Puzzle Time - Natives at the Crossroad
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PUZZLE: Three Natives at the Crossroads
A logician is visiting the South Seas and, as is usual for logicians in puzzles, she is at a fork, wanting to know which of two roads leads to the village. Present this time are three willing natives, one each from a tribe of invariable truth-tellers, a tribe of invariable liars, and a tribe of random answerers. Of course, the logician doesn't know which native is from which tribe. Moreover, she is permitted to ask only two yes-or-no questions, each question being directed to just one native. Can she get the information she needs? How about if she can ask only one yes-or-no question?
Only non-witches get due process.
- Cotton Mather, Salem Massachusetts, 1692
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PUZZLE: Three Natives at the Crossroads
A logician is visiting the South Seas and, as is usual for logicians in puzzles, she is at a fork, wanting to know which of two roads leads to the village. Present this time are three willing natives, one each from a tribe of invariable truth-tellers, a tribe of invariable liars, and a tribe of random answerers. Of course, the logician doesn't know which native is from which tribe. Moreover, she is permitted to ask only two yes-or-no questions, each question being directed to just one native. Can she get the information she needs? How about if she can ask only one yes-or-no question?
@jon-nyc said in Puzzle Time - Natives at the Crossroad:
Can she get the information she needs? How about if she can ask only one yes-or-no question?
Yes.
This is, from what I can tell, a variation of the "Fork in the road, one person tells the truth, and the other lies" question, with the added quirk of the random one.
For two tribes...
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"If I ask the other guy which road to take, what'll he say?"
And then, take the opposite.
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PUZZLE: Three Natives at the Crossroads
A logician is visiting the South Seas and, as is usual for logicians in puzzles, she is at a fork, wanting to know which of two roads leads to the village. Present this time are three willing natives, one each from a tribe of invariable truth-tellers, a tribe of invariable liars, and a tribe of random answerers. Of course, the logician doesn't know which native is from which tribe. Moreover, she is permitted to ask only two yes-or-no questions, each question being directed to just one native. Can she get the information she needs? How about if she can ask only one yes-or-no question?
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Is it possible for there to be “no answer”?
Do the three people from the three tribes know who is the truth-teller, invariable-liar, or random-answerer?:::
This is what I’m thinking:
Pick a person (A) to ask question to, pick another person (B) to ask question about. Ask (A) whether (B) will always say that the left fork is the correct way to go. For convenience, we label the third person (C).
If (A) gives no yes/no answer, then you know (B) is the random-answerer. You can eliminate (B) from consideration and the puzzle devolve into the well-known Knight vs. Knave puzzle. Problem solved.
Otherwise you know that (B) is NOT the random-answerer, and proceed with the following.
Ask (B) whether (C) will always tell you that the left fork is the correct way to go. From there you are left with two remaining possibilities:
Getting no yes/no answer from (B) means (C) is the random answerer. Eliminate (C) from consideration, treat (A) and (B) as the classic Knight and Knave, and use the answer you previously got from (A) to resolve this remaining classic Knight vs. Knave problem.
Getting a yes/no answer from (B) means (A) is the random answerer. Eliminate (A) from consideration, treat (B) and (C) as the classic Knight and Knave, and use the answer you already got from (B) to resolve this remaining classic Knight vs. Knave problem.
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"No answer" would be cheating because then you'd get 1.5 bits of information from a question, not 1 bit.
I don't even see how one can find out who the random guy is with two questions.
Usually, when a puzzle ask whether something is possible, it is possible, but maybe this is the exception to the rule. I think the random guy is just too annoying.
So my answer is: You really need three questions of the form @George-K suggested and then go by the majority answer.
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The only semi-solution I can imagine is one in which a person cannot answer because any answer would be contradictory.
For instance, if the Truth person is asked "Are you going to answer this question with No" he cannot answer.
So, can I use the full power of the English language for the questions or do I have to restrict the questions to not invoke contradictions and Russel/Gödel-esque paradoxes?
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I guess in the former case, two questions might be sufficient because I have three and not just two possible answers for each question.
For instance, the first question could be: "Would you give the same answer to the question whether Biden is POTUS as the random guy?" Only the random guy would give an answer, the other two could not answer. This lets you identify one person who is for sure not the random guy. Then proceed as in the well-known two-person variant and ask the non-random guy the question for that puzzle.
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Is it possible for there to be “no answer”?
Do the three people from the three tribes know who is the truth-teller, invariable-liar, or random-answerer?:::
This is what I’m thinking:
Pick a person (A) to ask question to, pick another person (B) to ask question about. Ask (A) whether (B) will always say that the left fork is the correct way to go. For convenience, we label the third person (C).
