  Let $G$ be a planar graph and let $C$ be a cycle in $G$. Inside of each
finite face of $G$, we write down the number of edges of that face which belong
to $C$. This is the signature of $C$ in $G$. The notion of a signature arises
naturally in the context of Slitherlink puzzles. The signature of a cycle does
not always determine it uniquely. We focus on the ambiguity of signatures in
the case when $G$ is a rectangular grid of unit square cells. We describe all
grids which admit an ambiguous signature. For each such grid, we then determine
the greatest possible difference between two cycles with the same signature on
it. We also study the possible values of the total number of cycles which fit a
given signature. We discuss various related questions as well.
