There is an island where every inhabitant is either of Type X or Type Y. One of the types always tells the truth, and the other type always lies, but you are not sure which is which. You meet two inhabitants of this island, A and B.
A: B is of Type X.
B: We are both of Type Y.
Which type tells the truth, which type lies, and what are A and B?

Suppose Type X always tells the truth and Type Y always lies. If A is of Type X, then A's statement is true, so B is of Type X. Then, B would be lying about them both being of Type Y, which is impossible since B is of Type X and tells the truth. If A is of Type Y, then A's statement is false, so B is really of Type Y. Then, they are both of Type Y, so B's statement is true, but B is of Type Y and always lies. We get a contradiction, so Type Y tells the truth and Type X lies. If A is of Type X, then A is lying, so B is of Type Y. Then, B is lying about them both being of Type Y, but B is of Type Y and always tells the truth. That is a contradiction. Therefore, A is of Type Y, so A's statement is true and B is of Type X. In summary, A is of Type Y and tells the truth, and B is of Type X and lies.

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