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You are in the intersection of two roads, there is a troll in the beginning of each road.

One of them is liar (always lies), and one of them always tells the truth.

One of the roads, ends to your destination and one of them ends to death :) .

The trolls know which way is the right one and which one ends to death.

How can you find the way if you can ask only one question?

it's your choice who to ask from and you don't know who tells the truth and who lies.

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    $\begingroup$ A harder question involves each of them always lying or telling the truth, but you don't know whether it's one, both, or none that tells the truth. $\endgroup$
    – user88
    Jun 10, 2014 at 14:00
  • $\begingroup$ that will be easy too, What would you tell me if I ask you whether this road ends to death? ( the exact answer of klm's) : the answer is always true. $\endgroup$
    – Alireza Fallah
    Jun 11, 2014 at 4:43
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    $\begingroup$ That's true; however, most people only know the "other guard" solution, and would be totally flummoxed by that formulation. $\endgroup$
    – user88
    Jun 11, 2014 at 5:45

9 Answers 9

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The idea is to ask question about question.
In that case Lie about Lie = Truth about Truth = Truth (if you ask person about question to himself)
or Lie about Truth = Truth about Lie = Lie (if you ask person about question to his antipode).
Therefore it won't be matter who you are asking.

An example of such a question is:

What would you tell me if I ask you whether this road ends to death?

If answer is "yes" you better do not take this road. If answer is "No" you should take this road.

Proof/explanation:

  1. If we ask a liar whether this road ends in death and the road ends in death, he must answer "No". Therefore to our question the liar must lie about his answer and say "Yes".
  2. If we ask a liar whether this road ends in death and the road doesn't end in death, he must answer "Yes". Therefore to our question the liar must lie about his answer and say "No".
  3. If we ask a truth-teller whether this road ends in death, then he must answer truthfully; he will answer the same truth to our question.
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  • $\begingroup$ yes, it's correct, but there is at least one more right answer $\endgroup$
    – Alireza Fallah
    Jun 10, 2014 at 11:51
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    $\begingroup$ @AlirezaFallah, there are millions of right answers:) But all should you the same idea. For example, you can ask "How your friend would answer if I ask him whether this road ends to death?". If you have different idea (without question about question) just make your own answer, it would be interesting to see. $\endgroup$
    – klm123
    Jun 10, 2014 at 11:59
  • $\begingroup$ I've been looking for a way to formulate the problem so that you can't use "the other person" and would have to go with klm123's answer. I asked about it in this math.SE post. $\endgroup$
    – user88
    Jun 10, 2014 at 13:55
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    $\begingroup$ @PM., lie+lie = truth. Do you know the basic mechanics of such type of puzzles? The liar finds the correct answer and then answers the opposite. $\endgroup$
    – klm123
    Jun 11, 2014 at 10:26
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    $\begingroup$ @WeckarE., no. It is common agreement, that in such puzzles the only possible answers are yes or no. Otherwise a puzzle would have no sence. $\endgroup$
    – klm123
    Aug 24, 2016 at 12:05
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Here's a solution without meta-questions:

Ask the left troll: "Does the liar guard the destination?" Iff yes, go right.

Explanation:

1. The left troll says the truth and guards the deathtrap:
The correct answer is yes. The troll will answer yes.
2. The left troll lies and guards the deathtrap:
The correct answer is no. The troll will answer yes.
3. The left troll says the truth and guards the destination:
The correct answer is no. The troll will answer no.
4. The left troll lies and guards the destination:
The correct answer is yes. The troll will answer no.

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  • $\begingroup$ you still don't know which is which. if the liar is guarding the destination and you ask the liar you get "No" and if you ask the truth teller you get a "Yes", but you don't know who's the liar. $\endgroup$
    – Marius
    Apr 8, 2016 at 14:26
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    $\begingroup$ So what? Your goal is to find the correct road, not who's lying. This method finds the correct road. In fact, you can't actually simultaneously find out the correct road and who's lying with a single yes/no question, since the two possible answers you can receive aren't enough to distinguish between the four possible combinations of road and liar. $\endgroup$
    – Anon
    Apr 8, 2016 at 14:30
  • $\begingroup$ @Kate Gregory I did so. $\endgroup$
    – Anon
    Apr 8, 2016 at 19:04
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The open questions in the other answers create loopholes that can lead to the answer not being discernable. - We have little assurances about what the lying troll answers to "what would you tell me if...". The only thing we know is that the lying troll would not say "I would say yes" (if the truthfull answer to the containeed question was no) or "I would say no" (if the truthfull answer to the containeed question was no).

By explicitly making the question a yes or no question you can avoid this;

If I asked you, does this road lead to death, would you answer yes?
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  • $\begingroup$ A lying troll could still answer "I don't know". $\endgroup$
    – Weckar E.
    Aug 24, 2016 at 12:01
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Select a random road and...

Ask either one of the trolls:

Would the other troll point me on this road in order for me to live?

Expanation:

1. You chose the wrong road and asked the liar.
He would answer Yes. Because the truth teller would tell you "No" when asked if this is the right road.
2. You chose the wrong road and asked the truth teller.
He would answer Yes. Because the liar would say "No" when asked if this is the right road.
3. You chose the right road and asked the liar.
He would answer No. Because the truth teller would tell you "Yes" when asked if this is the right road.
4. You chose the right road and asked the truth teller.
He would answer No. Because the liar would say "No" when asked if this is the right road.

Conclusion:

The answer is Yes from either one of the trolls: take the other road.
The answer is No from either one of the trolls: take the road you selected.

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Question I'd ask: "What would the other troll tell me is the road to my destination?"

Liar would tell me truth-teller would guide to death.
Truth teller would tell me liar would guide me to death.

I go the on the opposite road and reach my destination.

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The Troll that speaks and says, "IF Truthful Troll I be, then go though east, and be thou free. " that is the liar in my opinion. Because IF he were the Truthful Troll, he would always be Truthful so he would respond with, "Truthful Troll I am", instead of "IF Truthful Troll I be."

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    $\begingroup$ You seem to be misunderstanding what lying means in the context of these logic puzzles. See also this meta post. It's perfectly valid for a truthful person to make a statement beginning with "If I'm truthful, ..." $\endgroup$
    – Rand al'Thor
    Feb 15, 2017 at 1:01
  • $\begingroup$ This is a cool puzzle to work with...it gets your mind ticking..We are dealing with 2 absolutes which we already have the answer for...1 is an absolute liar and 1 is an absolute truth teller...so there is no room for "IF" at this part of puzzle because each one "IS" an absolute that has already been established from the onset..so then you can eliminate the liar and go on to the next part..which can include the "IF" statement..."IF the troll you choose = Truthful Troll, then You be set free <or> IF the troll you choose = Lying Troll, you will go to your death" $\endgroup$
    – izzie
    Feb 15, 2017 at 17:20
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"Which road would the other troll tell me to go to for this destination?"
Then take the opposite road.

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what is 2+2? Whichever one says 4 is the one that is telling the truth. the one that says fifty billion or something like that is lying.

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    $\begingroup$ But then you've used up your question, and you still don't know which road is the right one. $\endgroup$
    – f''
    Apr 8, 2016 at 4:11
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Why not just ask a general question you already know the answer to so you can determine if the guard you ask is telling truth or lies e.g "Am I human?"

Or would that be against the rules :)

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    $\begingroup$ How do you then find out the correct road after you've wasted your one and only question on determining which Troll is which? $\endgroup$
    – Jaap Scherphuis
    Mar 31, 2018 at 5:08

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