I gave up too quickly.

You paddle up to an island where you know there are two types of people: Liars and Truth Tellers. Liars ALWAYS lie, and Truth-Tellers ALWAYS tell the truth. You see 3 people on the beach. Liars and Truth Tellers often hang out together, so these 3 natives could be any combination of liars and truth tellers.

You ask, “Are you liars or truth-tellers?” and the first guy answers your question, but a big wave crashes on the beach just then. The natives hear the answer, but you do not.

So you repeat the question, and this time the second guy responds. He says, “The first guy told you that he is a truth-teller”. The second guy then continues, “The first guy actually is a truth-teller”. Finally, the second guy states, referring now to himself, “I am a truth-teller”.

At this point the 3rd guy decides to respond. He says, “No, the second guy is lying. I am a truth-teller.”

Who’s a liar and who’s a truth teller?

Here are the only possible accusations, where x=impossible and o=possible. It is assumed each person knows the status of the others.

[x] L<–(T)–L
[x] T<–(T)–L
[x] T<–(L)–T
[x] L<–(L)–T
[x] L<–(L)–L
[o] T<–(L)–L
[o] T<–(T)–T
[x] L<–(T)–T

T<–(T)–_<–(L)–_
1           2          3