The Liars' Club
A valuable painting was stolen from the Liars' Club, but the police are having a hard time identifying the culprit because every statement made by members of the Liars' Club is false. Only four members visited the club on the day that the painting was stolen. No two members were ever present at the same time. This is what they told the police:
- Ann : "None of us took the painting. The painting was here when I left."
- Bob : "I arrived second. The painting was already gone."
- Chuck : "I was the third to arrive. The painting was here when I arrived."
- Tom : "Whoever stole it didn't arrive after me. The painting was already gone."
Who stole the painting?
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12 solutions
The negation of "Whoever stole it arrived before me." is "Whoever stole it arrived after me OR at the SAME TIME as me." This shows that Tom could have stolen the painting.
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Not actually though
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What does that mean?
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@Siva Budaraju – I agree. I've submitted a report for you.
I changed the text to "didn't arrive after me" so the negation would properly be "arrive after me".
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Well I still had the same reasoning as Siva's, and the change wouldn't remove the ambiguity, it is true that it is either Tom (arrived second and painting was there) or Ann ( arrived third and painting was not there when she left) Note that in Tom's statement it doesn't indicate if the painting was there when he arrived or left, so the default is when he arrived. This adds the possibility that Tom arrived 2nd and stole it and Ann found it gone still her statement fits Which means that Tom's statement whether negation or affirmation remains ambiguous
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@Haitham Elatrache – "No two members were ever present at the same time."
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@Jason Dyer – indeed, thus it could be himself what I mean is that the statement "before me" or "after me" implicitly assumes the exclusion of the person telling the statement
But it is not just about negation (or is it?). I mean, Tom lied saying that the theif didn't arrive after him. So this can mean either the theif arrived after him OR he was the theif.
I made the same mistake and decided Tom had stolen the painting. Subsequently I realised that arriving at the same time would have made his statement true and therefore not an option.
You just copied the explanation from this site http://www.scientificpsychic.com/mind/liars.html
Please do not copy from sites, or site the source
Please do not copy, or at least, cite the source: http://www.scientificpsychic.com/mind/liars.html
As Hugo Posthuma noted.
This question is WRONG. The opposite of "None of us took the painting. The painting was here when I left." is "At least one of us took the painting. OR the painting was gone when I left."
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Well, they are not joined together with an AND, so they are two separate statements, which are individually both false.
This doesn't really work as a logic puzzle due to the vagueness of the English language. There are multiple statements that could be turned into those lies. For example, Tom's lie "Whoever stole it didn't arrive after me. The painting was already gone." doesn't imply he didn't take it. And Ann's lie "The painting was here when I left." could truthfully be "The painting was gone when I arrived."
Possible answers could be that Tom took it, or Ann took it, or Tom took it and Ann brought in back, and then Chuck took it...
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It is true that the English language is vague and ambiguous at describing logical propositions. However, for the sake of a logic puzzle, we take the simplest interpretation.
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And how do you know which is the simplest interpretation?
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@Paul Sinnett – The idea that someone took the painting, handed it to another member, and then brought it back starts to get into the "anything can happen" type of approach to a math problem. When you had some sort of physical aspect you can always think of some unmentioned exception that would affect it. (To pick a silly example, "Note: Assume space aliens did not shoot down anti-gravity rays in the area of the problem.") Adding every possible exception would make the text unwieldy and less comprehensible to solvers. Even if you think you have all exceptions covered, there is always another.
In this case, simplest reading here means you can assume if a problem implies four people do some task in some particular order, that they are in sequential order.
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@Jason Dyer – However in the question it states that "None of us took the painting." is a lie. There are at least 3 ways that could be a lie: "One of us took the painting," "Some of us took the painting," "All of us took the painting." None of these implies anything goes, and there's no apparent way to say which is simplest or which the question writer had in mind. And that's just the first statement. And it turns what should be a simple logic question into a game of guess the trick (which has very little to do with logic.)
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@Paul Sinnett – "Who stole the painting?" is multiple choice with only one person as a choice, so you can't assume the meaning to be multiple people.
Also, the police are looking for a "culprit" (singular).
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@Jason Dyer – True. But then you're relying on available answers to interpret the question. This is the problem. I did eventually figure out what was required in this question. But I spent more time on that than I did on the question itself.
@Paul Sinnett – One last thought: a version of one of these where there are multiple culprits would be really neat, and you should try writing one!
@Paul Sinnett – Here I (in a "logic problem solving" mindset) take that "None of us took the painting" means "Exactly zero of us took the painting", to which the simplest negation would be "Some number other than zero of us took the painting".
This is what stumped me the most: are we supposed to say both sentences are false for it to be considered a false statement. Or does one false sentence make the statement false.
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They make two statements. There are period marks and no "AND" connecting them. So they must be false separately.
The second statements imply that the painting was still there when Bob and Tom arrived, but was gone when Ann and Chuck left. Thus we establish the partial order (with X = the thief): B , T ≤ X ≤ A , C , which implies B , T < A , C , and the painting was stolen by the second or third person to arrive.
