Solution
Five Islanders and Their Conditionals
Show the problem again
Five islanders A, B, C, D and E are each a knight (always truthful) or a knave (always lying). Each makes one statement, read as an ordinary "if ... then ..." claim (it is false only when the first part is true and the second part false): - A: "If B is a knight, then I am a knave." - B: "If C is a knight, then E is a knight." - C: "If E is a knight, then A is a knight." - D: "If I am a knave, then B is a knight." - E: "If I am a knave, then D is a knight." How many of the five are knaves? Give a single integer.
Worked solution
Start with A. If B were a knight, A's statement would read "A is a knave". A knight cannot truthfully say that, and a knave saying it would be telling the truth, which a knave cannot do. So B is a knave. B's statement is then false, and a conditional is false only when its premise is true and its conclusion false: C is a knight and E is a knave. E's false statement requires the premise (E a knave, true) with a false conclusion, so D is a knave. D's false statement needs B to be a knave, which holds. C, a knight, speaks truly since E is not a knight, and A's statement is true because B is not a knight, so A is a knight. The unique consistent assignment is A and C knights, B, D and E knaves; a search over all 32 assignments agrees. Answer: 3 knaves.
Source: Original puzzle in the knights-and-knaves tradition; conditional statements are classical in Smullyan's collections. Statement written for AxiomIQ.