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I would like to ask if anyone has seen elsewhere this variation on the standard truth tellers/liars kind of puzzle.

In the usual repertoire it is assumed that every person questioned knows the answer to every question asked. We now assume that this need not be the case; and that if a truth-teller is asked a question to which he does not know the answer, he will truthfully reply "I don't know", while a liar will never admit to not knowing.

This can still be formulated in various ways: here is a specific example and a puzzle.

(But please note: my question is not to solve the puzzle but to ask if anyone has seen anything like it before.)

In a certain town each of the inhabitants is either a truth-teller or a liar; however this does not mean that everyone is actually able to answer every question they are asked. If a truth-teller is absolutely certain of the answer to a question, he will give that answer; if not, he will say, "I don't know." On the other hand, a liar will never truthfully admit to not knowing something: he will give an answer that he knows is false, if any, but if there is nothing that he is certain is false then he will give a randomly chosen answer (possibly even, by accident, the true answer). Moreover, everyone in this town can instantly deduce the logical consequences of any facts they know.

I meet four inhabitants of this town and ask them, "How many of you four are truth-tellers?"

Kevin says, "I don't know"; then Laura says, "One"; then Mike says, "None." Noela, however, is asleep. Fortunately I don't need to wake her up, since I can already tell whether she is a truth-teller or a liar. Which?

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    $\begingroup$ No, I have not.:) $\endgroup$
    – klm123
    Commented Aug 26, 2014 at 8:17
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    $\begingroup$ One clarification it could use - will liars ever say "I don't know"? You say they "will never truthfully admit to not knowing", but that leaves it ambiguous as to whether or not they will say "I don't know" when they do in fact know. $\endgroup$
    – Rob Watts
    Commented Aug 27, 2014 at 18:45
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    $\begingroup$ Hi @kaine, here is my solution. A liar will never admit to not knowing, so Kevin must be a truth-teller. Having heard this, Laura will also know that Kevin is a truth-teller. If she is also a truth-teller then she knows that the answer to my question is 2, or 3, or 4, or she doesn't know. In no case would she give the answer 1. So Laura is a liar. Mike also knows that Kevin is a truth-teller, so Mike has given a false answer and is a liar... to be continued $\endgroup$
    – David
    Commented Sep 3, 2014 at 6:30
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    $\begingroup$ ...Now suppose that Noela is a liar. Then Laura would have given the right answer: as she is a liar, the only way this could happen is if she didn't know any definitely false answer. But this is not the case: she knew that Kevin was a truth-teller and therefore that "None" would have been a false answer. It must therefore be that Noela is a truth-teller (and that Laura knew this, and therefore knew that "One" was a false answer). $\endgroup$
    – David
    Commented Sep 3, 2014 at 6:31
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    $\begingroup$ @RobWatts sorry I didn't reply earlier, for some reason I didn't get notified of your comment. In my scenario (of course it is possible to make up different scenarios) a liar will never admit to not knowing. I should have left out the word "truthfully" to make this more clear. Thanks! $\endgroup$
    – David
    Commented Sep 3, 2014 at 6:32

1 Answer 1

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Kevin = "I don't know" == True (Because he truthfully said, that he didn't know)

Laura = "One" == False (Because if she says truth, then there would be at least 2 truth tellers as Kevin is already a truth teller).

Mike = "None." == False (Because Kevin proves himself a truth teller)

Noela = == True (Because if Noela is False then there is only 1 (i.e. Kevin) and Laura also replied "One" and we know that Laura is false)

Hence, there are Two out of Four Truth-Tellers in town.

Now, I do remembered such a kind of question in a book titled as "Discrete Mathematics and its Applications" Author: "Kenneth H. Rosen" Ed:"7th".

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  • $\begingroup$ Could you site what was in the book? P.S. Note - the OP said explicitly "my question is not to solve the puzzle", so the solution is offtopic here. P.P.S. your solution is not complete, see the comments to the question. $\endgroup$
    – klm123
    Commented Sep 16, 2014 at 15:41
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    $\begingroup$ @waqas Can you give me a page reference to the question in Rosen? Much appreciated. $\endgroup$
    – David
    Commented Sep 16, 2014 at 22:49
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    $\begingroup$ yes but in the scenario that laura is a liar noela could be a liar.It just happens that laura said one by luck, giving the correct answer accidentaly $\endgroup$
    – libathos
    Commented Oct 3, 2014 at 11:19
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    $\begingroup$ @libathos OP specifies that if the liar has answers that she knows are false then she will give one of those answers. Laura had the option to answer zero and did not. $\endgroup$
    – Taemyr
    Commented Nov 28, 2014 at 10:07
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    $\begingroup$ @Cephalopod Remember that as a liar Laura will always, if possible, choose an answer that she knows to be false. If Laura did not know that there where excactly 2 truth tellers there was a chance that only one 1 truth tellers. This means that after she answers 1 we know that she knows that there are more than 1 truth teller, because she had the option to answer 0 or 5, both of which she would know to be false. $\endgroup$
    – Taemyr
    Commented Jan 1, 2015 at 23:36

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