The third person to arrive was not Chuck, so Ann must have arrived third. Bob did not arrive second, so he must arrived first. The order is B < T < A < C . Tom implies that the thief arrived after he did; therefore the thief must be Ann.
Tom's statement that the thief arrived after he did could also mean that it was Chuck, since his visit was also after Tom's (the statement is not conclusive). The reason that it must be Ann is that the painting was already gone when Chuck arrived. However, I like the way you described the solution!
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Your reasoning for Chuck is correct, but I didn't need it. I used the fact that Chuck did not arrive third, and must therefore have been the last to arrive.
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Oh, right, I missed the fact that you wrote: "the painting was stolen by the second or third person to arrive". However, you still need reasoning that it was Ann and not Tom who stole it. So my first reply was indeed wrong, but you would need to specify for your first "equation" that the painting was still there during Bob and Tom's visits, and it was missing during Ann and Chuck's visits. This was probably your thought process for setting up the first equation, but it isn't explained properly...
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@Giaan Kler – Okay, I'll insert an explanation to that effect.
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@Arjen Vreugdenhil – That looks much better. I still think that the last line "Tom implies that the thief arrived after he did; therefore the thief must be Ann" should be changed though, to something like "Ann's statement is the first in the order of visits to indicate that the painting is missing at some point during the visit, therefore the thief must be Ann", since the reasoning with Tom isn't correct.
Tom and Bob lied that the painting was already gone. This means that the painting was still there when the first two arrived. Since Bob lied about him arriving second which means that Bob arrived first and then Tom. So Bob didn't steal the painting. Tom might have stolen it, but he lied that whoever stole it didn't arrive after him, meaning the thief arrived after him. The two persons were Ann and Chuck who lied that the painting was still there meaning it was not there. The painting must have been stolen by the third person to arrive. Since Chuck lied that he was the third to arrive, then Ann must be the third to arrived. She took the painting and lied that it was still there when she left. Then Chuck had to lie that the painting was there when he arrived. Ann stole the painting.
I am totally distracted "every statement made by a member of the Liars' Club is false. "
Yeah Me too
Anne: "None of us took the painting". Anne can only be sure that statement is false if she knows that one of the four members took the painting. The only way she can know that, is if she was the one that took it. Hence by that statement she has incriminated herself.
If you were to look at the reverse statements in a table...
| 1st | 2nd | 3rd | 4th | |
| Ann | ? | ? | ? | ? |
| Bob | ? | - | ? | ? |
| Chuck | ? | ? | - | ? |
| Tom | ? | ? | ? | - |
...you can see that Ann could have been in the club as the first , second , third , and fourth person, Bob could have been in the club as first , third , and fourth and so on.
If we again look at the reverse statements we can see that the painting was in the room...
| 1st | 2nd | 3rd | 4th | Painting? | |
| Ann | ? | ? | ? | ? | "The painting was not here when I left." |
| Bob | ? | - | ? | ? | "The painting was not gone ." |
| Chuck | ? | ? | - | ? | "The painting was not here when I arrived." |
| Tom | ? | ? | ? | - | "The painting was not gone ." |
...after both Bob and Tom had been there. Therefore Bob has to have been there as the first person and Tom would - given that fact - have to be there as second . Chuck would then have to be fourth and hence Ann would be have to be third :
| 1st | 2nd | 3rd | 4th | Painting? | |
| Ann | - | - | X | - | "The painting was not here when I left." |
| Bob | X | - | - | - | "The painting was not gone ." |
| Chuck | - | - | - | X | "The painting was not here when I arrived." |
| Tom | - | X | - | - | "The painting was not gone ." |
With that information, Tom's reverse "Whoever stole it did arrive after me."-statement and Ann's reverse "The painting was not here when I left."-statement should make it clear that:
Ann stole the painting .
Nice work by creating an elimination grid !
Truthfully the second sentence is a giveaway and logic is not required. If we look at the second parts of the sentences:
- Ann: The painting was not here when I left. -> I took the painting and left
- Bob: The painting was not gone -> I saw the painting
- Chuck: The painting was not here when I arrived. -> I never saw the painting
- Tom: The painting was not gone. -> I saw the painting
Of course this is a preliminary inference and is only true because no 2 members were present at the same time. As the painting was not there when Ann was leaving then only she could have stolen the painting as no one else was in the club beside her. Thus Ann was the thief.
"Ann: The painting was not here when I left. -> I took the painting and left " This doesn't follow at all. It could have equally have already gone when she arrived.
Since all of them are liars, take the negation of every statement that they ever said. This tells us the following:
•The painting wasn't there when Ann left.
•Someone took the painting.
•Bob didn't arrive second.
•The painting wasn't gone when Bob came.
•Chuck didn't arrive third.
•The painting wasn't there when Chuck arrived.
•Whoever stole it arrived after Tom.
•The painting wasn't gone when Tim arrived.
Then, eliminate the suspects you know aren't guilty. From the statements above, Tom couldn't have stolen it because the person who stole it arrived after Tom. Chuck also isn't guilty because the painting wasn't there when he arrived. This leaves Ann and Bob.
