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 the first question, you have the following possibilities:
Person 1 is a knight, knave (telling lie), or liar
Person 2 is a knave (telling truth) or liar
Person 3 is a knave (telling lie)
For the second question, all the responses are false, or one and only one is true. The knave (telling lie) from the previous question will tell the truth this time, so one of the responses must be true.
Since the person telling the truth for question two has already been identified, Person 1 from question one can only be a liar. Now that we know that we have one knave and one liar, we know that the third response to question two is true.
For the third question, all the people are knaves or liars, so one of the first two responses is true. This make the second person from question one a knave (telling truth) as he is the only possibility to tell the truth on question three.
We now have two knaves and one liar. In the last question one knave will lie, one will tell the truth. Since there will be two lies, and the one truth must be one of the first two responses, "Different from Bert" must be false. Therefore, Alex and Bert are the same.
Alex - Knave
Bert - Knave
Carl - Liar