An Even Dozen

The surface of a standard soccer ball is covered with 20 hexagons and 12 pentagons. Interestingly, while we might vary the number of hexagons, the number of pentagons must always be 12.

That’s because the Euler characteristic of a sphere is 2, so VE + F = 2, where V is the number of vertices, or corners, E is the number of edges, and F is the number of faces. If P is the number of pentagons and H is the number of hexagons, then the total number of faces is F = P + H; the total number of vertices is V = (5P + 6H) / 3 (we divide by 3 because three faces meet at each vertex); and the total number of edges is E = (5P + 6H) / 2 (dividing by 2 because two faces meet at each edge). Putting those together gives

\displaystyle V-E+F={\frac {5P+6H}{3}}-{\frac {5P+6H}{2}}+P+H={\frac {P}{6}},

and since the Euler characteristic is 2, this means P must always be 12.