Solution
One Knight, One Knave, Two Normals
Show the problem again
Four people, Ana, Ben, Cal and Dee, are exactly one knight (always truthful), one knave (always lying) and two normals (each statement may be true or false, at will). They say: - Ana: "Dee is not a knave." - Ben: "Dee is of the same type as Ana." - Cal: "Exactly one of Cal and Dee is normal." - Dee: "Cal is of the same type as Ben." Who is the knight? Answer with the name.
Worked solution
First, Dee is not the knave. If she were, Ana's claim would be false, so Ana would be normal; Ben's claim (Dee and Ana alike) would be false, so Ben is not the knight; thus Cal is the knight. But then Cal's claim "exactly one of Cal and Dee is normal" is false (neither is normal), which a knight cannot say. So Ana's claim is true, and Ana is not the knave. Dee is not the knight either: then Ana would have to be normal (she is not the knave), so Ben and Cal would be the knave and the remaining normal, which differ in type, and Dee's claim "Cal and Ben are alike" would be false. Hence Dee is normal. Now Cal's claim is true if Cal is the knight and false if Cal is normal, so Cal cannot be the knave, since a knave cannot make a true statement (and with Dee normal, Cal being the knave would make his claim true). With neither Ana nor Cal the knave, Ben is the knave. Ben's claim is therefore false, so Dee (normal) and Ana differ in type, and Ana is not normal: she is the knight, leaving Cal normal. A search over all twelve role assignments agrees that this is the only consistent one. Answer: Ana.
Source: Original puzzle using Smullyan's 'normal' inhabitants from What Is the Name of This Book? (1978). Statement written for AxiomIQ.