A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet 6 inhabitants: Grace, Penelope, Mia, Noah, Matthew, and Lily. "Noah is a knave" - Grace. Penelope asserted: "Mia is a knight". In a statement by Mia: "Grace is a knave". Noah said, "Mia is a knight if and only if Matthew is a knight." Matthew said that Noah is not a knave. "Mia is not a knight" - Lily. So who is a knight and who is a knave?
Grace is a knave, Penelope is a knight, Mia is a knight, Noah is a knight, Matthew is a knight, and Lily is a knave.
(1) Grace is a knave
(2) Penelope is a knight
(3) Mia is a knight
(4) Noah is a knight
(5) Matthew is a knight
(6) Lily is a knave
Assume Grace is a knight. No contradiction is found in their claim that Noah is a knave. Noah cannot be a knight, because this would contradict the claim of Grace that Noah is a knave. Assume Noah is a knave. No contradiction is found in their false claim that Mia is a knight if and only if Matthew is a knight. Mia cannot be a knight, because this would contradict the claim of their own that Grace is a knave. Assume Mia is a knave. No contradiction is found in their false claim that Grace is a knave. Matthew cannot be a knight, because this would contradict the claim of their own that Noah is not a knave. Matthew cannot be a knave, because this would contradict the false claim of Noah that Mia is a knight if and only if Matthew is a knight. We have exhausted all possibilities for Matthew, Mia, and Noah, so let us go back and reconsider Grace. Assume Grace is a knave. No contradiction is found in their false claim that Noah is a knave. Assume Noah is a knight. No contradiction is found in their claim that Mia is a knight if and only if Matthew is a knight. Assume Mia is a knight. No contradiction is found in their claim that Grace is a knave. Assume Matthew is a knight. No contradiction is found in their claim that Noah is not a knave. Assume Penelope is a knight. No contradiction is found in their claim that Mia is a knight. Lily cannot be a knight, because this would contradict the claim of their own that Mia is not a knight. Assume Lily is a knave. No contradiction is found in their false claim that Mia is not a knight.