  A very old problem in campanology is the search for peals. The latter can be
thought of as a heavily constrained sequence of all possible permutations of a
given size, where the exact nature of the constraints depends on which method
of ringing is desired. In particular, we consider the methods of bobs-only
Stedman Triples and Erin Triples; the existence of the latter is still an open
problem. We show that this problem can be viewed as a similarly constrained
form of the Hamiltonian cycle problem (HCP). Through the use of special
subgraphs, we convert this to a standard instance of HCP. The original problem
can be partitioned into smaller instances, and so we use this technique to
produce smaller instances of HCP as well. We note that the instances known to
have solutions provide exceptionally difficult instances of HCP.
