# A death at a restaurant

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?

• the narrator did not do it,
• it is possible that the crime was done by more than one person.
• I hope this isn't based on which side Ulrich's glass on because there's no solid reason why it would have followed whatever social norm is in effect wherever they live. – Engineer Toast Aug 9 '16 at 11:55
• @EngineerToast No, it doesn't matter which side Urlich's glass was/is. That description is there just to make the puzzle nicer and not so sterile. – dtldarek Aug 9 '16 at 12:02
• This means we don't know exactly how many truth or lies each person spoke, right?? – Sid Aug 9 '16 at 12:14
• Only a single culprit or multiple? – The Dark Truth Aug 9 '16 at 12:15
• It was obviously Sansa and the dwarf! – DasBeasto Aug 9 '16 at 19:21

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.

• Good approach. Mistake: V says W innocent in her first sentence (refers to W as 'he'). – dtldarek Aug 9 '16 at 13:23
• just realized there's another statement of Violetta's that I missed. whoops, updating... – user812786 Aug 9 '16 at 13:24
• @dtldarek added that in, still had multiple answers but am I justified in the story-based assumptions? – user812786 Aug 9 '16 at 13:52
• Yes. The reason that 3 is not a possibility is correct, and thus also 1 cannot be true. 2 isn't true because there is a clear statement it is not a suicide and otherwise it doesn't make sense. Thus, only one answer. Good job. (You are the first one, but I will still wait a bit with an accept, esp. your answer has little "working it out" content.) – dtldarek Aug 9 '16 at 13:59
• Fair enough, I will try to explain it a bit more. – user812786 Aug 9 '16 at 14:09
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

• V says W innocent in her first sentence (refers to W as 'he'). – dtldarek Aug 9 '16 at 13:19
• So she does. Hmm... – LogicianWithAHat Aug 9 '16 at 13:51
• I hope this is the correct answer , same conclusion reasoning I came up with as well. – KBusc Aug 9 '16 at 14:08
• What about 3 or 4 of them working together? – dtldarek Aug 9 '16 at 14:30
• @dtldarek You've suggested that it's not all 4. I haven't checked the 3 person cases yet, though – LogicianWithAHat Aug 9 '16 at 14:34

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.

• There is a statement which implies that it could not have been them all together. – dtldarek Aug 9 '16 at 14:08
• @dtldarek - if it's the statement I think it is, that need not be an issue (see edit). – abligh Aug 9 '16 at 14:12
• +1 I have to admit that you are correct that my wording doesn't exclude two groups poisoning Urlich independently, but I will call that hair splitting. Nevertheless, even if it is not the intended solution, I like it very much! – dtldarek Aug 9 '16 at 14:19

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.

• I'm really glad you have posted this answer, because when constructing the puzzle that was what I intended in its first iteration. However, after rereading my story I thought that it is not clear enough, and also why then S. and V. didn't accuse themselves out of spite? Fortunately even without this inference the puzzle is still solvable, so I needed there only a few minor tweaks – the statement could be left for interpretation as it doesn't change the answer either way. – dtldarek Aug 10 '16 at 11:35

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.

• I'm not sure, but I don't see that Sam blames Willy. I think he blames 'you' so the mafia leader. He doesn't state anything about Willy. – Sechiro Aug 9 '16 at 12:39
• @Sechiro I think the "you" in their personal speeches is always directed at the previous speaker. Also makes more sense since Tom throws his blames at his "you" and below their speeches it states that Tom blames Sam. – The Dark Truth Aug 9 '16 at 12:42
• That's why it was doing of at least two of the suspects. – dtldarek Aug 9 '16 at 13:04

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.

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

• OP tells to assume that multiple people are involved.... Thus, your logic, though sound, might not be correct..& as @TheDarkTruth pointed out, You intends to the previous speaker. – Sid Aug 9 '16 at 12:44
• Should I delete the answer now the edit of the question states assumptions are false? – Sechiro Aug 9 '16 at 12:58
• Nice reasoning, but I intended for multiple culprits to be possible. Instead of deleting the answer, you can just make a disclaimer at the top saying that these assumptions are false. – dtldarek Aug 9 '16 at 13:00
• There is a statement which implies that it couldn't be them all together. – dtldarek Aug 9 '16 at 13:07