Soccer balls are usually covered with a design based on regular pentagons and hexagons.
How many pentagons/hexagons MUST there be, and why?
(In reply to A related question...
As it turns out, the number of hexagons must be 0 or 20. If there
was some other solid, then it would have been included as a 'Johnson
Solid'. Johnson solids are convex polyhedra which have all faces
as regular polygons of unit length and are NOT Platonic solids,
Archimedean solids, prisms or antiprisms. The number of Johnson
solids was proven to be 92, and none of those solids is composed of
solely pentagons and hexagons.