This puzzle type of questions can be easily solved by truth table and contradictions.
Assume 2 propositional variables P and Q where –
P: “A is Knight”, ~ P: “A is Knave”
Q: “B is Knight”, ~ Q: “B is Knave”
Given information:
- Knight always says truth means statement made by a Knight is always TRUE.
- Knave always says false means statement made by a Knave is always FALSE.
Based on the information, there can be only 4 possibility with 2 persons A and B.
Statement made by A: “The two of us are both Knights” $\equiv P\wedge Q$
Statement made by B: “A is a Knave” $\equiv \sim P$
P (Type of A) |
Q (Type of B) |
Statement made by A |
Statement made by B |
Meaning |
TRUE |
TRUE |
TRUE |
FALSE |
Assumption is B is Knight (Q = True) and Knight always says True but B (Knight) made a false statement which is a contradiction so this is not possible |
TRUE |
FALSE |
FALSE |
FALSE |
Assumption is A is Knight (P = True) and Knight always says True but A (Knight) made a false statement which is a contradiction so this is not possible |
FALSE |
TRUE |
FALSE |
TRUE |
Assumption is A is Knave (P = False) and Knave always says False and A (Knave) actually made a false statement which is consistent this is possible. Similarly we assumed B is Knight and B made a True statement so this is also consistent. |
FALSE |
FALSE |
FALSE |
TRUE |
Assumption is B is Knave (Q = False) and Knave always says False but B (Knave) made a true statement which is a contradiction so this is not possible |
Note that we got the contradictions in 3 cases as per the assumption so the assumption must be false in this 3 cases but 1 case is giving consistent results as per our assumption so this assumption must be true.
The answer would be Option C (A = Knave, B = Knight) corresponding to case 3 (P = False, Q = True).