  We show how two techniques from statistical physics can be adapted to solve a
variant of the notorious Unique Games problem, potentially opening new avenues
towards the Unique Games Conjecture. The variant, which we call Count Unique
Games, is a promise problem in which the "yes" case guarantees a certain number
of highly satisfiable assignments to the Unique Games instance. In the standard
Unique Games problem, the "yes" case only guarantees at least one such
assignment. We exhibit efficient algorithms for Count Unique Games based on
approximating a suitable partition function for the Unique Games instance via
(i) a zero-free region and polynomial interpolation, and (ii) the cluster
expansion. We also show that a modest improvement to the parameters for which
we give results would refute the Unique Games Conjecture.
