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# Knights and Knaves

On a special island, everyone is either a knight or a knave. Knights always tell the truth, and knaves always lie. However, there is no way to tell who is a knight and who is a knave just by appearance--you have to listen to what they say!

Recall that knights always tell the truth and knaves always lie. Suppose two island-dwellers make the following statements:

Alice: "Bob is a knave."
Bob: "Neither of us is a knave."

Which one is which?

Sometimes knowing the answers isn't even necessary!

Alice came across two island-dwellers, and asked them both the question, "Are you both knights?" After they answered (each one either "yes" or "no"), Alice knew how many of them were knights. How many of them were knights?

Alice came across two island-dwellers, and asked each of them the question, "Is the other person a knave?" After they answered, Alice knew how many of them were knights. How many of them were knights?

Alice came across 3 island-dwellers, and asked each of them how many of them were knaves. They all gave the same answer.

Afterwards, another island-dweller joined them, and Alice asked all four of them how many of them were knaves. Again, they all gave the same answer, but their answer was different from the first time. How many of them in this second group were knaves?

Alice came across 5 island-dwellers, all of whom knew one another, that would only answer yes-or-no questions of the form "Is X a knight?" What is the minimum number of questions Alice need to ask, if in any scenario she would be able to determine the number of knights?

Alice comes across 5 island-dwellers, none of whom knows one another. When Alice asks one of them about another person, they first ask that person (in private, without Alice hearing) before giving an answer.

For instance, if Alice asks person A if person B is a knight, person A will first ask person B if they are a knight and will then answer Alice (person A and person B, of course, might lie).

Note in this case that knights can "accidentally" lie--they tell the truth to the best of their knowledge based on their questioning.

What is the minimum number of questions Alice need to ask, if in any scenario she would be able to determine the number of knights?

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