Question

Three friends, P, Q, and R, are solving a puzzle with statements: 
(i) If P is a knight, Q is a knave. 
(ii) If Q is a knight, R is a spy. 
(iii) If R is a knight, P is a knave. Knights always tell the truth, knaves always lie, and spies sometimes tell the truth. If each friend is either a knight, knave, or spy, who is the knight?

• A. P
• B. Q
• C. R
• D. None

Hint:

For logic puzzles with knights, knaves, and spies, systematically test each person as a knight, use implications to assign roles, and ensure consistency with truth values.

Answer 1

The Correct Option is B

The objective is to identify the knight among individuals P, Q, and R by analyzing their statements, considering the defined behaviors of knights, knaves, and spies.

1. Definitions:

- Knight: Exclusively states truths. - Knave: Exclusively states falsehoods. - Spy: May state truths or falsehoods. - Each individual is precisely one of these three types.

2. Provided Statements:

(i) P is a knight implies Q is a knave. (ii) Q is a knight implies R is a spy. (iii) R is a knight implies P is a knave.

3. Scenario Analysis:

- Hypothesis: P is the knight. - If P is a knight, statement (i) is true, meaning Q is a knave. - If Q is a knave, statement (ii) made by Q must be false. Statement (ii): "If Q is a knight, then R is a spy." Since Q is not a knight, the antecedent ("Q is a knight") is false. A conditional statement with a false antecedent is logically true. This contradicts the requirement that a knave must lie. - Conclusion: P cannot be the knight.

- Hypothesis: Q is the knight. - If Q is a knight, statement (ii) is true, meaning R is a spy. - If R is a spy, statement (iii) made by R is not necessarily true or false, and thus provides no direct contradiction. - If Q is a knight, statement (i) implies that if P were a knight, Q would be a knave. Since Q is a knight, P cannot be a knight. Therefore, P must be a knave. - This assignment (Q=knight, P=knave, R=spy) is consistent with all statements and definitions.

- Hypothesis: R is the knight. - If R is a knight, statement (iii) is true, meaning P is a knave. - If P is a knave, statement (i) made by P must be false. Statement (i): "If P is a knight, then Q is a knave." Since P is not a knight, the antecedent ("P is a knight") is false, making the implication logically true. This contradicts the requirement that a knave must lie. - This hypothesis leads to a contradiction.

4. Determination:

The only consistent scenario where all statements align with the defined roles is when Q is the knight, P is the knave, and R is the spy.

Final Answer:

The knight is Q.