A brute force computer aided proof of an existence result about extremal hyperbolic surfaces

Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence of a second extremal packing in the same surface, but constructing explicit examples of this phenomenon is a complicated task. We present a brute force computational procedure that can be used to produce examples in all cases.