Sample solution to question #7 of the Exam Review

A said, “At least one of us is a knave.” (S1)

B said, “At most two of us are knaves.” (S2)

NOTE: S2 is equivalent to “At least one of us is a knight.”

S1 is false only when A, B, and C are all knights.

S2 is false only when A, B, and C are all knaves.

Knave: na Knight: I

Using a truth table to differentiate the cases:

	A	B	C	S1	S2	Contradiction?
Case 1	na	na	na	T	F	Contradiction
Case 2	na	na	I	T	T	Contradiction
Case 3	na	I	na	T	T	Contradiction
Case 4	na	I	I	T	T	Contradiction
Case 5	I	na	na	T	T	Contradiction
Case 6	I	I	na	T	T	No
Case 7	I	na	I	T	T	Contradiction
Case 8	I	I	I	F	T	Contradiction

Justifications:

· The first four cases result in contradiction, because A is a knave, who always tells a lie, but S1 is true.

· Cases 2, 5, and 7 result in contradiction, because B is a knave but S2 is true.

· Case 8 is a contradiction, because A is a knight but S1 is false.

· Therefore, only case 6 makes sense. That is, A and B are knights, and C is a knave.

Answer: A is a knight, B is a knight, and C is a knave.