Answer
Person A is the knight, person B is the spy, and person C is the knave.
Work Step by Step
Person A says "I am the knight," person B says "I am the knave," and person C says "B is the knight."
If person A is the knight, she is telling the truth. Person B is either the knave or the spy. She cannot be the knave or her statement would be true, which is impossible. Thus, she is the spy and person C must be the knave. As required, C's claim is false. This is a solution. Now we will examine whether there are others.
If person A is the spy, she is allowed to lie to say she is the knight. Person B must be the knave or the knight. She is telling the truth if she is a knave, which is impossible, and she is lying if she is the knight, which is also impossible. Thus, person A cannot be the spy.
If person A is the knave, she lies to say she is the knight. Person B can be either spy or knight, and she must be the spy because to say she is the knave is untrue. Person C must then be the knight. However, the statement that B is the knight is false, so she cannot be the knight. Person A cannot be the knave.
