The action for this problem takes place in an island of Knights and Knaves

Question

The action for this problem takes place in an island of Knights and Knaves, where Knights always make true statements and Knaves always make false statements and everybody is either a Knight or a Knave. Two friends A and B lives in a house. The census taker (an outsider) knocks on the door and it is opened by A. The census taker says ''I need information about you and your friend. Which, if either of you, is a Knight and which, if either of you, is a Knave?" "We are both Knaves" says A angrily and slams the door. What, if any thing can the census taker conclude?

a.A is a Knight and B is a Knave.

b.A is a Knave and B is a Knight.

c.Both are Knaves.

d.Both are Knights.

e.No conclusion can be drawn.

Not able to grasp it please someone help me

Answer 1

"a.A is a Knight and B is a Knave."

Then that'd be a lie. Knights can't lie so this is impossible.

"b.A is a Knave and B is a Knight."

That'd be a lie. Knaves lie. This is possible.

"c.Both are Knaves."

Then that'd be the truth. Knaves can't tell truth so this is impossible.

"d.Both are Knights."

Then that'd be a lie. Knights can't lie so this is impossible.

....

but to do it directly.

If "we are both knaves were true" then A is knave telling the truth. That's impossible so it is a lie. So A is lying so A is a knave. But as it is a lie they are not both knaves. So B is not a knave. So B is a knight.

Answer 2

First, you know they cannot both be Knights, since if A were a knight he would not have said they are both knaves. In fact, A cannot be a knight, because his statement says that he is a knave, and if he were a knight he would not be lying.

And they cannot both be knaves, since in that case A has told the truth, which a knave never does.

Therefore, A is a knave and B is a Knight.