1. Problem 28: A says “C is the knave,” B says “A is the knight,” C says “I am the spy.” - Assign roles: one knight (always truth), one knave (always lie), one spy (can lie or truth). - If A is knight (truth): then C is knave (true), B says A is knight (true), C says “I am the spy” (false, since C is knave). Consistent. - If A is knave (lie): then C is not knave, contradicts B’s statement if B is knight or spy. - If A is spy: more complex, but checking shows no consistent assignment. Conclusion: Unique solution: A = knight, B = spy, C = knave. 2. Problem 29: A says “I am the knight,” B says “I am the knave,” C says “B is the knight.” - B says “I am the knave” is a paradox if B is knight or knave. - If B is knave (lying), statement false, consistent. - A says “I am the knight,” if true, A is knight. - C says “B is the knight,” false since B is knave. Conclusion: Unique solution: A = knight, B = knave, C = spy. 3. Problem 30: A says “I am the knave,” B says “I am the knave,” C says “I am the knave.” - Knights cannot say they are knaves (would be false). - Knaves cannot say they are knaves (would be true). - Spies can say anything. - All three claim knave, impossible for all three. Conclusion: No solution. 4. Problem 31: A says “I am the knight,” B says “A is telling the truth,” C says “I am the spy.” - If A is knight, A’s statement true. - B says “A is telling the truth,” true if B is knight or spy telling truth. - C says “I am the spy,” if true, C is spy. Check assignments: - A knight, B spy (truth), C knave (lying) contradicts C’s statement. - A knight, B knight, C spy consistent. Conclusion: Unique solution: A = knight, B = knave, C = spy. 5. Problem 32: A says “I am the knight,” B says “A is not the knave,” C says “B is not the knave.” - If A is knight, A’s statement true. - B says “A is not the knave,” true. - C says “B is not the knave,” true. If both B and C say true statements, both cannot be knaves. Try A knight, B spy, C knave: - B’s statement true (spy can tell truth), C’s statement false (knave lying). Conclusion: Unique solution: A = knight, B = spy, C = knave. 6. Problem 33: A says “I am the knight,” B says “I am the knight,” C says “I am the knight.” - Only one knight. - If A is knight, B and C lying. - B and C say “I am the knight,” false. Conclusion: Unique solution: A = knight, B = knave, C = spy. 7. Problem 34: A says “I am not the spy,” B says “I am not the spy,” C says “A is the spy.” - If A is spy, A’s statement false. - C says “A is the spy,” true. - B says “I am not the spy,” true. Try A spy, B knight, C knave: - C lying contradicts C’s statement. Try A knave, B spy, C knight: - A lying about not being spy (true, since A knave), C telling truth. Conclusion: Two possible solutions: - A = spy, B = knight, C = knave - A = knave, B = spy, C = knight 8. Problem 35: A says “I am not the spy,” B says “I am not the spy,” C says “I am not the spy.” - Only one spy. - Two people claim not spy, one must be lying. Try A knight, B knave, C spy: - B lying about not being spy (true, since B knave), consistent. Try A knave, B knight, C spy: - A lying, B truth, consistent. Try A knave, B spy, C knight: - A lying, B truth, consistent. Conclusion: Multiple solutions: - A knight, B knave, C spy - A knave, B knight, C spy - A knave, B spy, C knight Final answers summarized.