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The Wikipedia article on Knights and Knaves says,
Knights and Knaves is a type of logic puzzle. On a fictional island, all inhabitants are either knights, who always tell the truth, or knaves, who always lie. The puzzles involve a visitor to the island who meets small groups of inhabitants. Usually the aim is for the visitor to deduce the inhabitants’ type from their statements, but some puzzles of this type ask for other facts to be deduced. The puzzle may also be to determine a yes/no question which the visitor can ask in order to discover what he needs to know. In some variations, inhabitants may also be alternators, who alternate between lying and telling the truth, or normals, who can say whatever they want (as in the case of Knight/Knave/Spy puzzles).
One important idea that is really helpful in solving such type of questions is the concept of Liar’s paradox. It essentially states that no one can make the statements like “I am a liar” or “This statement is false” as they are logical self-contradictions. Let me try to elaborate,
- I am a liar means that I always lie. This implies that the statement I made is a lie. This means that I am not a liar. As you can see, this is a logical contradiction.
- This statement is false means the statement is false, which would in turn mean that the statement is actually true. As you can see, we have again reached a logical contradiction.
A large type of such questions can be solved by making truth tables. Another popular technique is assuming one of the statements to be true and then figuring out the answer.
Let us look at some of the elementary examples.
Q1. Pankaj visits an island on which all inhabitants are either knights or knaves. He meets Ravi and Apurv there. Can you figure out who is what from their statements?
Type 1 – Ravi says: We are both knaves.
Ravi’s statement can’t be true because nobody can admit to being a liar (read the logical contradiction part above).
=> Ravi is a knave
=> Ravi must have been lying about them both being knaves
=> Apurv is a knight.
Type 2 – Ravi: We are the same kind.
Apurv: We are of different kinds.
Ravi & Apurv are making contradictory statements
=> Both of them cannot be of the same kind
=> One of them must be a knight and the other one must be a knave
=> They are of different kinds
=> Apurv is a knight and Ravi is a knave
Now let us look at some slightly more advanced examples.
Q2. There are three friends A, B and C. One of them is a knight; he always speaks the truth, and the other two are knaves; they always tell lies. One day, A says: “I am a knight.” B says: “What A says is true.” C says: “Both A and B are lying.” Who is what?
Solution. Let us try and draw up a table with this info keeping in mind that only one of them can be the knight. We will consider three cases, Case 1 – A is knight, Case 2 – B is knight, Case 3 – C is knight.
We need to have 1 True statement and 2 False statements which is happening only in the third case where C is the knight. So, A is knave, B is knave and C is knight.
Q3. Four jokers – A, B, C and D – are brought before Batman. He knows that two of them are lairs.
. A: “D robbed the bank”
. B: “C is always true.”
. C: “B is always true.”
. D: “B is a liar, but he didn’t rob the bank”
(1) D (2) B or C (3) B or D (4) C or D
Solution. Looking at the statements of B and C, either both of them are speaking the truth or both of them are lying. Let us consider two cases, Case 1 – B & C speak the truth and Case 2 – B & C are lying.
In case 1, from D’s statement – B is the robber.
In case 2, from A’s statement – D is the robber.
We cannot uniquely identify the robber. The robber is B or D. So, option 3.
Q4. While Balbir had his back turned, a dog ran into his butcher shop, snatched a piece of meat off the counter and ran out. Balbir was mad when he realised what had happened. He asked three other shopkeepers, who had seen the dog, to describe it. The shopkeepers really didn’t want to help Balbir. So each of them made a statement which contained one truth and one lie.
Shopkeeper Number 1 said: “The dog had black hair and a long tail.”
Shopkeeper Number 2 said: “The dog has a short tail and wore a collar.”
Shopkeeper Number 3 said: “The dog had white hair and no collar.”
Based on the above statements, which of the following could be a correct description? [CAT 2001]
(1) The dog had white hair, short tail and no collar.
(2) The dog had white hair, long tail and a collar.
(3) The dog had black hair, long tail and a collar.
(4) The dog had black hair, long tail and no collar.
Solution. We will have to assume some part of the statement to be true and proceed from there.
Case 1: Assumption is that the dog had black hair
The dog had black hair
=> Short tail (from Shopkeeper number 1)
=> No collar (from Shopkeeper number 3)
Case 2: Assumption is that the dog had white hair
The dog had white hair
=> Long tail (from Shopkeeper number 1)
=> Collar (from Shopkeeper number 3)
So, from the two cases we get two possible conclusions which are:
The dog had black hair, short tail and no collar.
OR
The dog had white hair, long tail and a collar.
The second case is given to us in Option 2.
Q5. Out of three people X, Y and Z, one is a knight, one a knave and the third a spy, who can either speak the truth or lie. X says: “Y is not a spy.” Z says: “X is a spy.” Which of the following statements is definitely true?
(1) X is a knave
(2) Y is a knave
(3) X is a spy
(4) Y is a knight
(5) Z is a knight
Solution. Let us consider cases with various possibilities for X
Case 1:
X is a knight so he is speaking the truth
=> Y is not a spy
=> Y is a knave
=> Z is a spy and he is lying.
Case 2:
X is a knave so he is lying
=> Y is a spy
=> Z has to be a knight but there is logical inconsistency
=> Case 2 is not possible
Case 3:
X is a spy and we do not know whether he is speaking the truth / lying
=> Z is a knight
=> Y is a knave
So, the possible combinations are
X (Knight), Y (Knave) & Z (Spy)
OR
X (Spy), Y (Knave) & Z (Knight)
In both cases Y is a knave. Option 2 is our answer.
Ravi Handa has taught Quantitative Aptitude at various coaching institutes for seven years. An alumnus of IIT Kharagpur where he studied a dual-degree in computer science, he currently runs an Online CAT Coaching and CAT Preparation Course on the website Handa Ka Funda.