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It's a little-known fact that the guards on Liar Island have to continue their slavish devotion to the individual rules dictating the veracity of their statements. Of course, some always lie, and some always tell the truth, but most have a little more nuance.

I had escaped from a previous encounter (you'll have read of my exploits in numerous posts) and was journeying across the island to the port to catch a ship home. But it grew dark, and I cautiously approached a fire, around which sat three guards. I could tell they were guards because they were still in uniform and had name tags.

Art: Welcome! I'm Art, this is Tony, and this is Xavier.
(I could tell from their name badges that this was true)
Tony: I always lie.
Xavier: That's not true.
Tony: Is too!
Art: Tony's quite right.
Tony: Wait! That's not true.
Xavier: Tony! You're lying in front of our new friend!
Tony: No, I wasn't.
Me: Well, it really is a pleasure to meet you.
(Awkward pause)
Art: Ha! Tony and Xavier don't know what to say to that!
Me: So, you're Art, you're Tony, and you're Xavier.
Xavier: You got it in one!
Tony: Art was wrong. We did know what to say.
Xavier: Don't listen to him, kid. We really didn't. And don't listen to whichever of the three of us speaks next either. It's going to be a lie.
Art: There's no way you could know that.
Tony: Well, any of the three of us would have told you the truth now, that's for sure.
Art: Not true this time. I'm the only one who'd give it to you straight. But whoever speaks next is going to lie.
Xavier: I've already displayed all four of my conditions.
Tony: No he hasn't. And I'm the only one who'd have told you that.
Art: He has displayed all his conditions.
Xavier: Okay, fine, I haven't displayed them all.
Art: Actually, he has now. He just hadn't previously. He said it wrong both times.
Xavier: Well, the other two have simpler rules. Mine is a mixture of both of theirs in a way.
(I'd figured it out by now)
Me: I'm guessing any of you is going to lie to me next.
Art: Well, it's been a wonderful conversation.

Can you tell which statements are true and which are false? And, more importantly, what are the rules dictating the guards' statements?

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1 Answer 1

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Here's what I've got assuming the guards know each other's rules:

  1. Art: Welcome! I'm Art, this is Tony, and this is Xavier. (I could tell from their name badges that this was true) TRUE -- as given.

  2. Tony: I always lie. FALSE -- If true it would be nonsense.

  3. Xavier: That's not true. TRUE -- If (2) was true it would be nonsense.

  4. Tony: Is too! FALSE -- If (2) was true it would be nonsense.

  5. Art: Tony's quite right. FALSE -- If (2) was true it would be nonsense.

  6. Tony: Wait! That's not true. TRUE -- Tony realizes he was wrong.

  7. Xavier: Tony! You're lying in front of our new friend! FALSE -- (6) is true.

  8. Tony: No, I wasn't. TRUE -- (6) is true.

  9. Me: Well, it really is a pleasure to meet you. (Awkward pause) UNKNOWN -- May be lying or telling the truth.

  10. Art: Ha! Tony and Xavier don't know what to say to that! TRUE -- Tony's pattern is to lie when the previous speaker told the truth and to tell the truth when the previous speaker lied. At this point Xavier's pattern appear to be something similar and will be shown later. The main point is that the guards don't know if (9) is true.

  11. Me: So, you're Art, you're Tony, and you're Xavier. TRUE -- From (1)

  12. Xavier: You got it in one! TRUE -- From (1)

  13. Tony: Art was wrong. We did know what to say. FALSE -- From (10)

  14. Xavier: Don't listen to him, kid. We really didn't. And don't listen to whichever of the three of us speaks next either. It's going to be a lie. TRUE -- Tony's pattern is to lie when the previous speaker told the truth and to tell the truth when the previous speaker lied. Xavier's pattern is the similar at this point. Art's pattern is to alternate telling the truth with lying.

  15. Art: There's no way you could know that. FALSE -- Art's pattern is to alternate telling the truth with lying.

  16. Tony: Well, any of the three of us would have told you the truth now, that's for sure. TRUE -- All have told a truth at this point.

  17. Art: Not true this time. I'm the only one who'd give it to you straight. But whoever speaks next is going to lie. TRUE -- Art's pattern is to alternate telling the truth with lying.

  18. Xavier: I've already displayed all four of my conditions. FALSE -- From (17). A clear pattern for Xavier has not been shown.

  19. Tony: No he hasn't. And I'm the only one who'd have told you that. TRUE -- From (18) and Tony's pattern is to tell the truth when the previous speaker lied. It's Art's time to lie. Xavier's pattern is not fully apparent at this point but will be shown later.

  20. Art: He has displayed all his conditions. FALSE -- Art lies. Xavier's pattern is not fully apparent at this point but will be shown later.

  21. Xavier: Okay, fine, I haven't displayed them all. TRUE -- Xavier's pattern is not fully apparent at this point but will be shown later.

  22. Art: Actually, he has now. He just hadn't previously. He said it wrong both times. TRUE -- Art's time to tell the truth because his previous response was false.

  23. Xavier: Well, the other two have simpler rules. Mine is a mixture of both of theirs in a way. (I'd figured it out by now) TRUE -- Xavier's pattern is to tell the truth when the previous speaker lied and to alternate truth with lie when the previous speaker told the truth.

  24. Me: I'm guessing any of you is going to lie to me next. TRUE -- Tony and Art's time to lie. Xavier's pattern is to tell the truth when the previous speaker lied and to alternate truth with lie when the previous speaker told the truth.

  25. Art: Well, it's been a wonderful conversation. FALSE -- Art's time to lie.

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  • $\begingroup$ 16 was meant to refer to the 16th statement, ie the statement is that whichever of the three spoke that particular statement, they'd have told the truth. Great job, BTW. $\endgroup$
    – Dr Xorile
    Commented May 19 at 15:06
  • $\begingroup$ Yes, everything still works for what you say about the 16th statement. $\endgroup$
    – Bob Bixler
    Commented May 20 at 16:56
  • $\begingroup$ 21 has to be FALSE if 22 is TRUE. So I think that's the only other thing. $\endgroup$
    – Dr Xorile
    Commented May 20 at 19:52
  • $\begingroup$ This is a hard one. You just described one way to look at it. The other is to realize that 21 is TRUE as he's speaking it. It doesn't become FALSE until he's finished and the next speaker is up. $\endgroup$
    – Bob Bixler
    Commented May 21 at 11:42

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