You are on an island inhabited only by knights, who always tell the truth, and knaves, who always lie.

Question

You are on an island inhabited only by knights, who always tell the truth, and knaves, who always lie. You meet two women who live there and ask the older one,

"Is at least one of you a knave?"

She responds yes or no, but! you do not yet have enough information to determine what they were. So you then ask the younger woman,

"Are you two of the same type?"

She answers yes or no and after that you know which type each is. What type is each?

• both knight
• both knave
• older knight, younger knave
• older knave, younger knight
• not enough information

I thought it should be "not enough information" but that seems wrong

Answer 1

Just write a truth table: You've got 4 possibilities:

• Woman 1 is a knight and woman 2 is a knight, answers are no and yes.
• Woman 1 is a knight and woman 2 is a knave, answers are yes and yes.
• Woman 1 is a knave and woman 2 is a knight, answers are no and no.
• Woman 1 is a knave and woman 2 is a knave, answers are no and no.

This means they are both knights. If the first answer was yes, it would be clear the first woman is a knight and the second a knave, so the first answer was no.

If the second answer was no, it would still be unclear what both women are, so the second answer was yes.

Answer 2

They are both knights. Here's one approach.

The answer to question one can't be "yes." If it's "yes," the only logical possibility is that the older one is a knight and the younger is a knave. But then you already have enough information to know which is which, and the riddle says you don't. So the answer to question one is "no."

Now the answer to question two can't be "no." If it's "no," then it's possible that the older is a knave and the younger is a knight, or that both are knaves. But then you don't have enough information to know which is which, and the riddle says you do. So the answer to question two is "yes."

Now that we know the answer to both questions, it then follows that they are both knights.

Answer 3

$a$: older woman is knight
$b$: younger woman is knight

Behavior of older woman can be represented with the boolean function $$f(a,b)=a(a'+ b') + a'(ab) = a b'$$ Behavior of younger woman can be represented with the boolean function $$g(a,b)=b(ab+ a' b') + b'( a'b +a b' ) = a$$

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With older woman the output has to beno, that is $f(a,b)=0$, so that we can eliminate just one case $(a, b')$, and the remaining possibilities are $$(a,b), (a',b), (a',b')$$

Above $a$ is true in only one case.
This means the younger woman has to output yes, that is $g(a,b)$ has to be $1$, then you can conclude it is $(a,b)$.