A death at a restaurant

Question

You are a mafia leader and because of mysterious recent developments, you had to call the meeting of heads which is happening right now at your's cousin restaurant. The problem is, after the opening toast, Urlich the Unusual who was the key person today fell dead from poison in his glass. It's your turf and to avoid bad name and escalation you have to find who did it.

There are four immediate suspects:

• Sam the Sommelier, who poured the wine
• Tom the Trickster, who sat to the left of Urlich
• Violetta the Violent, who sat to the right of Urlich
• Willy the Waiter, who served the food to Urlich

Certainly all of them will try to avoid getting on your bad side, yet, they also want to appear defiant in the eyes of the others. In other words, at least one claim about another person is true and at least one will be false. This is what they say:

Sam the Sommelier:

Tom did not do anything. And neither Miss Violetta. It was that damn waiter, it is his fault.

Tom the Trickster (replying to Sam):

Yeah, of course, you're just angry at the kid, it was you, you sneaky, tasteless murderer. Me, Viola, sure are innocent. But Willy, come on, he's 15 or 16 or whatever, he didn't kill anybody!

Violetta the Violent (replying to Tom):

\$#%@ Of course he #@\$# didn't! Because it's you ^@#\$@#$, who is true culprit, you %%^@#\$! Leave the %#% staff alone %#\$@#\$, Sam's alright, I'm #\$% certain he's innocent, you %^@#%!

Willy the Waiter (to the narrator, the mafia leader):

I'm sorry sir I didn't react in time, I saw it, I should have told you, it was the lady over there, she put the poison in the glass. Sam is innocent. The gentlemen at the table is innocent too, I'm sorry sir, please don't kill me!

As for yourself, you know that it was not a suicide, and from everyone present only the four suspects could have done it (in particular it could not have been you, the mafia leader). All had reasons not to, but nevertheless, Urlich was murdered. Of course, Sam would whine about Willy, because since the kid was hired, the tips had to be split even more. Naurally, due to his nature, Tom would defend the boy and blame Sam in turn. However, one thing bugged you: you know that Sam and V. detest each other so much that it would be impossible for them complete a single task together, and yet, she still had defended him. This doesn't sound right. What is going on? Who is the culprit?

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Additional clarifications (because comments might be missed):

• the narrator did not do it,
• it is possible that the crime was done by more than one person.

Answer 1

I wrote down their statements symbolically:

S1: !T
S2: !V
S3: W

T1: S
T2: !T
T3: !V
T4: !W

V1: T
V2: !S
V3: !W (missed this the first time)

W1: V
W2: !S
W3: !T

There's not really any obvious starting point, so my new attempt at a solution was more empirical... trying each of the true/false combinations for each statement and finding which resulted in a valid solution (>0 true and >0 false statements for each person.)

To be more explicit, I used a grid like this:

 
            S   T   V   W
 Sam            f   f   t
 Tom        t   f   f   f
 Violetta   f   t       f
 Willy      f   f   t

 

And then tried assigning truth values to each suspect (columns) like increasing binary values (TTTT, TTTF... FFFF), tracking which resulted in valid counts of true and false statements for each person (rows).
For example, the potential solution S, T, !V, !W would look like:

           (t   t   f   f)
            S   T   V   W
 Sam            f   f   t       -> f, t, f
 Tom        t   f   f   f       -> t, f, t, t
 Violetta   f   t       f       -> f, t, t
 Willy      f   f   t           -> f, f, f
 

So, this solution is invalid, because Willy has no true statements. Rinse and repeat for all possibilities...

I ended up with four solutions:

1. Everyone did it
2. Nobody did it
3. Sam and Violetta did it
4. Tom and Willy did it

Based on the storyline,

1 and 2 are nonsensical, and 3 is unlikely (based on the narrators assertion that they would never work together.)

Therefore,

I guess that Tom and Willy are the murderers.

Answer 2

Sam says:  

T innocent, V innocent, W guilty

Tom says:  

S guilty, T innocent, V innocent, W innocent

Violetta says:  

T guilty, S innocent, W innocent

Willie says:  

V guilty, S innocent, T innocent

Everyone's said at least one lie and at least one truth. This doesn't work if we assume a single party's guilt, so (at least) two people are responsible.

