1, 2, Or 3 Of Us Are Knaves

Logic Level 1

Edgar Abercrombie was an anthropologist who was particularly interested in the logic and sociology of lying and truth-telling. One day, he visited the island of Knights (who always tell the truth) and Knaves (who always lie), and met three people, Andrew, Bernard and Charlie.

Andrew: "Exactly one of us is a knave."
Bernard: "Exactly two of us are knaves."
Charlie: "All of us are knaves."

Which of them are knaves?

This problem is taken from Raymond M. Smullyan's book Logical Labyrinths.
None of them Andrew only Bernard only Charlie only Andrew and Bernard only Andrew and Charlie only Bernard and Charlie only All of them

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Calvin Lin by Calvin Lin

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2 solutions

How many knights are there?

If there were 0 knights, then Charlie would be telling the truth and this would lead to a contradiction. Therefore, there is at least 1 knight. Since Andrew, Bernard and Charlie are saying different things about themselves, this means that then two are knaves and one is a knight. This means that Bernard is the knight, and Andrew and Charlie are knaves.

5 Helpful 1 Interesting 1 Brilliant 0 Confused
Saya Suka
Apr 25, 2021

In a list of n contradictory statements, there's at most 1 truth among them. So the possibilities are 0 or 1 truth, equivalent to 0 or 1 Knight speaking it. If there's no Knights among the three, then the unspoken truth should be "Exactly 0 of us is a knave", with both suppositions contradicting each other (all 3 are either a Knight or a knave). Therefore, it should be the case where one of them is telling the truth as a Knight, and that would be Bernard who claims the existence of 2 knaves among them, talking about knave Andrew and knave Charlie.

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