Having just won His prize, Mr. Smullyan was enjoying his visit - that is, until

Question

Having just won His prize, Mr. Smullyan was enjoying his visit - that is, until he got captured by a ferocious gang of local brigands! They grabbed his prize too, and promised to kill him unless he could solve another puzzle: Among three gang members A, B, C, exactly one was a witch doctor. Mr. Smullyan had to point at one of the three. If he pointed at the witch doctor, he would be killed! If he pointed at one that was not a witch doctor, he would go free. But he would have his prize back only if he could give a rigorous explanation. Here is what those three said to Mr. Smullyan: A: I am a witch doctor. B: I am not a witch doctor. C: At most one of us is a knight. Can you do it? Perform formal analysis as in Prob 3.

Explanation / Answer

Mr.Sullyan must point at C to go free.

Reason:

There are 2 cases:

(i) C is a knight. Then his statement that "At most one of us is a knight" must be true implying he is the only knight. This would mean both A and B are knaves. Since A said he is a witch doctor, he is lying and he is not the witch doctor. Similarly B's claim that he is not the witch doctor is false and so he is indeed the witch doctor. Therefore, Mr.Smullyan must point at A or C to go free.

(ii) C is a knave. Then his statement that "At most one of us is a knight" must be false indicating there are two or three knights. But since C is a knave there are exactly two knights A and B. Now since A says he is the witch doctor, he is indeed the witch doctor and since B says that he is not the witch doctor, he is not the witch doctor. So in this case Mr.Sullyan must point at B or C to go free.

In both cases, C is not the witch doctor. So Mr. Sullyan must point at C to go free.

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