This is the planet Rhombicosidodecahedria:
This lovely planet has 62 countries, each with its own distinct history and culture. By an amazing coincidence, the countries all happen to coincide perfectly with the faces of a rhombicosidodecahedron. You, as a tourist, are naturally hoping to visit all of them.
How many different ways can you travel though each of the countries on this planet, visiting each exactly once?
- Your tour may start in any country you wish.
- You must visit each country exactly once. Once you have visited a country, you cannot visit it again. (The word "tour" should not be taken to imply a requirement that you return to your starting point.)
- You may only move between countries that are adjacent. In other words, each new country you visit must share a border with the previous one. (If they just touch at the corners that doesn't count.)
- For purposes of this puzzle, every country should be treated as unique, and reflections and rotations all count as different paths. (In other words, the orange pentagon counts as a different starting point from the purple pentagon.)