  Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved
that the Tantrix(TM) rotation puzzle problem is NP-complete. They also showed
that for infinite rotation puzzles, this problem becomes undecidable. We study
the counting version and the unique version of this problem. We prove that the
satisfiability problem parsimoniously reduces to the Tantrix(TM) rotation
puzzle problem. In particular, this reduction preserves the uniqueness of the
solution, which implies that the unique Tantrix(TM) rotation puzzle problem is
as hard as the unique satisfiability problem, and so is DP-complete under
polynomial-time randomized reductions, where DP is the second level of the
boolean hierarchy over NP.
