Logic Puzzle. 10 points for first correct answer!!!?!


Question: A logician is on the island where the inhabitants are either knights (they always tell the truth) or knaves (they always lie). George and Cat are there. The logician approaches them and asks: “Is at least one of you a knight? ”. Either George answers the question or Cat answers the question (and not both). The answer was either “yes” or “no”. The logician receives an answer that lets her figure out who is who. What answer did she get (did she get a “yes” or a “no”)?


Answers: A logician is on the island where the inhabitants are either knights (they always tell the truth) or knaves (they always lie). George and Cat are there. The logician approaches them and asks: “Is at least one of you a knight? ”. Either George answers the question or Cat answers the question (and not both). The answer was either “yes” or “no”. The logician receives an answer that lets her figure out who is who. What answer did she get (did she get a “yes” or a “no”)?

no

yes

no

you did not specify if george or cat was a knight or knave.

it was yes. it can't be no, because if neither was a knight then they would both be knaves. and if they were knaves they'd be lying. so they wouldn't be able to answer no, they'd have to say yes. and if they answer yes, that means at least one is a knight or both are knaves.

Case 1: Assume that the answer to the logician’s question was ‘yes’. This response was either true or false, so we have:
case A: Assume that the response was the truth. Then at least one of the two is a knight. Further, the person who responded (i.e. the respondent) to the question is a knight. The person who did not respond the question could be either a knight or a knave.

B: Assume that the response was a lie. Then neither of them are knights—they are both knaves.

From the response ‘yes’ we are able to infer that either they are both knights, the respondent is a knight and the other is a knave, or they are both knaves. But we do not know which of these possibilities is the case. However, the logician was able to determine who was who from the response that s/he received, thus we can conclude that ‘yes’ was not said.

Case 2: Assume that the answer to the logician’s question was ‘no’. This response was either the truth or false, so we have the following sub-cases.
Sub-case A: Assume that the response was the truth. Then neither of them are knights, so both of them are knaves. But knaves always lie, so the response was a lie. This is a contradiction, so sub-case A is eliminated as a possibility.

Sub-case B: Assume that the response was a lie. Then at least one of them is a knight. But we know that the respondent is a knave since s/he lied. So the person who did not respond is the knight.
We have shown that the answer ‘no’ could only have been a lie, so the person who responded to the question is a knave and the other is a knight. Thus, the logician would have been able to determine who was who from the answer ‘no’.

Since case 2 is the only possibility, the person who answered the logician’s question is a knave and the other is a knight.



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