The Troll Brothers

There are four troll brothers – Wudhor, Xhaqan, Yijlob, and Zrowag.

  • Wudhor always says the truth.
  • Xhaqan always lies.
  • Yijlob lies or says the truth unpredictably.
  • Zrowag is deaf and never answers.

You must ask these brothers four YES/NO questions (one troll per question), and figure out their names. What questions would you ask?

Source: Puzzling StackExchange

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Responses

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  1. Q: Tell the name of each of you?
    Use this Q to ask 3/4 trolls, then we have 1 time left to ask. There are 4 possiblitities of those 3 trolls (without regards of order):
    WXY
    WXZ
    WYZ
    XYZ
    And 5 possibilities of their answers (True = says the truth about Z; False = lies about Z):
    1. 1T + 2F (WXY): T is W. Done
    2. 2T + 1F (WXY): F is X. Point to one of the 2T and ask X “Is this W? Choose yes or no”. His answer will reveal W. Done
    3. 2F (XYZ): the 4th one is W, ask him. Done
    4. 2T (WYZ): the 4th one is X. Ask similar to case 2. Done
    5. 1T + 1F (WXZ, WYZ, XYZ): T can be W or Y, F can be X or Y
    – If T calls F=X: then F cannot be Y because F=Y => T=W => T must calls F=Y. Thus F is X. Ask similar to case 2. Done
    – If T calls F=Y: ask the 4th one. If he lies then T is W. Done. If he tells the truth then F must be X => T lies about F => he is W. Done
    – Otherwise, if T calls F someone else (not X and not Y): then T must be Y and the 4th one must be W, ask him. Done

    1. I just noticed that the problem was missing the condition that all questions must be YES/NO. Still, you can replace your first question with, for example, “Is Earth round?”. Outcomes 1, 2, 3, 4 seem to be working fine, but can you explain better outcome 5? I don’t quite understand the subcases.

    2. Excuse my bad english
      My question is to ask each troll the names of all 4 trolls. Then in case 5 we consider the answer of T, what name did he assume to F? There are 3 possibilities that he assumed F=X, F=Y, and F=something else.
      – If T calls F=X then F cannot be Y. Because if F=Y leads to T=W and W lies about F (invalid). So F must be X. Ask him and done
      – If T calls F=Y then ask the 4th one
      + If the 4th one lies then we have 2 liars. Done
      + If the 4th one tells the truth then we have 1 liar, that is F. F must be X, so T has lied about F => T must be Y, and the 4th one is W. Done
      – If T calls F=something else, then T has lied about F. So T must be Y, and the 4th one must be W. Done
      If all questions must be yes no then the problem is far harder. I have to rework on it 🙂

      1. I see… But even then, I think it is possible that everyone says the truth about Z but possibly lies about someone else. For example, Xhaqan can say that Zrowag’s name is Zrowag but switch the names of Wudhor and Yijlob. Asking too complex questions allows a liar to lie about a single part of the question but say the truth about the rest.
        Anyway, it is an interesting problem. I need to put some time into it and solve it as well:)

      2. I don’t think X may lies about a part of the answer while the rest is true. His characteristic is to lies, not to cheat nor to fool. If he wants to fool us he can keep silent like Z, and the problem becomes unsolvable
        Y is not the same, his characteristic is to say random and in 1 answer there could be the truths and the lies together

    1. How did you know that one of them is? Best to suppose though, that the question really meant that you had been told the names of the trolls (not in fact stated) and their attendant characteristic (also not stated in the question) and that you had merely to correctly allocate a name to each one.
      Wudhor would say that he was (Wudhor).
      Xhaqan would say that any one of them was, other than Xhaqan, including himself, but were he intelligent he would exclude Zworag from his lying identification.
      Yijlob may or may not tell the truth. If he names Wudhor as Wudhor you would have a strong indication that either he or Wudhor was Wudhor, but you would not know which was which. If he lied and he matched his lie to that of Xhaqan, you would be no better off. If he named Zworag, you might assume he was Xhaqan, once you found the identity of Zworag.
      There is no point asking Zworag anything, as you would have found out had he been one of the first three you questioned. In the end you would only know the identity of Zworag with any certainty and this does not solve the puzzle. Not that I myself know nor, perhaps, does the question setter, given the length of time that has passed since the posting.