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.
(In reply to Summing up - spoiler
by Ady TZIDON)
a. (0,1,2,3,4,6) - This set is a logical impossibility, as would be any set containing the integer 6 with no integer 5. If all six were liars, then the villager who answered 6 would have been telling the truth. By the given, a village liar could never tell the truth. A truthteller would not provide that answer, as by the exclusion of himself there could be no more than five others, and logic would dictate all five others would be liars.
c. (0,1,2,3,5,6) - As in all sets that contain both the integers 5 and 6, the villager who answered 5 must be a truthteller.
For sets containing no integer 6 element, but does contain the integer element 5, the number of liars is either 5 or 6. The villager who answered 5 could be either a truthteller or liar - for which only villagers could ascertain the truth of which by the givens to the problem.
Thus, there are not three groups (a,b,c), but four (a,b,c,d).
a. If there is no 5 or 6 -- six liars.
b. If there is a 5 and a 6 -- five liars.
c. If there is a 5 and no 6 -- five or six liars, the exact number indeterminable without any other givens.
d. If there is no 5 but there is a 6 -- impossible set of answers.
I've added bullet d, as "If there was no 5 - then there were 6 liars" did not specify the exclusion of the condition where the integer 6 existed in the set - and was incorrectly given in the example as representing 6 liars, when, in fact, such a case is an impossibility.
Edited on November 17, 2015, 9:17 am
Posted by Dej Mar
on 2015-11-17 03:43:08