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