Alex, Bert, and Carl are each a knight, knave or a liar. Three people asked them what their types were.

The first person got the following responses:
Alex:Carl is a liar.
Bert:Alex is a knight.
Carl:Bert is a knave.

The second person got the following responses but forgot who made which ones:
Exactly one of us is a knight.
Exactly one of us is a knave.
Exactly one of us is a liar.
(Alex, Bert, and Carl each made one of the statements.)

The third person got the following responses:
Alex:Bert is a knave.
Bert:Carl is a liar.
Carl:Alex is a knight.

What types are Alex, Bert, and Carl?

By the nature of this problem, anyone giving a true statement to the first person (Knight or Knave) will also make a true statement to the third person.  Likewise, anyone making a false statement to the first person (Knave or Liar) will make a false one to the third person.

Begin by considering whether Carl is the liar.  If so, then both Alex and Bert's known statements (1 & 3) are true.  Yet Bert's allegedly true statement in 1 matches Carl's allegedly false statement in 3.  Likewise, Alex's allegedly true statement in three matches Carl's allegedly false statement in 1.  Thus, Carl cannot be a liar, which means that both Alex and Bert gave false statements in 1 & 3.  And since Carl in 3 claimed Alex to be a Knight, his statements in 1 & 3 must be false as well.  By this, none of the three can be Knights.

Since Carl is falsely accused of being a Liar by both Alex and Bert, he must be a Knave.  And since Bert is falsely accused of being a Knave by both Alex and Carl, he must be a Liar.  This reduces the problem to two possibilities:

{A = Liar, B = Liar, C = Knave}  OR  {A = Knave, B = Liar, C = Knave}

If there are two Knaves, then two of the statements to the second person would be true.  However, this cannot be since both the first and second statements to the second person would be false.  Thus, there is only one Knave and two Liars.

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Alex = Liar
Bert = Liar
Carl = Knave

Posted by hoodat on 2007-09-30 00:04:13