Everyone Gossips

Suppose A is telling the truth, then C and D are both liars. Then E must be lying. So we have 3 liars already. Since we're given that at least 3 of them are truth-tellers, and we have found 3 liars already, then the remaining two people -- B and F -- are telling the truth. But by B 's account, E is telling the truth, which contradicts our previous findings. Hence A must be lying.

Since we know that A is a liar, then F must be lying as well. And so, B must be lying as well. Since we're given that at least 3 of them are truth-tellers, and we have found 3 liars already then the remaining three people -- C , D and E -- are telling the truth. Double checking shows that this is indeed true.

In short, C , D and E are only telling the truth.

Observe that what B says can't be true because supposing that it is true then E and F should be both truth tellers but since E and F make contradictory statements about D , E saying it is a truth and teller while F says that D is a liar supposing they are both truth tellers would also imply to agree that D is both a liar and also a truth teller at the same time (as a result of what E and F say) which however would mean to agree with an absurdity. Therefore B is a liar and persons E and F will be one a truth teller and the other a liar.

Now , consider the statements of E and F where either E , either F should be truth tellers and analyze where each of this cases leads by the way they are consistent or not with the set of statements of the others. To simplify the analysis and make a little bit more synthetic understanding about the way things are determined observe that since B is a liar one of the statements of D and C will be true and what would make them false would be for D that A wouldn't be a liar but a truth teller and for C that E wouldn't be a truth teller. Observe further that by telling if E is or not a truth teller then for any value it has A's value is also determined and therefore consistent. And because A and F are consistent with each other it is possible to conceive that they will have the same different value from E.

Considering this observe that if E is a truth teller then the values of A , F and D are already determined remaining to check just the case for D which will say a true thing since A is a liar. This reasoning would imply therefore that A , F and B are liars and E , D and C are truth tellers and as there are at least 3 truth tellers is a possible configuration.

For the case in which E is a liar then there will be 3 liars determined by E and just 2 truth tellers. This is another possible case of the configuration but as it is known there are 3 truth tellers it doesn't check the last condition of the problem therefore remaining that the only configuration possible is that in which E , D and C are truth tellers and A , F and B liars therefore the answer being 112221 anyways.

"You know that at least 3 of them are truth-tellers", so at least 3 of the statements should support each other and not being contradictory between themselves (because they share and bear the same truth equally among all the truth tellers).

We heard accusations among these statements, so there exists some contradictions proving that some or all of the other three are liars.

Since the word 'both' is inclusive and the truth depends on the value of two different objects, let's focus on what C and F said first, because they didn't use the 'both' word.

C : B is a liar and E tells you the truth.
==> C associated themselves with E and being detached from B.

F : D is a liar and A is a truth-teller.
==> F associated themselves with A and being detached from D.

So we have at least two different parties here. By A's and E's statements that's polar opposite to each other, we can say that A, F and B is on one pole while E, C and D is on another. Further investigation shows that E, C and D are totally supportive of each other while A, F and B have discrepancies between their statements.

TEAM E (✓ indicates the relative in-team intra-truthfulness, not as a universal truth)
C : B is a liar and E tells you the truth ✓
D : A and B are both liars ✓
E : C and D are both truth-tellers ✓

TEAM A (✓ indicates the relative in-team intra-truthfulness, not as a universal truth)
A : C and D are both liars ✓
B : E and F are both truth-tellers X
F : D is a liar and A is a truth-teller ✓

Therefore, Team E with internally consistent statements must be the truth tellers while Team A are full of liars.

Answer = 112221