Mr. Smullyan visited a famous island where knights always tell the truth and kna

Question

Mr. Smullyan visited a famous island where knights always tell the truth and knaves always lie and every inhabitant is either a knight or a knave. He was introduced to three inhabitants A, B, and C of which at least one was a knave and one a knight. One of them had a prize that Mr. Smullyan could win (if and) only if he could determine correctly which one had it. The three spoke to him. A said: B does not have the prize. B said: I don’t have the prize. C said: I have the prize. Formalize the problem using six propositional variables, one for each of the locals being a knight (or not) and one for each having the prize or not. Write down a compound proposition that takes the value T (true) exactly when the values of all propositional variables describe the solution to this puzzle. Which local has the prize? Can you argue formally using your formula? Mr. Smullyan visited a famous island where knights always tell the truth and knaves always lie and every inhabitant is either a knight or a knave. He was introduced to three inhabitants A, B, and C of which at least one was a knave and one a knight. One of them had a prize that Mr. Smullyan could win (if and) only if he could determine correctly which one had it. The three spoke to him. A said: B does not have the prize. B said: I don’t have the prize. C said: I have the prize. Formalize the problem using six propositional variables, one for each of the locals being a knight (or not) and one for each having the prize or not. Write down a compound proposition that takes the value T (true) exactly when the values of all propositional variables describe the solution to this puzzle. Which local has the prize? Can you argue formally using your formula? Mr. Smullyan visited a famous island where knights always tell the truth and knaves always lie and every inhabitant is either a knight or a knave. He was introduced to three inhabitants A, B, and C of which at least one was a knave and one a knight. One of them had a prize that Mr. Smullyan could win (if and) only if he could determine correctly which one had it. The three spoke to him. A said: B does not have the prize. B said: I don’t have the prize. C said: I have the prize. Formalize the problem using six propositional variables, one for each of the locals being a knight (or not) and one for each having the prize or not. Write down a compound proposition that takes the value T (true) exactly when the values of all propositional variables describe the solution to this puzzle. Which local has the prize? Can you argue formally using your formula?

Explanation / Answer

Background for the question-

Firstly let P, Q and R be the propositional variables for A, B and C. Also, the result column would denote if they cab result into a solution.

Now the statement given by A and Bush are the same and hence can only be TRUE or FALSE together.

Thus, the cases not having both as True or False can be taken as not possible directly.

P Q R Result T T F Answer - A has the prize T F T Not Possible (A & B should be same) F T T Not Possible F F T Both B and C can't have the prize F T F Not Possible T F F Not Possible

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