# Knights and liars

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?

• Welcome to Puzzling SE! Please take the tour to see our community's expectations (you'll also get a badge!) Commented Mar 21, 2018 at 15:42
• Is this your own creation? If not, please cite where you got this riddle. Commented Mar 21, 2018 at 15:44
• How many liars of the 12 or on the island? Commented Mar 22, 2018 at 7:31
• What exactly "us" means? Is it among the 12 people or among the one's who are speaking? Commented Mar 22, 2018 at 10:12
• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it :)
– Rubio
Commented Mar 23, 2018 at 12:41

There are

Either 6 liars or 12 liars

Because

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.

• @MontyHarder: I see what you mean, but note that the question says "One day 12 islanders gathered together" - i.e. there's no guarantee that all the islanders are represented in the group... Commented Mar 21, 2018 at 16:37
• @MontyHarder even if there must be at least one (arguably two, given the plural construction) of each on the island, this answer is still correct, as it's never stated that the 12 islanders are the entire population. Commented Mar 21, 2018 at 16:37
• Given that the "island is inhabited by both liars and knights", I suspect there are both knights and liars, not knights and/or liars Commented Mar 22, 2018 at 14:49
• If all 12 lie, there can be even more liers. Maybe the whole population
– user46960
Commented Mar 22, 2018 at 18:26

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

• I received two downvotes, why? is it because i don't explain enough? is it because it's the same answer as hexomino, in the same time? Downvotes are here to show to others that this answer is bad, bud also to send a message to the author, but i don't see what I've done wrong here Commented Mar 21, 2018 at 18:23
• it maybe that you say it is simple...
– tom
Commented Mar 21, 2018 at 18:33
• It could also be that you start by saying "Answer is X", give your thought process, and then right at the end say "Or, it might be Y instead" - you should probably group your solutions in one part of the Answer, in case people stop reading after the first line of the spoiler! Commented Mar 22, 2018 at 13:10
• Your answer assumes there are 6 liars... why? Commented Mar 23, 2018 at 18:25