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An island has two kinds of inhabitants, knights, who always tell the truth, and knaves, who always lie. You encounter two people $A$ and $B$. What are $A$ and $B$ if: $A$ says “$B$ is a knight” and $B$ says “The two of us are opposite types”?
1. $A$ is knight, $B$ is knight
2. $B$ is knight, $B$ is knave.
3. $A$ is knave, $B$ is knave.
4. $A$ is knave, $B$ is knight

Assume $A$ is knight then $B$ is also knight But now according to $B$, “The two of us are opposite types” should be true But it is Not true.
So, $A$ is knave, so, according to $A$ “$B$ is a knight”, should be a lie, so $B$ is knave.

Knights : x : TRUE wala.

Knaves : y: FALSE wala.

A: “B is x”

B: “Opposite”

Option A :

A is x & B is x?

As A is x: which means he is true → B is x

As B is given/proven to be x → A must be y: CONTRADICTION: HENCE OPTION A IS FALSE

Option B :

A is x & B is y?

As A is x: which means he is true → B is x: CONTRADICTION: HENCE OPTION B is also FALSE

Option C :

A is y & B is y?

A is y → which means A is false → B is y

B is y → which means B is false → A is y

HENCE TRUE

@Deepak Poonia sir : Please confirm this.

@Udhay_Brahmi

It’s correct But making two cases is more efficient. Being efficient is also important.