  Ice systems are prototypes of locally constrained dynamics. This is
exemplified in Coulomb-liquid phases where a large space of configurations is
sampled, each satisfying local ice rules. Dynamics proceeds through `flipping'
rings, i.e., through reversing arrows running along the edges of a polygon. We
examine the role of defect rings in such phases, with square-ice as a testing
ground. When placed on a curved surface, the underlying square lattice will
form defects such as triangles or pentagons. We show that triangular defects
are statistically more `flippable' than the background. In contrast, pentagons
and larger polygons are less flippable. In fact, flippability decreases
monotonically with ring size, as seen from a Pauling-like argument. As an
explicit demonstration, we wrap the square ice model on a sphere. We start from
an octahedron and perform repeated rectifications, producing a series of
clusters with sphere-like geometry. They contain a fixed number of defect
triangles in an otherwise square lattice. We numerically enumerate all
ice-rule-satisfying configurations. Indeed, triangles are flippable in a larger
fraction of configurations than quadrilaterals. The obtained flippabilities are
in broad agreement with the Pauling-like estimates. As a minimal model for
dynamics, we construct a Hamiltonian with quantum tunnelling terms that flip
rings. The resulting ground state is a superposition of all ice configurations.
The dominant contribution to its energy comes from localized resonance within
triangles. Our results suggest local dynamics as a promising observable for
experiments in spin ice and artificial ice systems. They also point to
hierarchical dynamics in materials such as ice V that contain rings of multiple
sizes.
