A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet 6 inhabitants: Matthew, Harper, Luke, James, Sebastian, and Alexander. "Luke is a knave" - Matthew. Harper stated, "Alexander is a knight". Luke was heard saying, "Harper is a knight or Luke is a knight". "Alexander is a knight and Matthew is a knight," James claimed. Sebastian said that Luke is a knight if and only if Matthew is a knave. Alexander said, "Matthew is not a knave." So who is a knight and who is a knave?
Matthew is a knave, Harper is a knave, Luke is a knight, James is a knave, Sebastian is a knight, and Alexander is a knave.
(1) Matthew is a knave
(2) Harper is a knave
(3) Luke is a knight
(4) James is a knave
(5) Sebastian is a knight
(6) Alexander is a knave
Assume Matthew is a knight. No contradiction is found in their claim that Luke is a knave. Luke cannot be a knight, because this would contradict the claim of Matthew that Luke is a knave. Assume Luke is a knave. No contradiction is found in their false claim that Harper is a knight or Luke is a knight. Harper cannot be a knight, because this would contradict the false claim of Luke that Harper is a knight or Luke is a knight. Assume Harper is a knave. No contradiction is found in their false claim that Alexander is a knight. Alexander cannot be a knight, because this would contradict the false claim of Harper that Alexander is a knight. Alexander cannot be a knave, because this would contradict the false claim of their own that Matthew is not a knave. We have exhausted all possibilities for Alexander, Harper, and Luke, so let us go back and reconsider Matthew. Assume Matthew is a knave. No contradiction is found in their false claim that Luke is a knave. Assume Luke is a knight. No contradiction is found in their claim that Harper is a knight or Luke is a knight. Assume Harper is a knight. No contradiction is found in their claim that Alexander is a knight. Alexander cannot be a knight, because this would contradict the claim of their own that Matthew is not a knave. Alexander cannot be a knave, because this would contradict the claim of Harper that Alexander is a knight. We have exhausted all possibilities for Alexander, so let us go back and reconsider Harper. Assume Harper is a knave. No contradiction is found in their false claim that Alexander is a knight. Alexander cannot be a knight, because this would contradict the false claim of Harper that Alexander is a knight. Assume Alexander is a knave. No contradiction is found in their false claim that Matthew is not a knave. James cannot be a knight, because this would contradict the claim of their own that Alexander is a knight and Matthew is a knight. Assume James is a knave. No contradiction is found in their false claim that Alexander is a knight and Matthew is a knight. Assume Sebastian is a knight. No contradiction is found in their claim that Luke is a knight if and only if Matthew is a knave.