # 2 for Alex and # 3 for Dexter are false.