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Your kingdom has been aided, for as long as anyone remembers, by three great seers. These seers are all-knowing and will truthfully answer one question of any person that seeks them. This question must be addressed at one seer and that seer will answer the question. It is not possible to seek out the seers more than once or to ask a second question of a second seer.

One of the seers has gone mad recently. Instead of answering any question truthfully, he now randomly tells either the truth or a lie. Your king has tasked you with executing the mad seer to prevent him from affecting the sanity of the other seers. Unfortunately, the Seers are an identical triplet (they look and sound identical) and they are always together in the same cave hence figuring out who the mad seer is is problematic in itself. As multiple seers don't serve a purpose anyway, the king has ruled that, if needed, you are allowed to kill one sane seer if it ascertains the death of the mad seer. If you kill both sane seers or fail to kill the mad seer then you will be executed.

You arrive at the seers' cave and go in. Before you stand all three seers. As a person that sought out the seers you are allowed to ask one seer one question.

Which question must you ask and must you kill two seers, or is it possible to kill only the mad seer?

EDIT:: As pointed out in the comments: I forgot to mention its only yes/no questions.

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  • $\begingroup$ Any question? Not just yes/no? $\endgroup$ – Ben Frankel Apr 15 '15 at 21:37
  • $\begingroup$ My bad I missed that detail. Just yes/no. $\endgroup$ – Poelie Apr 15 '15 at 21:39
  • $\begingroup$ How will the sane seers react to being asked a question with no valid answer (e.g. Will you answer no to this question?) How will the mad seer react? $\endgroup$ – Lynn Apr 15 '15 at 21:43
  • $\begingroup$ No seer will be able to answer an invalid question. To take your example: the truth speaking seers must answer truthfully, but whatever they answer causes a paradox. The mad seer must (randomly) tell a truth or a lie, no matter which one he selects, it will still cause a paradox. Note the phrasing for the mad seer is that he randomly tells the truth or a lie, not he gives a random answer. $\endgroup$ – Poelie Apr 15 '15 at 21:57
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Solution to killing one sane Seer:

Pick Seer 1 and ask him this question: "Is Seer 2 the mad Seer?". After he answers, kill him. If he said yes, kill Seer 2 as well, otherwise kill Seer 3.

Solution to killing no sane Seers:

Not really. We can say WLOG that we ask Seer 1 our question. There are 3 possibilities: 1 is mad, 2 is mad, 3 is mad. We must be able to choose between these three options based only on the answer to a yes/no question. This is of course impossible.

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    $\begingroup$ Well I guess this was too easy o.O $\endgroup$ – Poelie Apr 15 '15 at 21:41
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    $\begingroup$ I thought it was a very cute puzzle! (also I haven't answered if it's possible only to kill the mad seer yet). $\endgroup$ – Ben Frankel Apr 15 '15 at 21:42
  • $\begingroup$ To respond to your last edit: to my knowledge you are right; it is impossible to not kill a sane seer. $\endgroup$ – Poelie Apr 15 '15 at 21:59
  • $\begingroup$ Given that the insane seer randomly tells the truth/lies, I think it's probably impossible to not kill any sane seers, but I'm working on it. I'm also thinking about if you knew that the insane seer always lied... it seems like that should be possible, though it would require a more complex question. $\endgroup$ – Duncan Apr 15 '15 at 22:39
  • $\begingroup$ Using psychic powers, yes you can! Also with Telekensis... $\endgroup$ – Anthony Pham Apr 16 '15 at 21:30
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It is actually possible to not kill a sane seer, because there are three possible states for the question - yes, no, and paradox. If it's a paradox, the seer can't answer, giving us three possible responses - yes, no, or no-response. So we need to construct a question that can do all three. And because the mad seer's response is untrustworthy, we need the paradox to occur if seer 1 is the mad seer (assuming we ask seer 1 the question).

This isn't trivial, but it's certainly doable. The neatest question I can think of is...

"Is it simultaneously true that seer 2 is not the mad seer, and either seer 3 is the mad seer or your answer to this question is 'no' when you tell the truth?"

This works because if seer 2 is the mad seer, then...

since "seer 2 is not the mad seer" is false, the answer is "no".

while if seer 3 is the mad seer, then...

since seer 2 is not the mad seer, we pass to the "either" case. Since seer 3 is the mad seer, the answer is "yes".

but if seer 1 is the mad seer...

we have that seer 2 is not the mad seer, so it passes to the "either" case, and since seer 3 is not the mad seer, the question boils down to "is your answer to this question 'no'?" This is a paradoxical question, and thus seer 1 cannot answer.

So no response indicates that seer 1 is the mad seer, "no" indicates that seer 2 is the mad seer, and "yes" indicates that seer 3 is the mad seer.

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    $\begingroup$ "Is your answer to this question no?" is not a paradoxical question, as the mad Seer could simply answer yes as a lie. As a paradoxical question you can use "Is your answer to this question a lie xor no?". I think. $\endgroup$ – Ben Frankel Apr 16 '15 at 18:00
  • $\begingroup$ A minor flaw, you're right. I need to specify that I'm referring to the answer he gives when he tells the truth, which can be achieved with a slightly neater expression (edited into the solution). $\endgroup$ – Glen O Apr 17 '15 at 0:19
  • $\begingroup$ If truth-tellers cannot give answers that yield paradoxes, but must answer whenever doing so would not create a paradox, an arbitrary amount of information may be gleaned from a single "yes-no" question by making it so that for each possible state of interest the only way to avoid a paradox would be for the truth-teller to perform a different action in addition to answering yes or no. To my mind, however, that is rather unsatisfying. A better formulation would be to say that a truth-teller may answer "yes" if doing so would be no less truthful than answering "no", or vice versa. $\endgroup$ – supercat Aug 28 '17 at 20:51
  • $\begingroup$ @supercat - that's only if they are permitted non-verbal forms of communication. Failure to respond is not a non-verbal form of communication, it's just the "empty set" of verbal communication. $\endgroup$ – Glen O Sep 11 '17 at 4:43
  • $\begingroup$ @GlenO: If truth-tellers aren't required to respond whenever possible, then none of these problems are solvable. If one formulates a question such that a truth-teller could only respond between 12:01am and 12:02am if the answer is door #1, could only respond between 12:02am and 12:03am if it's door #2, etc. $\endgroup$ – supercat Sep 11 '17 at 14:19

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