Count the paths along the edges from a vertex of a polyhedron to its antipode. You may take a detour, but may not pass through the same vertex twice.
| Polyhedron | Length of the shortest path | Number of paths |
|---|---|---|
| Cube | 3 | 18 |
| Octahedron | 2 | 28 |
| Cuboctahedron | 3 | 684 |
| Truncated cube | 6 | 1888 |
| Truncated octahedron | 6 | 2568 |
| Small rhombicuboctahedron | 5 | 484088 |
| Great rhombicuboctahedron | 9 | 4889088 |
| Dodecahedron | 5 | 780 |
| Icosahedron | 3 | 5170 |
| Icosidodecahedron | 5 | 11461272 |
| Truncated dodecahedron | 10 | 83250048 |
| Truncated icosahedron | 9 | 307611158 |
| Small rhombicosidodecahedron | 8 | 162567930895168 |
| Great rhombicosidodecahedron | 15 | 34460645968551616 |