227. Of the three inhabitants K, M, and P of the island of truth-tellers and liars, two said:

K: We are all liars.

M: One of us is a truth-teller.

Who among the inhabitants K, M, and P is the truth-teller and who is the liar?
$\triangle$ If K were a truth-teller, then based on his statement, all three islanders, including K, would be liars, which contradicts our assumption. Therefore, K is a liar. Since his statement is false, among M and P there is a truth-teller.

Now let's consider M. If M were a liar, then based on the previous statement, P would be a truth-teller. In that case, M's statement would be true, which is impossible by our assumption. Therefore, M must be a truth-teller.

Since M's statement is true, P is a liar.

Answer: K and P are liars, M is a truth-teller. $\triangle$