An island is inhabited by both Liars and Knights. Knight always tells the truth and Liar always lies. One day 12 islanders gathered together and issued a few statements. 2 of the islanders said "Exactly 2 of us are liars". Another 4 said "Exactly 4 of us are liars". The remaining 6 islanders said "Exactly 6 of us are liars". How many liars are there?
This can be solved by elimination,
First note that each group of 2, 4, 6 has to be all Liars or all Knights.
Now two of the groups can't be Knights because they would not be saying different things. That means at most only one of the groups can be Knights. So the only possibilities are:
246 --- KLL LKL LLK* LLL*But the first two can be eliminated because in both cases there are more liars than the Knights are saying.
That leaves us with the two possibilities of either 6 or all Liars.