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I read this puzzle in an Italian newspaper some days ago. It reported it was asked at one of the latest Mathematical Olympiads. It's not difficult, but it requires some logical thinking. I'm posting it here as I'd like to know if you reach the same conclusion as me.

In a village live two kinds of people: those who always lie and those who always tell the truth. Everybody knows each other in the village and therefore knows whether a particular villager is a liar or a truth-teller.
Strolling around the village we meet a group of four villagers. We ask to each one of them: "How many liars are in this group?" and we get these answers: "0", "1", "2", "3".
How many liars are in that group?

EDIT: There is probably an error in the riddle as published in the newspaper. If the answer we get from the group of villagers is instead "1", "2", "3", "4" then there is an unique solution (as reported in the accepted answer below).

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  • $\begingroup$ Alternately, instead of the answers being wrong, it's possible that the question should have been "How many truth-tellers are in this group?"; that would have given a unique solution. $\endgroup$ Sep 11, 2022 at 19:53

3 Answers 3

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The answer is...

...either 3 or 4:
As all four answers are different, at most one of them can be true.
Hence either all of them are liars, or the guy with the answer "3" is a truth-teller.

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    $\begingroup$ What about using the four answers 1234? (or 0134, or 0234) $\endgroup$
    – Gamow
    Sep 13, 2015 at 14:31
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    $\begingroup$ You're right. If the answers of the villagers are e.g. "1", "2", "3", "4" then the puzzle has one unique solution. Probably there was an error in the newspaper. Bonus points for this comment. $\endgroup$
    – dr_
    Sep 13, 2015 at 14:39
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    $\begingroup$ There is also a weird version of the puzzle, where the answers of the villagers are "0", "1", "2", "4". (Perhaps you could reformulate your puzzle text, and discuss all variants.) $\endgroup$
    – Gamow
    Sep 13, 2015 at 14:43
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    $\begingroup$ @Nautilus: A liar can say 4 if not all four of them are liars. $\endgroup$ Sep 13, 2015 at 19:12
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    $\begingroup$ The point is that a truth speaker can't say 4, so that marks someone who says 4 as a liar. $\endgroup$
    – LukStorms
    Sep 13, 2015 at 22:01
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Let's assume that the villagers answered, "1", "2", "3", "4" .

Possibilities for number of liars: 0 1 2 3 4 . The total number of liars has to be one of these numbers.


  1. If the person who is saying "1" were a truthteller, then it would mean that there are 3 truthtellers. So, there should be 3 people saying "1" when asked how many liars are there. But this is not the case. So, the person whose answer is 1, is a liar. 

      We now know that the total number of liars > 0 and that the number of liars is not equal to 1.

       Possibilities for number of liars:   0  1  2 3 4




  1. If the person who is saying "2" were a truthteller, then it would mean that there are 2 truthtellers. So, there should be 2 people saying "2" when asked how many liars are there. But this is not the case. So, the person whose answer is 2, is a liar.

       Possibilities for number of liars: 0 1 2 3 4

       We now know that the number of liars is either 3 or 4.




  1. Let's look at the person who says 3. If the person who is saying "3" were a truthteller, then it would mean that there is 1 truthteller. This means that there is exactly 1 person who says. "3" when asked how many liars are there, which is indeed the case.

     So, it is possible that the person saying 3 is indeed a truthteller. It is also possible though that he's lying and there are 4 liars. 




  1. The last person says "4" . He is saying  that all 4 of them are liars. He cannot be a truthteller because he is saying that all 4 of them, including himself, are liars. He cannot be a truthteller because a truthteller will never call himself a liar. This means that he is a liar and he is lying when he says that there are 4 liars.

    Possibilities for number of liars : 0 1 2 3 4


   

Therefore, we now know that there are 3 liars and 1 truthteller. We also know that the person who said "3" is the truthteller.

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I agree with Gamow's reasoning but I think the answer is a bit wrong, why??

...either 3 or 4: As all four answers are different, at most one of them can be true. Hence either all of them are liars, or the guy with the answer "3" is a truth-teller

That much is true, but come to think again , if he told the truth then the other three are all liars.

  • Now on answering the answer about other three villagers non of the liars would have answered 2 (i.e 2 liars and 1 truth teller) because by that he would be telling the truth which makes the reasoning invalid

    So the only possible answer here is that all of them (all 4 villagers) are liars

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    $\begingroup$ The question asks how many liars are in the whole group, including the person asked. Your variant is a bit more interesting though :-). $\endgroup$
    – Tibos
    Sep 14, 2015 at 6:13
  • $\begingroup$ Yeeah but when asked, each one would have to talk about the other three without including himself, from his description you are able to deduce if he is a truth teller or not and also the other three left...just think about it $\endgroup$
    – AguThadeus
    Sep 14, 2015 at 6:26
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    $\begingroup$ Probably for the same reason @Tibos outlined: the question villages were asked was How many liars are in this group?, not How many liars are in the rest of this group?. $\endgroup$
    – Cthulhu
    Sep 14, 2015 at 10:37
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    $\begingroup$ No, when you're asked "How many X are in your group?" you include yourself in the count (if you meet criterion X). If you went to a restaurant with 3 friends and the hostess asked you "How many in your party?" would you say 3? $\endgroup$
    – cjm
    Sep 14, 2015 at 15:02
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    $\begingroup$ Okay I get it, the puzzle required inclusive condition and not exclusive as I had reasoned $\endgroup$
    – AguThadeus
    Sep 14, 2015 at 15:17

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