If (A) gives no yes/no answer, then you know (B) is the random-answerer. You can eliminate (B) from consideration and the puzzle devolve into the well-known Knight vs. Knave puzzle. Problem solved.
Otherwise you know that (B) is NOT the random-answerer, and proceed with the following.
Ask (B) whether (C) will always tell you that the left fork is the correct way to go. From there you are left with two remaining possibilities:
Getting no yes/no answer from (B) means (C) is the random answerer. Eliminate (C) from consideration, treat (A) and (B) as the classic Knight and Knave, and use the answer you previously got from (A) to resolve this remaining classic Knight vs. Knave problem.
Getting a yes/no answer from (B) means (A) is the random answerer. Eliminate (A) from consideration, treat (B) and (C) as the classic Knight and Knave, and use the answer you already got from (B) to resolve this remaining classic Knight vs. Knave problem.
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@axtremus said in Puzzle Time - Natives at the Crossroad:
This is what I’m thinking:
Pick a person (A) to ask question to, pick another person (B) to ask question about. Ask (A) whether (B) will always say that the left fork is the correct way to go. For convenience, we label the third person (C).
If (A) gives no yes/no answer, then you know (B) is the random-answerer.I think that last statement isn't true. Both truth teller and liar would answer yes or no, and so would random guy.
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Ah, sorry.
SOLUTION: We can dispose of the one-question case easily: if the question is directed to the random answerer, the logician gets no information, thus can never guarantee identifying the right road. (This argument does not apply if you assume the random answerer first flips a mental coin to determine whether to lie or tell the truth; you could then gain information with a well-chosen self-referential statement, for example, "Out of the other two guys, if I pick the one whose response's truthfulness will least likely match your response's truthfulness and ask him if Road 1 goes to the village, will he answer 'Yes'?" But we assume the random answerer just randomly answers "yes" or "no" regardless of the question, so no information is imparted.)
Similarly, if the logician doesn't know the right road after one question, and her second question is directed to the random answerer, she is in trouble. It follows that after the first answer, she must be able to identify a native who is not the random answerer.
If she can do that, she's in business, because she can then use a traditional one-native query as her second question: for example, something like, "If I were to ask you whether Road 1 goes to the village, would you say yes?"
To attain the objective, she'll need to ask Native A something about Native B or Native C, then use the answer to choose between B and C. Here's one that works: "Is B more likely than C to tell the truth?" Curiously, if A says "yes," the anthropologist picks C, and if he says "no," she picks B! If A is the truth-teller, then she wants to query the companion who is less likely to tell the truth, namely, the liar. If A is the liar, she queries the more truthful of his companions, namely the truth-teller.
Of course, if A is the random answerer it doesn't matter which of B and C she turns to next.
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PUZZLE: Three Natives at the Crossroads
A logician is visiting the South Seas and, as is usual for logicians in puzzles, she is at a fork, wanting to know which of two roads leads to the village. Present this time are three willing natives, one each from a tribe of invariable truth-tellers, a tribe of invariable liars, and a tribe of random answerers. Of course, the logician doesn't know which native is from which tribe. Moreover, she is permitted to ask only two yes-or-no questions, each question being directed to just one native. Can she get the information she needs? How about if she can ask only one yes-or-no question?
@jon-nyc said in Puzzle Time - Natives at the Crossroad:
PUZZLE: Three Natives at the Crossroads
A logician is visiting the South Seas and, as is usual for logicians in puzzles, she is at a fork, wanting to know which of two roads leads to the village. Present this time are three willing natives, one each from a tribe of invariable truth-tellers, a tribe of invariable liars, and a tribe of random answerers. Of course, the logician doesn't know which native is from which tribe, but the natives do. Moreover, she is permitted to ask only two yes-or-no questions, each question being directed to just one native. Can she get the information she needs? How about if she can ask only one yes-or-no question?
Fixed it for whomever wrote the question.
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I guess in the former case, two questions might be sufficient because I have three and not just two possible answers for each question.
For instance, the first question could be: "Would you give the same answer to the question whether Biden is POTUS as the random guy?" Only the random guy would give an answer, the other two could not answer. This lets you identify one person who is for sure not the random guy. Then proceed as in the well-known two-person variant and ask the non-random guy the question for that puzzle.
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OK, that solution is better than what I proposed, but at least I got the overall strategy right:
@klaus said in Puzzle Time - Natives at the Crossroad:
This lets you identify one person who is for sure not the random guy. Then proceed as in the well-known two-person variant and ask the non-random guy the question for that puzzle.
