# The village of the liars and truth-tellers

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).

...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.

• What about using the four answers 1234? (or 0134, or 0234) Sep 13, 2015 at 14:31
• 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.
– dr_
Sep 13, 2015 at 14:39
• 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.) Sep 13, 2015 at 14:43
• @Nautilus: A liar can say 4 if not all four of them are liars. Sep 13, 2015 at 19:12
• The point is that a truth speaker can't say 4, so that marks someone who says 4 as a liar. Sep 13, 2015 at 22:01

3:

As all four answers are different, at most one of them can be true.
So there are either 3 or 4 liars in the group.

If there are 4 liars, a liar could not say 4 because it would be the truth.
So the presencer of 4 means that there is at least 1 truth teller and all the
answers being different means there is at most 1 truth teller.

4-1 = 3 so there are 3 liars in this group.

• The puzzle in the question does not have anyone who says "4", so this is in effect answering a different question. Mar 14 at 21:14

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

• The question asks how many liars are in the whole group, including the person asked. Your variant is a bit more interesting though :-). Sep 14, 2015 at 6:13
• 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 Sep 14, 2015 at 6:26
• 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?. Sep 14, 2015 at 10:37
• 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?
– cjm
Sep 14, 2015 at 15:02
• Okay I get it, the puzzle required inclusive condition and not exclusive as I had reasoned Sep 14, 2015 at 15:17