From there, we know Bob didn't arrive second and the painting wasn't gone when he did. It's most likely that he came first since the painting being there signifies an early arrival. We also know Chuck wasn't third to arrive and that the painting was gone when he did, so it would be reasonable to assume he came fourth. Since the person who stole the painting arrived after Tom, this is the order of people who came:
-
Bob
-
Tom
-
Ann
-
Chuck
After verifying this order by going through the statements and noticing that the person who stole the painting arrived after Tom and that the painting was gone when Chuck arrived, it is clear that Ann is the thief.
Can the wording be changed in the prompt to say, "every statement made by [the or these] members of the Liars' Club is false." as it ambiguous as it can be interpreted to mean one of the four members is lying.
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You have a point. I have edited the wording.
Information obtained:
Ann --> One of them was the thief. Painting was absent.
Bob --> Arrived as 1 or 3 or 4. Painting was present.
Chuck --> Arrived as 1 or 2 or 4. Painting was absent.
Tom --> Thief arrived after him. Painting was present.
Data from Tom:
Bob is 1st. Tom is 2nd.
Thief would say the painting was absent.
So it should be Ann or Chuck.
Data from Chuck:
He arrived last.
He did not see the painting.
Data from Ann:
He arrived 3rd.
But did not see the painting.
So the thief is Ann.
I don't know why people are going so mathy about this question...
Let us break the question statement by statement... Keep in mind that every statement they make is false. So read the statement and negate it. Ann : Painting ( Not present ) and ( The other statement made by Ann is irrelevant as we know someone obviously took the painting ) Bob : 2nd ( False ) and Painting ( Present ) Chuck : 3rd ( False ) and Painting ( Not Present ) Tom : 4th ( False ) and Painting ( Present )
Now from the above statement let us try to deduce the order in which things happened. Logically
Basically, the people who saw the painting when they arrived have to be on top of the list. Bob was 1st, Tom was 2nd, Chuck was fouth but when he came he saw no painting... Leaving us with Ann.
"When you have eliminated the impossible, whatever remains, however improbable, must be the truth" - Sir A. C. Doyle
Converting the liars' statements to true statements:
Ann: One of us stole the painting. It was gone when I left.
Bob: I didn't arrive second. The painting was still here when I arrived.
Chuck: I didn't arrive third. The painting was already gone when I arrived.
Tom: Whoever stole the painting arrived after me. It was here when I arrived.
From these statements, clearly neither Chuck or Tom stole the painting, so it was Ann or Bob. Moreover, the thief must have been the second or third visitor.
If it was Bob, he arrived third. But this is impossible, since then both Ann and Chuck would have both arrived after he did.
Therefore it was Ann .
Furthermore, we can readily determine the order in which the four visited the club: Tom, Ann, Bob, Chuck.
I solved this by analyzing who has suspect. Bob and Tom obviously has no suspect since the painting was still there when they arrived. Chuck also has no suspect since the painting is gone when he arrived. This leaves with Ann, which makes sense since the painting is gone when she left, which leaves time for her to steal.
Going through all the statements systematically, it's clear Ann stole the painting. Bob and Tom had to be consecutive visitors to the club. This is because their statements imply that the painting was present when they were at the club, as they always tell lies, and they claim that the painting was already gone, so it can be inferred that it was still there. It's not logically possible for the thief to have arrived between Bob and Tom, otherwise Bob or Tom would not have seen the painting. This means they were consecutive visitors, as I stated above. The only logical conclusion is that Bob and Tom were V1 and V2 to the club.
Ann and Chuck, therefore, where also consecutive visitors. If Chuck's statement implies, using the same logic as above, that the painting was still there when he left, then that would mean that he could not have stolen the painting. As neither Bob nor Tom could have stolen it, given that they saw the painting when they left, the only logical suspect is Ann.
Relevant wiki: Truth-Tellers and Liars
Since every statement is false, let us convert them into true statements, and number each statement:
Ann: 1) One of us took the painting. 2) The painting was gone when I left.
Bob: 3) I arrived first, third, or fourth. 4) The painting was still here.
Chuck: 5) I arrived first, second, or fourth. 6) The painting was gone when I arrived.
Tom: 7) Whoever stole the painting arrived after me. 8) The painting was still here.
According to statement #7, Tom is not the thief. #8: Since the painting was there when Tom arrived, he could not have been the last to arrive. Tom must have gone there first, second, or third. #6: The painting was gone when Chuck was there, so he didn't arrive first. #5: So Chuck got there second or fourth. #4 and #8: As two other members (Bob and Tom) arrived to see the painting, Chuck didn't get there second, either. So Chuck arrived fourth. #3: This means Bob arrived first or third. #2: Since the painting was gone when Ann left, she didn't arrive first. Otherwise, no member after her would have seen the painting. So Ann went there second or third and Chuck arrived fourth. But since two other members (Bob and Tom) saw the painting when they arrived, Ann didn't go there second, either. So Ann arrived third. #3: Therefore, Bob arrived first, and Tom arrived second.
In summary, Bob arrived first. Tom got there next and the painting was still there, so Bob was not the thief, and neither was Tom. When Ann arrived, the painting was still there, but it was gone when she left. So Ann was the one who stole the painting. Chuck arrived last and discovered that the painting was gone.