Let me quickly re-check the logic:

1. Let T = Truth-teller, L = Trickster (liar).

X says: “Y is a Trickster.”
Y says: “Exactly one of us (X, Y, Z) is a Truth-teller.”

Case 1: Assume X is T.
Then Y must be L.
Y’s statement must be false: “Exactly one of us is a Truth-teller” is false.

• Truth-tellers so far: X (1).
  If Z were L, then exactly one (X) would be T → Y’s statement would be true, contradiction.
  So Z must be T.
  Configuration: (X T, Y L, Z T) works.

Case 2: Assume X is L.
Then “Y is a Trickster” is false → Y is T.
Now Y’s statement must be true: exactly one of X, Y, Z is T.

• We already have Y as T, so X and Z must both be L.
  Configuration: (X L, Y T, Z L) works.

Both configurations are consistent:

• In one, Z is T.

• In the other, Z is L.

So Z can be either a Truth-teller or a Trickster