Ethel , Felicia and Gabrielle live on an island inhabitated by three types: the Knights, the Liars and the Weirdos.
Each is either a Knight who always tells the truth, a Liar who always lies , or a Weirdo who may do either - that is, a Weirdo chooses whether to speak truth or lie for each statement.
Ethel says : "If we all belong to the same type, then that type is the Liar."
Felicia says: "If just one of us belongs to a different type from each of the others, then that one is a Liar."
Gabrielle says : "If each of us belongs to a different type from each of the others, then I am a Liar."
Whose type can you deduce with absolute certainty?
If they all belong to the same type, then statements by F and G are automatically true, so F and G cannot be liars. And They cannot all be knights, based on E's statement, so one possibility is
(E,F,G) = (W,W,W)
If all but one are the same type, then statements by E and G are automatically true, so E and G cannot be liars. And F cannot be a liar, because that would made F the odd man out, and her statement impossible. So F's statement is a lie, but F is not a liar, making F a Wierdo.
(E,F,G) = (W,W,K) or (K, W, W) or (K, W, K)
If they are of three different types, then statements by E and F are automatically true, so E and F cannot be liars. That leaves G to be the only Liar, but that is impossible given G's statement.
Therefore, the only possibilities are
(E,F,G) = (W,W,W) or (W,W,K) or (K,W,W) or (K,W,K)
Nobody is a liar, and F is known Wierdo.