Liar!

Classwork

1)     There is an island with exactly two types of people--truthtellers who always tell the truth and liars who always lie. Any statement by a liar is false, taken as a whole. Any statement by a truthteller is true. Xavier, Yancy and Zach make the following statements about each other:

Xavier: "Yancy is a liar."
Yancy: "Zach is a liar."
Zach: "Those two are both liars."

2)     A stranger meets a group of three people, A, B, and C on the island. The following conversation ensues.

Stranger to A: "How many truthtellers are among you ?"
A: mumble, mumble
Stranger to B: "What did A say?"
B: "A said that there is exactly one truthteller among us."
C: "Don't believe B; he is lying!"

3)     There is a group of three people, A, B, and C on the island. A and B make the following statements:

A: All of us are liars;
B: Exactly one of us is a truthteller.

4)     There is a group of three people, A, B, and C on the island. The conversation goes:

A: "B and C are the same type."
Stranger: "Are A and B of the same type?"

5)     There is an island with exactly three types of people -- truthtellers who always tell the truth, and liars who always lie, and normals who sometimes tell the truth and sometimes lie. We are given three people, A, B, C, one of whom is a truthteller, one a liar, and one a normal (but not necessarily in that order ). They make the following statements:

A: I am normal;
B: That is true.
C: I am not normal .

6)     Liars are said to be of the lowest rank , normals are middle rank, and truthtellers of the highest rank. Two people A and B on the island make the following statements:

A: I am of lower rank than B
B: That's not true!

7)     A queen, wishing to know which of her three advisers is the wisest, paints a white spot on each of their foreheads, tells them the spots are black or white and there is at least one white spot, and asks them to tell her the color of their own spots. After a time the first advisor says : "I don't know whether I have a white spot." The second, hearing this, makes the same statement. The third advisor then says: "My spot must be White". How is it possible that third advisor can figure it out when the first two could not? Is the third advisor wiser or luckier?