Six persons lived in a castle wherein each occupant had his own apartment. One morning, two of the occupants were dead:

A died because he sat on a poisoned needle that was hidden in his bed. In the last moment of his life, he scratched into the bedsheet with the needle that he took out of his backside: M = C
It was understood to mean:

A: "My murderer is C."

B was suffocated by a tampered Baroque bed canopy that crashed down to him. In the last seconds of his life, he carved a message on the wooden bedstead with a cork-screw in the one hand hanging out:

B: "My murderer is F."

Collection and evaluation of evidence by the police proved that each victim had been murdered by one of the other occupants. C, D, E, and F refused to answer police questions. There was hope that they would speak at the trial. But since the court room was bombed near the end of the proceedings and no one survived inside, the outcome of the court case remained unknown until now.
Seventy years later, two half burnt pieces of the court record were found in a huge forgotten box full of stones and ashes. The first piece of paper contains the statements of C, D, E, and F:

C: "F is a murderer or D is lying as always."

D: "A's or B's statement is erroneously false, or C's statement is true."

E: "If what I say is true, then the murderer of A and the murderer of B are not the same."

F: "If C murdered B or E is telling the truth, then I killed A."

The second piece of paper contains a snippet of the court decision:

"...the A case and the B case are closed and confirm the rule: No murderer without a lie..."

Can you prove that there is a unique solution for one of the victims and that there are two solutions for the other one?

(The word "or" was used to mean "at least one, and maybe both".

Stumbled across this one the other day and really wanted to solve it.  The self-referential nature of E’s statement was messing with me though, and I’m not 100% comfortable that I’ve resolved it the right way.  I also don’t know what “no murderer without a lie” means but after working on the problem for a while I’ve decided to believe it means that the murderer(s) must be lying.  Given those assumptions I think I got it.  So…

If we assume E’s lying, then we negate his statement.  The negation of P->Q is P and not Q.  In this case, “What I say is true, and the murderers are the same.”  This is a contradiction since we assumed his statement was a lie, so we conclude that E is telling the truth, and the murderers are not the same. 

If F is telling the truth, then F killed A.  Since I’m operating under the assumption that a murderer would not tell the truth, this is a contradiction.  Therefore F is lying.  His statement is (P or Q) -> R.  The negation of this is (P or Q) and not R.  So, “C murdered B, or E is telling the truth.  And I didn’t kill A.”  So we know F didn’t kill A. 

If D is lying then A & B’s statements would be true and C’s statement would be false.  But then both of C’s clauses would be true, which is a contradiction.  Therefore D is telling the truth.

If C is telling the truth, then F is a murderer.  Since F didn’t kill A, then F killed B, and B is telling the truth.  Then the only liars are F and A, and since F didn’t kill A then A killed himself.  But this contradicts the fact that each victim was murdered by another occupant.  Therefore C is lying.  The negation of his statement is “F is not a murderer and D is not lying.”  Therefore F is not a killer, and B is lying. 

If A is telling the truth, then C murdered A.  Since there are two different murderers and B is the only remaining possible killer, then B would have killed himself which isn’t allowed.  So A is also lying and C didn’t murder A. 

So to recap:

- There are two different murderers.
- E and D are telling the truth and are not murderers.
- F is lying, but is also not a murderer.
- A, B, and C are all lying. 

So we have two victims and three possible murderers.  We know C didn’t kill A, so B must have killed A. This is the one with the unique solution.  And B was killed by either A or C, this is the one with two possible solutions.

Edited on February 24, 2020, 1:40 pm

Posted by tomarken on 2020-02-24 13:39:58