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A truth teller's Love Triangle


Here is the story of three unhappy people: Angelica, Bernardo, and Cameron. Angelica and Cameron have been friends since childhood. Cameron is hopelessly in love with Angelica, but Angelica has always thought of Cameron as "just a friend".

During high school, a new boy, Bernardo, moves into the area. Bernardo immediately catches the attention of Angelica, who falls head-over-heels for him. However, Bernardo is not interested in women--he is strongly attracted to Cameron. Cameron, of course, is jealous of Bernardo, because he has stolen Angelica's love away. Angelica is angry at Cameron, because she feels that Bernardo's lack of attention to her is Cameron's fault. Bernardo is jealous of Angelica, who recieves all of Cameron's attention. What are these three to do?

(this is a rhetorical question--there is no need to answer it)

One day, you meet them, all together in a chat room. They are all using nicknames: Uberkewl, Vaxxipaxx, and Willywutang (in no particular order). Because they are so jealous and screwed-up, each of them will only answer a question truthfully only if:

  1. one of your last two questions was to their sweetheart, and
  2. your last question wasn't to the person with whom they are upset.

Otherwise, they will answer spitefully--giving you the answer that will confuse you the most.

Your task is to figure out who is who. Is it possible? If so, how many questions might you have to ask?

And of course you cannot ask them for their name, gender, or where they are from.

Edited by BMAD
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Assuming they won't lie and give correct answers even if they're not compelled to tell the truth.

6 Questions

Ask U Two Questions that don't matter.
Ask V if the sky is blue.
If he tells the truth the U < V < W otherwise U > V > W where > denotes love.
then ask the person who loves them if they are Bernado.
Once someone answers truthfully, who ever they love is Cameron and the remaining person is Angelica.
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I believe that a spiteful answer could be truthful if the person you're asking thinks that telling the truth will screw up your strategy. (Otherwise the problem would be rather trivial.) I'll also assume that you can't pull shenanigans like "Ask both V and W such-and-such a question simultaneously, and then ask U whether the sky is blue, and then you can be assured that U must answer the next question truthfully." But correct me if I'm wrong about that.

The best I can do is determine one person's identity. And not the person of my choosing, just some person out of the three.

There are initially six different ways to distribute A, B, and C's identities among U, V, and W. You can narrow those six down to two.

Ask V what time it is twice.

Ask U what V's and W's names are.

Ask W what time it is (you only need to ask once at this point).

Ask U what V's and W's names are.

One of those two sets of identities for V and W (and by process of elimination U) must be true, so you've narrowed things down from six possibilities to two.

Unfortunately, if U is really trying to confuse you with spiteful answers, he/she can do so. Suppose U is A, V is B, and W is C. U must have answered truthfully the first time around (when you asked V first) and have given you the correct identities -- in this scenario U loves V, V loves W, and W loves U. When you ask U for identities after talking to W, U would know that he would have to give you a set of identities where U is in love with W or else you would easily be able to tell that he's answering spitefully. So it must be one of {U=A, V=C, W=B} or {U=B, V=A, W=C} or {U=C, V=B, W=A}. Whichever one of those possibilities he picks, there must be one person out of {U, V, W} who was assigned to the same identity out of {A, B, C} on both of U's responses, and since one of the two responses must be true, you could know for sure that the person with the same identity in both of U's answers really is that identity.

However, after getting to that point, I see no way of succeeding if everyone gives spiteful answers by responding as if whichever false arrangement U proposed is the real arrangement. For that matter, no matter what strategy you choose, I can't see a way of preventing them from choosing an alternate arrangement that they would all use as a basis for spiteful answers that would be indistinguishable from the truth, as long as their alternate arrangement has the reverse of the true love-hate polarity of each person's relationship.

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I ask U whether V has a crush on them. Since I have not asked U's sweetheart a question yet, U must answer spitefully (untruthfully). If U says no, then in fact U is V's sweetheart (and V is W's and W is U's) and V will answer my next question (What is U's name?) honestly, from which I can deduce everyone's identity. On the other hand, if U says yes, then U is not V's crush; therefore W is V's crush, V is U's crush, and (more importantly) U is W's crush. So in that case W will answer my next question honestly (What is U's name?) from which I can deduce everyone's identity.

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Maybe I am misinterpreting the OP, but...

Upon entering the chat room the very first two questions directed to the same person must draw spiteful responses. Therefore, ask U:

1) Do you love V?
2) Do you love W?

And what is a spiteful response in this situation, but lying?
This way we establish love/hate relations between U, V, and W. Thereafter, we could just ask the one who loves U about the true identity of U -- and the mystery is solved.
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If a spiteful answer cannot be factually correct, then the problem is easy.

If a spiteful answer can be correct as long as the person thinks they can confuse you by giving a correct answer, AND if you're not allowed to do stuff like address a question to multiple people simultaneously, or ask a question using a real name instead of a username (for example, ask "O Angelica, where art thou?" and then ask Uberkewl "Who is Cameron?" knowing that the previous question went to Angelica), then I suspect that it's not possible to determine everyone's identity.

My suspicion of impossibility is based on what looks like an unbeatable strategy to conceal the truth. Suppose (without loss of generality) that Angelica is Uberkewl, Bernardo is Vaxxipaxx, and Cameron is Willywutang. If you ask a question and they are forced to tell the truth, then they will of course tell the truth. However, if they are not forced to tell the truth, then they will instead consider the alternate scenario where Angelica is still Uberkewl, but Bernardo is Willywutang and Cameron is Vaxxipaxx. If they would be forced to tell the truth if this alternate scenario were true, then they will tell what would be a truthful answer under that alternate scenario. If they would not be forced to tell the truth under either scenario, then they will just answer with "lol".

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