You are in a popular tourist town in the land of Liars and Knights. You happen to overhear a conversation another tourist is having with three of the locals: Alex, Bert and Carl. Each of the three could be a knight, a knave or a liar. You know that for each question the tourist asks, Alex, Bert and Carl each give one response, but you don't know who said what. The conversation is as follows:
 What type are each of you?
 I am a knight.
 I am a knave.
 I am a liar.
 How many of you are the same type?
 We are all the same type.
 We are all different types.
 Exactly two of us are the same type.
 What type is Alex?
 A knave.
 A liar.
 Different from Bert.
Can you determine which type Alex, Bert and Carl are?
{For clarity, each response will be labeled by question number and response number. (e.g. the second response to question 3 will be labeled
Q3,R2).}
Q1,R3 could only be asked by an untruthful Knave.
The responses to
Q2 are totally exclusive of each other. They also complete the set of all responses. Therefore, one must be true and the other two must be false. Since a Knave answered
Q1,R3 untruthfully, he must be the one answering truthfully to
Q2. Since the other two responses are false, there can be no Knight. Thus,
Q1,R3 is also false. With at least two false responses to
Q1 &
Q2, there must be at least one Liar. Thus,
Q2,R3 is true.
Looking at
Q3,
R1 &
R2 are exclusive of each other, meaning that one must be true and the other must be false since there can only be Knaves or Liars. Since the Knave that answered
Q1,R3 lied to
Q1, he must also lie to
Q3. Therefore, one of the remaining two cannot be a Liar, but instead is a Knave who answered truthfully to
Q1 &
Q3.
Thus, there are two Knaves and one Liar. There is also one truthful and two untruthful responses to each question.
Since the truthful response to
Q3 is either
R1 or
R2,
R3 must be untruthful. Thus Alex and Bert are the same which makes them both Knaves.
Alex  Knave
Bert  Knave
Carl  Liar

Posted by hoodat
on 20070517 18:38:21 