Extending the idea of W. Jagy, this is a Mathematica code visualizing that 33 spheres with radius 1/2 centered at the origin and the midpoints of the faces of a soccerball with circumradius 3/4 cover the unitsphere.
coord = PolyhedronData["TruncatedIcosahedron", "VertexCoordinates"];
faces = PolyhedronData["TruncatedIcosahedron", "FaceIndices"];
f6 = Select[faces, Length[#] == 6 &];
f5 = Select[faces, Length[#] == 5 &];
len = Norm[coord[[1]]Norm[coord[[1]]] // Simplify;
Graphics3D[{{Opacity[0.3], Sphere[{0, 0, 0}, len*4/3],
Sphere[ Mean[coord[[#]]], len*2/3] & /@ f6,
Sphere[ Mean[coord[[#]]], len*2/3] & /@ f5, Opacity[1]Opacity[1],
Sphere[{0, 0, 0}, len*2/3]}, {Opacity[0.4],
PolyhedronData["TruncatedIcosahedron" , "Faces"]}}, Boxed -> False]
[Graphic from the above code added by J.O'Rourke:]
Plotting only three outer spheres as in
Graphics3D[{{Opacity[0.3], Sphere[{0, 0, 0}, len*4/3],
Sphere[ Mean[coord[[#]]], len*2/3] & /@ Take[f6, 2],
Sphere[ Mean[coord[[#]]], len*2/3] & /@ Take[f5, 1], Opacity[1]Opacity[1],
Sphere[{0, 0, 0}, len*2/3]}, {Opacity[0.4],
PolyhedronData["TruncatedIcosahedron" , "Faces"]}}, Boxed -> False]
[Graphic from the above code added by J.O'Rourke:]
shows that the two intersection points of three neighboring outer spheres lie either outside the sphere with radius 1 or inside the sphere of radius 1/2. This can easily be made rigorous by calculation:
mcl = Chop[ Join[Mean[coord[[#]]] & /@ Take[f6, 2], Mean[coord[[#]]] & /@ Take[f5, 1]]]; erg = Solve[(Norm[{x, y, z} - #] == len*2/3) & /@ mcl, {x, y, z}] // N; ((Norm[{x, y, z}]/len*3/4) /. #) & /@ erg
that yields the distances
{0.216794, 1.06042}
