A very special island is inhabited only by knights and liars. Knights always tell the truth, and liars always lie.

Nine inhabitants of the island: Mel, Bart, Sue, Betty, Rex, Zeke, Sally, Zoey and Homer are busy at a conversation. A visitor from a neighboring island stops by and asks each of the nine inhabitants their respective identities. They say:

• Mel: "Only a liar would say that Sally is a liar."
• Bart: "Rex is a liar."
• Sue: "Mel and Homer are liars."
• Betty: "I know that I am a knight and Sally is a liar."
• Rex : "Betty and I are both knights."
• Zeke: "At least one of the following is true: that Sally is a knight or that Sue is a knight."
• Sally:"It's false that Betty is a liar."
• Zoey: "It's not the case that Sue is a liar."
• Homer: "Betty is a liar or Zeke is a liar."

Determine the type of each of the inhabitants from the abovementioned statements.

(In reply to computer solution by Charlie)

Homer stated that "Betty is a liar or Zeke is a liar."
One of the problems with a statement such as this is that the operative word, "or", may be interpretted as the operator for either an exclusive or a logical disjunction. (The 'or' used in the last statement is an exclusive disjunction as it is used with the word 'either'. The absence of the word 'either' does not exclude 'or' from being interpreted as an exclusive disjunction, as in common speech it is often still to be interpreted as such.)

As an exclusive disjuntion, one of the following would be TRUE, none of which leads to a solution:

• Homer is a knight   Betty is a liar       Zeke is a knight
• Homer is a knight   Betty is a knight   Zeke is a liar
• Homer is a liar       Betty is a knight   Zeke is a knight
• Homer is a liar       Betty is a liar       Zeke is a liar

As a logical disjunction, one of the following would be TRUE, of which, the third does lead to the solution:

• Homer is a knight   Betty is a liar       Zeke is a knight
• Homer is a knight   Betty is a knight   Zeke is a liar
• Homer is a knight   Betty is a liar       Zeke is a liar
• Homer is a liar       Betty is a knight   Zeke is a knight

Thus, the only solution would be, as did Charlie deduce with his interpretation using Homer's 'or' as a logical disjunction, is that Homer and Bart are knights and the other of the seven conversing inhabitants are liars.

Edited on February 7, 2010, 8:13 am

Posted by Dej Mar on 2010-02-06 21:49:47