Step-by-step Solution:

\[ \text{Let }T=\text{Truth-teller},\quad L=\text{Trickster (liar).} \] \[ \text{Statements:}\quad X:\ "Y\ \text{is a Trickster}"\qquad Y:\ " \text{Exactly one of }(X,Y,Z)\text{ is a Truth-teller.}" \] \textbf{Case 1: } Assume \(X\) is \(T\).\\ Then \(X\)'s statement is true, so \(Y\) is \(L\). Since \(Y\) is lying, the statement “exactly one of \((X,Y,Z)\) is \(T\)” is false. But \(X\) is \(T\), so to make “exactly one” false, \(Z\) must also be \(T\) (giving 2 truth-tellers). Hence in this case \(Z\) is \(T\), i.e. \(Z\) has the same type as \(X\). \medskip \textbf{Case 2: } Assume \(X\) is \(L\).\\ Then \(X\)'s statement is false, so \(Y\) is \(T\). Since \(Y\) is telling the truth, “exactly one of \((X,Y,Z)\) is \(T\)” is true. But \(Y\) is the single truth-teller, so \(X\) and \(Z\) must both be \(L\). Hence in this case \(Z\) is \(L\), i.e. again \(Z\) has the same type as \(X\). \bigskip \[ \boxed{\text{Therefore }Z\text{ is of the same type as }X\ (\text{both Truth-tellers or both Tricksters}).} \]