In a small village there are two kinds of people: liars and truthtellers.

Everybody knows everybody and everybody knows as well who is a liar and who’s a truth-teller.

I approach six villagers and pose the same question to each of them:

"How many liars are among you?"

I get six distinct answers (integers, of course) and deduce the true one.

How many liars are in that group?

Liars always lie and truthtellers never do.

Short answer:

The 6 distinct answers imply that there were **either 5 or 6** liars in the group.

Although the asker could deduce the correct partition (T, L) the puzzle solvers can't determine the number of liars in the group - **either 5 or 6.**

Explanation (**revised** following Dej Mar's comment):

The asker based his decision only upon presence of numbers 5 and/or 6 in the 6-tuple of answers:

If there was neither 5 nor 6 - then there were 6 liars.

Both 5 and 6 - 5 liars.

5 and no 6 - no possibility to deduce.

obviously 6 without 5 is not possible, forcing a liar to tell the truth - ...

Examples:

(0,1,2,3,4,11) or (3,4,7,8,55,99) …..6 liars.

(1,2,3,4,5,6) or (3,4,5,6,7,8,) …..5 liars.

(0,1,2,3,4,5) or (5,7,8,11,23,25) …5 or 6 liars.

(0,1,2,3,4,6) or (1,6,7,9,22,77)

If one considers “logically consistent” liars who use neither negative numbers nor integers over 6 (the 1st sets in the above examples) as their answer then in 7 possible cases of a “logically consistent” 6-tuple (one of the 0,1,2,3,4,5,6 absent):

a.(0,1,2,3,4,6) ….…..**not an option**.

b.(1,2,3,4,5,6); (0,2,3,4,5,6); (0,1,3,4,5,6); (0,1,2,4,5,6) ;(0,1,2,3,5,6) ........…**5 liars**.

c.(0,1,2,3,5,6) **no way to tell** - **…5 or 6 liars.**

So if the author deduced correctly and the consistent 6-tuple was a randomly chosen set then the odds are **5.5 to .5** in favor of 5 liars, (5 implied absolutely by logic and .5 by luck) i.e. 11 to one.

- If there is no consistency and any set of 6 distinct numbers is allowed the odds move heavily in favor of 6 liars and no truth tellers.

P.S.

Minor remark: it is beyond my comprehension why somebody (Steve ?) rated this puzzle as level 2, - IMHO it is certainly easy, interesting, **thought provoking** and should not be downgraded by someone who even did not solve it correctly.

No hard feelings, bro. Just wondering...

*Edited on ***November 16, 2015, 1:52 pm**

*Edited on ***November 17, 2015, 10:53 am**