Soccer balls are usually covered with a design based on regular pentagons and hexagons.

How many pentagons/hexagons MUST there be, and why?

Your standard soccer ball has twelve pentagons and (I think) 20 hexagons.  Does there HAVE to be this many? I don't think so.  If there were zero hexagons, the 12 pentagons would form a dodecahedron which you could kick around.  If you wanted to, you could stick with 12 pentagons and increase the number of hexagons to make Buckyballs of various sizes (some hexagons would not be regular, but that is not a constraint of this problem as worded).  At first blush it seems you need 12 pentagons, and a certain number of hexagons that follow a geometric progression that I won't figure out just now.

Posted by Bryan on 2004-04-08 15:26:09