Answer
Person A is the knight, person B is the spy, and person C is the knave.
Work Step by Step
A says "C is the knave," B says, "A is the knight," and C says "I am the spy."
One knight, one knave, one spy.
If A is the knight, C is the knave. In this case, she does indeed lie. This would require B to be the spy. Since she says that A is the knight, she is choosing to tell the truth.
If A is the knave, then C is not the knave. B's statement is untrue. Because this is a lie and the knave spot is taken, B must be the spy. However, C also is lying so none of the three are knights. Therefore, this situation is impossible.
If A is the spy, then her statement provides no conclusive information about the identity of person C. However, person B is known to be lying and thus must be the knave (because the spy position is taken). By elimination, person C must be the knight who tells the truth. Since this identity contradicts the statement of person C, she cannot be the knight.
Therefore, A must be the knight. Person B is telling the truth, so she must be the spy making the choice to tell the truth. Therefore, person C is the knave.
