There are three kinds of people on an island: knights who always tell the truth, knaves who always lie, and spies who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. A says “I am the knight,” B says “A is not the knave,” and C says “B is not the knave". Is there a unique solution to determine the knight, the knave, and the spy? If yes, determine who the knight, knave, and spy are.

We know that there are one knight, one knave and one spy among three people, A, B, and C.

a) If A is a knight, he tells the truth that he is a knight.

Then B tells the truth that A is not the knave. Hence B is a spy, and C is a knave. But C tells the truth that B is not the knave. We have a contradiction.

b) If B is a knight, he tells the truth that A is not the knave. Hence A is a spy, and C is a knave. But C tells the truth that B is not the knave. We have a contradiction.

c) If C is a knight, he tells the truth that B is not the knave. Hence B is a spy, and A is a knave.

Check

A is a knave and A lies that he is a knight. True.

B is a spy and he lies that A is not the knave. Possible.

C is a knight and he tells the truth that B is not the knave.

C is the knight, A is the knave, B is the spy.