# The thief, liars, and the honest

There are four people: A, B, C, D. One person is a thief, the thief and two other people are lying. The other person is telling the truth. Who is telling the truth?

• A: I am not the thief.

• B: D is the thief.

• C: B is the thief.

• D: I am not the thief.

• Ok everyone, I think I've waited long enough, pretty much all of you were correct, so here is the answer. A is the thief. D is telling the truth. B and C are lying but they are innocent. Nov 23 at 6:26

Even simpler solution, with no hypotheticals needed:

B and D precisely contradict each other, so one of them must be the single truth-teller. So the others are both liars; in particular, A is lying, so A is the thief. Knowing that, we can easily check all the statements and see that D is the truth-teller.

Simple solution.

If A is telling the truth, then A is not the thief by A's statement. Since B and C are lying, D and B cannot be the thief by B's and C's respective statements, and thus C is the thief. However, this is a contradiction, as D also lies, and thus D is also the thief by D's statement. So A is not the truth-teller. Thus, A is lying, and thus A is the thief by A's statement. This means B and C are lying by their respective statements, and D is the truth-teller.

D!

• There's a lot of solutions. Oct 30 at 0:23

Suppose D is lying. Then D is the thief and so A and B are both telling the truth, but there can only be one truth-teller.

Therefore

D is telling the truth.

• This is a lovely simple answer. Oct 29 at 9:31
• Good point that we don't need to know who's the thief, just who is telling the truth! Oct 29 at 23:41

Only the truth-teller or the thief can say "I am not the thief". Two people (A and D) said that, so those two roles are both accounted for between them. This means B and C must both be the only role that's left, liars. Since B is lying, D can't be the thief and so must be the truth-teller.

Another line of thought:

Setting either B or C as truth-tellers implies that both A and D are liars. However, there is only one thief and A and D are saying the same thing: "I am not a thief". If they both were indeed lying, there would be two thieves. Therefore, either A or D is telling the truth.

If A is telling the truth, the true statements would be

A: I am not the thief (the assumed truth)
D: I am the thief (ok, does not contradict with A's statement)
B: D is not the thief (invalid, contradicts with D's statement)

If D is telling the truth, the true statements would be

D: I am not the thief (the assumed truth)
A: I am the thief (ok, does not contradict with D's statement)
B: D is not the thief (ok, reinforces D's statement)
C: B is not the thief (ok, reinforces A's statement)

Therefore,

D is telling the truth.

Another line of reasoning:

A and D cannot both be lying because that would require there to be two thiefs. One of them must therefore be the truth-teller, and the other (because they lie when they claim otherwise) must be the thief.

We can discriminate between those because in that case,

by elimination, B and C are the (ordinary) liars, and in particular, B's claim that D is the thief is a lie.

That means

A must be the thief, and D the honest person.

?Even simpler solution:

Maybe A is the thief. Then they're lying, so their statement is consistent, and two of the remaining three will be lying.
If A is the thief, then B's statement must be a lie. Fine.
If A is the thief, then C's statement must be a lie, in which case D must be telling the truth.
D's statement is consistent with all of the above, so, supposing there is a unique solution to the problem, we have found it already, and D is the truth-teller.

• I'd call this "The Pessimist's solution" Oct 29 at 23:39
• Re "supposing there is a unique solution to the problem", see puzzling.stackexchange.com/q/49557/34791 Oct 30 at 5:46

A and D make the same claim. They cannot both be telling the truth (since there is only one truth-teller), and they cannot both be lying (otherwise there would be two thieves). Thus, either A or D is the truth-teller while the other is lying.

If A is telling the truth, then D must be lying. This would imply that D is the thief. However, B also claims that D is the thief, which would make B's statement true, and he'd be a second truth-teller. Since there are not two truth-tellers, this situation does not work.

Thus, D is the truth-teller; A is the thief (though this proof is not required); and B and C are regular liars.

Another simple solution:

A and D both say they are not the thief. Since there is only one thief, one of them must tell the truth. If A tells the truth --> D is the thief --> B is telling the truth --> 2 people tell the truth --> conflict with precondition. Therefore, D must tell the truth, and A is the thief.