33. Who Is the Politician?
In a certain mythical land, every politician always lies, whereas every non-politician always tells the truth.
A traveller meets three natives—A, B and C—and asks native A:
“Are you a politician?”
You do not hear A’s reply, but you do hear what the other two natives then say:
- Native B reports: “A denied being a politician.”
- Native C states : “A is a politician.”
From this information alone, decide for each native whether he is certainly a politician, certainly a non-politician, or whether his status cannot be determined.
Added 15 May 2014
Hint: Consider the two possible cases for A (politician or not) and check each case against B’s and C’s statements.
Solution:
Let us analyse the two possibilities for native A.
Case 1: A is a politician (and therefore lies).
Asked whether he is a politician, a politician would have to answer “No.” Thus A does deny being a politician. B’s report, “A denied being a politician,” is therefore true, so B must be a truth-telling non-politician. C’s statement, “A is a politician,” is also true, so C too must be a truth-telling non-politician. This produces a consistent scenario (politician A; non-politicians B and C).
Case 2: A is a non-politician (and therefore tells the truth).
If A tells the truth, he must answer “No, I am not a politician.” Again A denies being a politician, so B’s report is true and B is a non-politician. This time C’s statement, “A is a politician,” is false, hence C must be a lying politician. This is also consistent (non-politicians A and B; politician C).
In both cases B is a non-politician, while A and C swap roles. Because both complete assignments are self-consistent, we cannot tell which of them is the actual one.
Conclusion: Only native B is certainly a truth-telling non-politician; the statuses of natives A and C cannot be deduced from the information given.
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