# Who stole the jewels?

Maybe you guys can help me with a puzzle.

Only one of the following is telling the truth. Who stole the jewels?

Frank: Marcus didn't steal the jewels.

Marcus: Leon didn't steal the jewels.

Leon: Marcus and Sylvester were at the bar together when the jewels were taken.

Sylvester: Leon stole the jewels

My guess:

I think the answer is that either Marcus and Leon or just Marcus stole the jewels. If what Frank is saying is true then either Marcus or Sylvester must also be telling the truth and only one person can tell the truth. If Leon is telling the truth you have the same problem and Frank must also be telling the truth. If Marcus is telling the truth then he must be a thief while Leon isn't because Frank is lying and Sylvester is lying. But there are no contradictory statements there. Same with Sylvester, if his statement is true then Leon must be a thief and Marcus must be a thief as well but there are no contradictory statements. Perhaps I'm missing something.

The solution can easily be obtained using a truth-table.

Let's symbolize using the following atomic sentences:

M: Marcus stole the jewels

L: Leon stole the jewels

S: Sylvester stole the jewels

Assuming "Marcus and Sylvester were at the bar together when the jewels were taken" implies that that neither of them stole the jewels, the relevant assertions of Frank, Marcus, Leon, and Sylvester become:

Frank: ~M

Marcus: ~L

Leon: ~M & ~S

Sylvester: L

where '~' represents negation and '&' represent conjunction.

We can now look at all possible assigments of Truth and Falsity to the atomic sentences and how these assignments affect the possible truth-values of the complex sentences asserted by Frank, Marcus, Leon, and Sylvester:

Because only one person can be telling the truth (ex hypothesi), we want to look at rows where only one of Frank, Marcus, Leon, or Sylvester says something true. Rows 1, 2, 5, and 6 are the only such rows.

That there are multiple rows indicates that your question is poorly constrained: There are multiple circumstances consistent with only one person telling the truth and in which possibly multiple people steal the jewels.

In case one, Sylvester is telling the truth that Leon stole the jewels, and in fact, Leon, Marcus, and Sylvester all stole the jewels.

In case two, Marcus is telling the truth that Leon did not steal the jewels, and in fact Marcus and Sylvester both stole the jewels.

In case five, Sylvester is telling the truth that Leon stole the jewels, and in fact Leon and Marcus both stole the jewels.

In case 6, Marcus is telling the truth that Leon didn't steal the jewels, and in fact only Marcus stole the jewels.

If we assume that the jewels were stolen by only one person, case 6 is the only option.

If we assume that Leon's statement that "Marcus and Sylvester were at the bar together when the jewels were taken" says nothing about whether or not they stole the jewels, we end up with the following truth-table:

where Q stands for Leon's irrelevant statement. In such a situation, only cases 5 and 6 are ones in which a single person is telling the truth.

In case 5, Sylvester is telling the truth that Leon stole the jewels, and in fact Leon and Marcus both stole the jewels.

In case 6, Marcus is telling the truth that Leon did not steal the jewels, and in fact only Marcus stole the jewels.

You will notice that Marcus steals the jewels in every one of these possible scenarios, although he is not necessarily the only one!

• In the first case I don't think that Leon's statement being false necessitates that Sylvester and Marcus stole the jewels, it just says that they weren't at the bar. Dec 7, 2016 at 5:30
• I think this answer overcomplicates an otherwise simple problem.
– user20
Dec 7, 2016 at 7:18
• @Emrakul I disagree, because only this answer shows that the problem is under-specified. This answer shows that your answer is only partially correct even though it contains no logical errors (but we all know that logical consistency is necessary but not sufficient for a true inference). Dec 7, 2016 at 11:57
• Those graphics desperately need (a) numbers for the rows you are referring to and (b) shading to show some properties of the rows so it becomes easy to follow along with your explanation. Mar 6, 2017 at 18:40

Either Marcus or Sylvester is the one telling the truth, because Leon can't both have stolen and not stolen the jewels. One of the two must be lying, and the other one must be telling the truth.

As a result, you know that Frank is lying, because either Marcus or Sylvester is telling the truth, and there is only one truth-teller. Therefore, because Frank says Marcus didn't steal the jewels, and Frank is a filthy liar, Marcus must have stolen the jewels.

• This was my solution as well. Simple and elegant. Dec 7, 2016 at 8:21
• Yup, that was my solution too. And for the record: Leon's statement does not even matter, because the jewels could have been at the bar. Dec 7, 2016 at 9:46
• This answer is based on the (not given!) fact that only one person stole the jewels. If this can be assumed, this answer is great. Otherwise it lacks completeness.
– Alfe
Dec 7, 2016 at 15:55
• This is great, I was trying to wrap my head around it, and then Alfe said (not given!) and boom it all made sense. Dec 21, 2016 at 12:37

Both Frank and Leon say Marcus didn't steal the jewels. One of these must be a liar, so Marcus did steal the jewels.

Assuming everybody lies, the first two statements already conflict. If both were lies, both Marcus and Leon were guilty. So one of the first two statements must be true, which leaves only Marcus and Leon as suspects.

The last two statements must be false, so it was not Leon and it must have been Marcus or Sylvester.

This leaves Marcus as the only solution.

Answer of the Quiz can be explained by the following diagram :)

Sorry my bad English :B How only one person is telling the truth, and Marcus and Sylvester said opposite answers:

Marcus: Leon didn't steal the jewels.

Sylvester: Leon stole the jewels

Or Marcus or Sylvester told the truth, so, we can know that Frank and Leon told a lie.

If Frank told a lie, Marcus stole the jewels!