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Who Said What? (Posted on 2007-05-17) Difficulty: 3 of 5
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:

  1. What type are each of you?
    • I am a knight.
    • I am a knave.
    • I am a liar.
  2. 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.
  3. What type is Alex?
    • A knave.
    • A liar.
    • Different from Bert.

Can you determine which type Alex, Bert and Carl are?

See The Solution Submitted by Brian Smith    
Rating: 4.5000 (4 votes)

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Solution solution | Comment 6 of 7 |
{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 R2R3 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 2007-05-17 18:38:21
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