A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet 6 inhabitants: Joseph, Liam, Lily, Isabella, Samuel, and Avery. Joseph noted, "Lily is not a knave". Liam commented, "Samuel is a knight if and only if Isabella is a knight". Lily was heard saying, "Liam is a knave if and only if Joseph is a knave". Isabella said, "Joseph is a knave or Avery is a knight." Samuel was heard saying, "Avery is a knave and Avery is a knight". According to Avery, "Isabella is not a knave". So who is a knight and who is a knave?
Joseph is a knight, Liam is a knight, Lily is a knight, Isabella is a knave, Samuel is a knave, and Avery is a knave.
(1) Joseph is a knight
(2) Liam is a knight
(3) Lily is a knight
(4) Isabella is a knave
(5) Samuel is a knave
(6) Avery is a knave
Assume Joseph is a knight. No contradiction is found in their claim that Lily is not a knave. Assume Lily is a knight. No contradiction is found in their claim that Liam is a knave if and only if Joseph is a knave. Assume Liam is a knight. No contradiction is found in their claim that Samuel is a knight if and only if Isabella is a knight. Samuel cannot be a knight, because this would contradict the claim of their own that Avery is a knave and Avery is a knight. Assume Samuel is a knave. No contradiction is found in their false claim that Avery is a knave and Avery is a knight. Assume Avery is a knight. No contradiction is found in their claim that Isabella is not a knave. Isabella cannot be a knight, because this would contradict the claim of Liam that Samuel is a knight if and only if Isabella is a knight. Isabella cannot be a knave, because this would contradict the false claim of their own that Joseph is a knave or Avery is a knight. We have exhausted all possibilities for Isabella, so let us go back and reconsider Avery. Assume Avery is a knave. No contradiction is found in their false claim that Isabella is not a knave. Isabella cannot be a knight, because this would contradict the claim of Liam that Samuel is a knight if and only if Isabella is a knight. Assume Isabella is a knave. No contradiction is found in their false claim that Joseph is a knave or Avery is a knight.