# Four Doors, One Escape

You are trapped in a room with four adjacent doors upon a wall. On each door is one of the first letters of the alphabet, in order, from left to right. You know that one of the doors leads to freedom, while the other three will immediately plunge you into the abyss. You then hear four voices...

From door A you hear a voice say, "Aringarosa is on the edge. Alithea's door is the correct way.”

From the next door, door B, you hear someone declare, “Veritas is not next to Aringarosa. Veritas is not on the edge.”

From door C another voice murmurs that, “Aringarosa is not on the edge. The correct door is next to Veritas.”

And again, from door D you hear a new voice proposing that, “Aringarosa is not on the edge."

Although you do not know who is who, you have already been told that:
-Veritas and Alithea always speak the truth.
-Dolos always speaks the opposite of the truth.
-Aringarosa speaks without regard to lies or truth.

Which door is the correct way?

Taking "the correct door is next to X" to mean "the correct door is next to X's door", note first of all that

C and D both say Ar is not on the edge. Suppose Ar is on the edge. Then they're both saying something false, hence one is Do and the other is Ar. Specifically, Ar (being on the edge) must be D and Do must be C. Now A,B are our two truthtellers in some order. B says Ve is not on the edge so A is Al and B is Ve; our order is A:Al, B:Ve, C:Do, D:Ar. But this doesn't work, because the correct door is next to Ve as C says it is, but C is Dolos who always lies.

So,

we have found that Ar is not on the edge: C and D are either our two truthtellers or one of the truthtellers and Ar. Let's consider the second of those possibilities: Ar is then one of C,D and also not on the edge, so C is Ar and D is a truthteller. A says something false (that Ar is on the edge) and is not Ar (because C is Ar), so A is Dolos and B is a truthteller. But this doesn't work, because Ve is either B or D, hence is next to C, hence is next to Ar, so B has said something false despite being a truthteller.

The only remaining possibility is:

Ar is not on the edge, C and D are the two truthtellers; hence Ar is B and Do is A. We learn from C that the correct door is next to Ve and from A that it isn't Al; so Ve is C, Al is D, and the correct door is B.

Conclusion:

The correct door is B. (And the order is A:Dolos, B:Aringarosa, C:Veritas, D:Alithea.)

[Note: There was an entirely different earlier version of this answer, before OP made a correction to the question. And then another one, before OP made another correction. Third time lucky?]

• yup. third time's a charm. May 19, 2019 at 21:26