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?
Shorthand:
T = knighT
L = Liar
LTL = Knave who starts and ends with lie
TLT = Knave who starts and ends with truth
For question 1:
Who could say I am a knight?
Could be T, L or LTL
Who could say I am a knave?
Could be L or TLT
Who could say I am a liar?
Only an LTL
So there are 3 possibilities for the type of the first person and 2 for the second, making 6 possibilities altogether:
1 T; 1 L; 1 LTL
1 T; 1 LTL; 1 TLT
2 L; 1 LTL
1 L; 1 LTL; 1 TLT
1 L; 2 LTL
2 LTL; 1 TLT
For question 2:
Exactly 1 of these statements must be true. How many true statements would you expect from each of the six lineups made possible by question 1?:
1 T; 1 L; 1 LTL 2
1 T; 1 LTL; 1 TLT 2
2 L; 1 LTL 1
1 L; 1 LTL; 1 TLT 1
1 L; 2 LTL 2
2 LTL; 1 TLT 2
So it would seem that there are either 2 liars and 1 knave who starts and ends with lies, or a liar and one of each type of knave.
For question 3:
So there is at least one liar and at least one knave. There are either 2 or 3 lies told for the last question. If A is indeed a liar, then the middle answer is the only true one, so B is also a liar. But if there are 2 liars (as opposed to 1, the only other possibility), then all three statements would have to be lies, as the knave would have to be in his lying phase.
So there's one liar and one of each kind of knave. That means two lies were told and one truth, the truth being told by one of the knaves. If A were the liar, only the first statement would be a lie, but there must be two lies. So A is one of the knaves, and the first statement is the one that's true. So A is no different from B, so he's also a knave, leaving C as the liar.

Posted by Charlie
on 20070517 15:36:02 