Options:
S&T guilty

Doesn't work for Willie - three lies

S&V guilty

Works for Sam - two lies, one truth
Works for Tom - one lie, three truths
Works for Violetta - two lies, one truth
Works for Willie - two lies, one truth

S&W guilty

Doesn't work for Sam - three truths

T&V guilty

Doesn't work for Sam - three lies

T&W guilty

Works for Sam - one lie, two truths
Works for Tom - three lies, one truth
Works for Violetta - one lie, two truths
Works for Willie - two lies, one truth

V&W guilty

Doesn't work for Willie - three truths

S&T&V guilty

Doesn't work for Sam - three lies

S&T&W guilty

Doesn't work for Willie - three lies

S&V&W guilty

Doesn't work for V - 3L

T&V&W guilty

Doesn't work for Tom - four lies

All 4:

Works for all, trivially

So:

S&V or T&W or S&T&V&W all work. We know that Sam would accuse Willie regardless, Tom would accuse Sam and defend Willie regardless, and that Sam and Violetta struggle to work together, precluding it from being both S&V or S&T&V&W. That last part seems to suggest that Tom and Willie are guilty

Answer 3

There seems to be a trivial solution by inspection (unless I've got the wrong end of the stick):

They each claim they are innocent (statement X), and each claim at least one other person is guilty (statement Y), and make some other claims we can ignore

So

Assume all statement Xs are false, and all statement Ys are true, satisfying the preconditions

So

One solution is that all of them are guilty.

Note

I have not shown this is the only possible solution.

However, there phrasing of the question suggests

there is a single solution

in which case

that all of them are guilty must be the only solution

In the comments, it was suggested this is a problem as:

there is a statement suggesting that it could not have been all of them together.

However, this is not necessarily an issue, as the problematic statement is:

"you know that Sam and V. detest each other so much that it would be impossible for them complete a single task together, and yet, she still had defended him."

and

this does not preclude Sam plus others, and V. plus others, separately and successfully working to poison the victim, and even know about each other's plans, i.e. they don't have to cooperate.

Answer 4

As is already confirmed by the OP, the answer is

Tom and Willy committed the murder together

However, none of the answers so far presented seem to catch an important detail that cracked the case for me:

The narrator says that "Sam and V. detest each other so much that it would be impossible for them complete a single task together." This rules out the possibilities that all four committed the murder together and that any three including both Sam and Violetta did it. It also rules out the possibility that either Sam or Violetta is guilty and the other lied to protect them, the prospect of which the narrator indeed found implausible himself. We can conclude, therefore, that Sam and Violetta are each truthful when they say that the other is innocent.

Having determined that, consider:

At least one of Sam's remaining statements must be a lie. They cannot both be lies, however, because then Violetta's statements would all be truthful. Sam's claim that Willy is the murderer cannot be the lie, because that would leave everyone innocent, therefore he is truthful about Willy being guilty. That leaves Sam's lie being that Tom is innocent -- in other words, Tom must be guilty, too.

That last part could also be argued the other way around to reach the same conclusion.

Answer 5

If we only have a single culprit and see the problem strictly logical none of those 4 could have done it.

Why:

Each of the 4 persons is blaming 1 of the other 3 and protecting the remaining 2.

Since each person is lying at least once we can deduce that the person they blame is not the culprit.
If the person they blame were the culprit they would tell the truth about them and would therefore lie about one of the other 2 giving us 2 culprits which should not be possible.

From this we know the following:
Sam blames Willy so it wasn't Willy.
Tom blames Sam so it wasn't Sam.
Violetta blames Tom so it wasn't Tom.
Willy blames Violetta so it wasn't Violetta.

If we leave the strict logic out there is a single point where you can break this:

Tom doesn't exactly say that Willy is innocent in this case.
He siply states that Willy didn't kill anybody.
If we take this as a lie and Willy did kill before (just not this time) then suddenly Tom blaming Sam could be the truth.

Which would mean that:

Sam is the culprit.

Answer 6

I think that:

They all are guilty.
They all state that one of the others is guilty and one is innocent which makes at least a truth and a lie for each of them.

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Previous answer

Here is what I think, but I make some assumptions:
- There is only one culprit.
- 'you' in the text refers to me, the mafia leader (count as a claim about another).
- When someone talk about him/herself it doesn't count as a claim about another person.

Sam claims:

Tom & Violetta innocent and Willy guilty.
if he is telling only one lie:
- T or V is guilty -> impossible (2 guilty persons)
- W innocent
if he tell 2 lies:
- he lies about T & V -> impossible (3 guilty)
- He lies about (T & W) or (V & W)
So Willy is innocent.

Willy claims:

Sam & Tom innocent Violetta guilty
with the same logic as before Violetta is innocent.

Tom claims:

Violetta & Willy innocent & I am guilty with the same logic as before I am innocent (what a relief).

Violetta claims:

I am guilty, Sam is innocent
she lies about me, so she tells the truth about Sam who's innocent.

So:

Tom is the culprit