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# The Liar, the Truthteller, and the Trickster

## Question

You come across three people at a fork in the road. One path leads to a fiery dragon, and the other leads to your destination. You know that one of the three people always tells the truth, another always lies, and the third alternates between lying and truthtelling. ( for example truth, lie, truth, lie, etc.) You only have time to ask one person two questions. What should they be to ensure that you find the right road?

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Depending on the behavior of the "trickster",

Ask any person, "If, instead of this question, I had asked you which way leads to my destination, what would you have said?" Both the liar and truth-teller will point you the right way (since the liar must lie about what he would have said, which would have been a lie). Similarly, regardless of whether the "trickster" is in liar mode or truth-teller mode when queried, he must also point in the correct direction.

Edited by ThunderCloud
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Depending on the behavior of the "trickster",

Ask any person, "If, instead of this question, I had asked you which way leads to my destination, what would you have said?" Both the liar and truth-teller will point you the right way (since the liar must lie about what he would have said, which would have been a lie). Similarly, regardless of whether the "trickster" is in liar mode or truth-teller mode when queried, he must also point in the correct direction.

OP says the trickster randomly lies or tells the truth. That kind of "trickster" can be modeled.

The responder who randomly says Yes or No cannot be modeled.

You might sharpen the question by asking: If I were to ask "Is THIS the road [pointing to one of the roads] that leads to my destination?", how would you respond? This constrains the reply to be Yes or No. As stated, an indifferent truth-teller is free to reply in a non-responsive manner: "I would have said, 'Iam not in a mood to answer', or 'The sky is an unusual shade of blue today, isn't it?'"

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Depending on the behavior of the "alternator"

Ask any person, "If, instead of this question, I had asked you which way leads to my destination, what would you have said?" Both the liar and truth-teller will point you the right way (since the liar must lie about what he would have said, which would have been a lie). Similarly, regardless of whether the "alternator" is in liar mode or truth-teller mode when queried, he must point in the correct direction.

Depending on the behavior of the "trickster",

Ask any person, "If, instead of this question, I had asked you which way leads to my destination, what would you have said?" Both the liar and truth-teller will point you the right way (since the liar must lie about what he would have said, which would have been a lie). Similarly, regardless of whether the "trickster" is in liar mode or truth-teller mode when queried, he must also point in the correct direction.

OP says the trickster randomly lies or tells the truth. That kind of "trickster" can be modeled.

The responder who randomly says Yes or No cannot be modeled.

You might sharpen the question by asking: If I were to ask "Is THIS the road [pointing to one of the roads] that leads to my destination?", how would you respond? This constrains the reply to be Yes or No. As stated, the person is free to reply in a non-responsive manner: "I am not in a mood to answer," or "The sky is an unusual shade of blue today, isn't it?"

The question can certainly be sharpened. But be careful: in your phrasing, you lost one of my own "sharpenings":

The trickster is given by the OP to alternate between lying and truth-telling, so I felt it was important to specify that my hypothetical question was in place of the question I am currently asking, thus preserving the trickster's "state". Otherwise, as given, the answer could be interpreted to depend upon the timing of when the question was asked. The trickster might read the question as "If my next question to you was '

xyz...'."
The case of the random responder is more interesting... I have not solved that yet.

Also, note that the questions were not constrained by the OP to the yes-no variety, so I think my original answer makes sense so far as it goes.

Edited by ThunderCloud
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TC, Thanks.

I did not fully appreciate the "timing" aspect.

handle the random yes-no responder with a first question of the type

[asking A] "would B respond more truthfully than C would, to the question ....."

Using that question to identify the random person, your question asked of either of the others does the trick.

But I have not thought it through, either.

I assume the OP meant to include the fact that each knows the type of responder the other two are.

But reading OP again, you have to ask both questions, not knowing which, of a single person.

What if you chose the random yes-no person? Hmmmm.

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To Thundercloud: Nice job! The " two questions" was to try to confuse people, but you figured out how to obtain the correct road with just one question. Great problem solving skills. That dragon will just have to go without its meal!

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