Home > Logic > Liars and Knights
| De Ja True? (Posted on 2007-09-29) |
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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?
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Submitted by Brian Smith
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Solution:
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hoodat provides a solution here. My solution follows:
All the responses to the first and third statements are either all true or all false. This can be demonstrated by the circular chain A1=A3=C1=C3=B1=B3=A1. The implications A1=A3, C1=C3, and B1=B3 are from the fact that every other statement any of the three make must be equivalent. The implications A3=C1, C3=B1, and B3=A1 are from the fact that those statements are identical pairs.
If all the statements are true, that contradicts the statements "Carl is a liar." so all the statements must be false. Then there are no knights, so Bert is then a liar, Carl is a knave and Alex is a knave or a liar.
The knave(s) must be telling the truth when answering the second question. Since the split of knaves to liars is 2-1 or 1-2, only one of the responses to the second person is true. Then there is only one knave, namely Carl. That leaves Alex and Bert as liars. |
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