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?
Either 6 liars or 12 liars
The statements are mutually contradictory which means at one most one is true. Only the third can be true but they can also all be False meaning they are all liars.
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.
the solution seems simple:
There are 6 liars and 6 knights
As there are 6 liars, the people saying there are exactly 2 liars lie. same for the other 4 people.
an alternative is 12 liars