Binary Logic

We are given the following statements:
- X says: "Y is a Trickster."
- Y says: "Exactly one of us is a Truth-teller."
Let's consider the possible cases:
1. Case 1: If X is a Truth-teller:
- If X is a Truth-teller, then X's statement is true, so Y must be a Trickster.
- If Y is a Trickster, then Y's statement "Exactly one of us is a Truth-teller" must be a lie, meaning that both X and Z are either both Truth-tellers or both Tricksters.
- Therefore, Z must also be a Trickster.
2. Case 2: If X is a Trickster:
- If X is a Trickster, then X's statement "Y is a Trickster" is false, meaning that Y must be a Truth-teller.
- If Y is a Truth-teller, then Y's statement "Exactly one of us is a Truth-teller" must be true, meaning that X and Z must both be Tricksters.
In both cases, we conclude that Z could either be a Truth-teller or a Trickster, so we cannot definitively determine Z's identity based on the given statements.
Thus, the correct answer is option (a).