Alex, Bert and Carl are all different types. One is a knave, one is either a knight or liar, and one is a doubleton. A doubleton is a type similar to a knave, except that a doubleton's truth pattern is two true statements followed by two false statements repeatedly.

From the statements below, determine who is the doubleton.

Alex:
1. I am a knight.
2. Bert is the doubleton.
3. Carl is the knave.

Bert:
1. I am the knave.
2. Carl is the doubleton.
3. Alex is a liar.

Carl:
1. I am a liar.
2. Alex is the doubleton.
3. Bert is a knight.

The best approach is to determine who is the Knight/Liar.  Upon first inspection, it is clear that Carl is neither since his first statement could not be made by either. 

One thing known about Knaves is that their 1st and 3rd statements must agree (i.e. both truthful OR both lies).  But with Doubletons, their 1st and 3rd statements must not agree (i.e. one is a lie and the other one truthful).

Examining Alex, if he is a Knight, then Bert must be a Doubleton.   Yet this case makes Bert's 1st and 3rd statements both false.  Thus, Alex cannot be a Knight.

If Alex is a liar, then Bert must be the Knave and Carl the Doubleton.  Yet this makes all three of Bert's statements false.  Thus, Alex cannot be either a Knight or a Liar.  This means that only  Bert can qualify as Knight/Liar.

By his first statement, it is obvious that Bert cannot be a Knight.  Therefore, he is a Liar, and all his statement are false.  Thus Alex is the Doubleton.

Alex = Doubleton  (Lie,Lie,Truth)
Bert = Liar  (Lie,Lie,Lie)
Carl = Knave  (Lie,Truth,Lie)

Posted by hoodat on 2007-09-20 12:39:53