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# The Thief in 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 a member of the Liars' Club is false. Only four members visited the club on the day that the painting was stolen. 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 the painting arrived before me. The painting was already gone.

Who of these four liars stole the painting?

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.

Marc Cramer had the following comment:

In the answer for "The Thief in the Liars' Club", the negation of Tom's statement "Whoever stole the painting arrived before me." was given as "Whoever stole the painting arrived after me." So, using this statement, you disqualify Tom as the thief.

This is not correct. The negation of the statement should be "Whoever stole the painting did not arrive before me." Tom could have stolen it himself.

Jason Karns had the following comment:

I disagree with the reasoning behind the solution for the Liars' Club puzzle. It requires one of two assumptions in order to narrow it down to 1 thief, and the assumption taken by the solution seems to be the less reasonable of the two. First, using the list of deductions below, without any assumptions, we can narrow down the arrival order to three, using initials for each person:

```ATBC
TABC
BTAC
```

This ordering reveals that the thief is either Ann or Bob. In order to charge Ann with theft, the assumption MUST be made, that no more than one person can be at the Liars' Club at once. For it is possible that Ann arrives first or second, and Tom arrives first or second. Then Bob (who steals the painting) and then Chuck. This ordering supports all of the statements made (once converted to truths), assuming that Ann then leaves after Bob (which is not an unreasonable assumption seeing that it is a club).

Further, to charge Ann with theft would imply that an item is 'gone' at the time a theft is in progress. I would argue that saying an item is gone at the time a person leaves implies that it is gone 'before' they leave and that it is not their 'going' that makes the item 'gone'. If so, then at the time they leave the item is still technically 'there' (just not in its original place, but still on the premises.) It is not until the person is gone that the item is gone as well.

For these reasons I believe that of the two assumptions needed, the posted solution takes the less reasonable of two assumptions. Without any assumptions, we can deduce no further than that either Bob or Ann steals the painting.

Deductions:

Ann: The painting was gone when I left.
Bob: I didn't arrive second. The painting was still here when I arrived.
Chuck: I wasn't the third to arrive. The painting was gone when I arrived.
Tom: Whoever stole the painting arrived after me. The painting was still here when I arrived.

Possible arrival orderings:
ATBC (Thief is Bob)
TABC (Thief is Bob)
BTAC (Thief is Ann)

Thief is either Bob or Ann.

Not possible because Bob didn't arrive second:
ABCT
ABTC
CBAT
CBTA
TBAC
TBCA

Not possible because Chuck didn't arrive third:
ATCB
BACT
BTCA
TACB

Not possible because Chuck had to arrive after Bob (The painting was gone when Chuck arrived. The painting was there when Bob arrived)
ACBT
ACTB
CABT
CATB
CTAB
CTBA
TCAB
TCBA

Not possible because stealer arrived after Tom:
BCAT

Not possible because Chuck had to arrive after Tom (painting still there when Tom arrived, but gone when Chuck arrived):
BCTA

Not possible because stealer arrived after Tom, but painting was gone when Chuck arrived.
BATC