Abe, Ben and Cal live on a remote island inhabited by three groups: the Knights, the Liars and the Knaves. Precisely one of them is a Knight, who always tells the truth, another is a Liar, who always lies, and the remaining one is a Knave, who alternately tells the truth and a lie.

One of the three men is the President of the island.

Abe says:
(1) “The President belongs to a different group from each of the other two of us".
(2) “Ben is not the President".

Ben says:
(1) "The President is a Liar".
(2) "Abe is not the President”.

Cal says:
(1) "Exactly two of us belong to the same group".
(2) "I am not the President".

Who is the President?

We know that Abe's first statement is true as we're told that initially. Likewise Cal's first statement is false.

Either Ben or Cal is the liar.

If Cal is the liar he is president and both Ben's statements are true, making him the knight, but also making Abe's statements both true and therefore also a knight, contradicting the assumption that all are different. So Cal is not the liar.

Therefore Ben is the liar, and thus also not the president. In fact Ben's second statement makes Abe the president, which answers the puzzle.

In sum:

Abe is the president and a knight.
Ben is the liar.
Cal is a knave, telling a lie first and then the truth.

Posted by Charlie on 2012-05-15 17:21:12