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, 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. (There are always at least one knave and one knight. Please solve this through using compound propositions) 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, 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. (There are always at least one knave and one knight. Please solve this through using compound propositions) 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, 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. (There are always at least one knave and one knight. Please solve this through using compound propositions)

Explanation / Answer

The member C is a knave. This is proved by the following argument.

Suppose C is knave. This means he is lying when he says that at most one of them is a knight, so the truth must be the opposite of what C says ( assuming C is a knave). This means the truth under this assumption is that there can be two or three knights as well. And this scenario is possible, because A and B both can be knights and C can be a knave.

Thus, C is a knave.

This means both A and B are knights, which means both of them are telling the truth.

So Mr. Smullyan should point at B ,because B is not a witch doctor.

Hope this helps !

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