Knave Dilemma
In the island of knights and knaves, knights always tell the truth and knaves always lie.
Three islanders, who are each either knights or knaves, make the following statements about going to a movie:
- Alice: I'll go if and only if Bob does not go.
- Bob: I'll go if and only if Carol does not go.
- Carol: I'll go if and only if Alice does not go.
Among the 3 people, what is the minimum possible number of knaves present in this group?
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Suppose there are no knaves in the group. In that case, all of their statements would have to be true. Now suppose Alice goes. According to Alice's statement, this means Bob does not go, and according to Carol's statement, it also means Carol does not go. But if Neither Bob nor Carol goes, then Bob's statement is false, which is a contradiction. Now instead, suppose Alice does not go. Based on Alice and Carol's statements, we can conclude that Bob and Carol both go, which again conflicts with Bob's statement. So regardless of whether Alice goes or not, we arrive at a contradiction if we assume all three people are knights. So the answer must be at least 1 knave in the group.
Now to show that it's possible for this to work with one knave, suppose Alice is the knave, and Bob and Carol are knights. If Alice goes, then Carol's statement implies that Carol will not go. Since Carol doesn't go, Bob's statement implies that Bob will go. If Alice and Bob both go, then Alice's statement is false, which it should be, since we're assuming Alice is the knave.