Hamilton's Puzzle

Thomas from New York sent in a very clear solution to the Hamiltonian problem. He wrote:

One route around the dodecahedron is: Rio, Pretoria, Washington DC, Canberra, London, Paris, Baghdad, Ottawa, Nairobi, Beijing, Wellington, Berlin, Mexico City, Seoul, Tripoli, Tokyo, Delhi, Madrid, Bangkok, Cairo, Rio.
There are other routes, such as: Rio, London, Canberra, Washington DC, Pretoria, Berlin, Wellington, Beijing, Nairobi, Ottawa, Baghdad, Paris, Bangkok, Madrid, Delhi, Tokyo, Tripoli, Seoul, Mexico City, Cairo, Rio.

Other solutions were sent in by Thomas from Heversham St Peter's Primary School, Michael from North Sydney Boys High School and Katherine from The Hills Grammar School. These included:

Rio de Janeiro to London to Canberra to Ottowa to Nairobi to Beijing to Washington DC to Pretoria to Berlin to Wellington to Tripoli to Tokyo to Delhi to Baghdad to Paris to Bangkok to Madrid to Seoul to Mexico City to Cairo and back to Rio de Janeiro.
1. Rio de Janeiro 2. Cairo 3. Mexico City 4. Berlin 5. Pretoria 6. Washington DC 7. Canberra 8. Ottowa 9. Baghdad 10. Delhi 11. Tokyo 12. Nairobi 13. Beijing 14. Wellington 15. Tripoli 16. Seoul 17. Madrid 18. Bangkok 19. Paris 20. London 21. Rio de Janeiro
Rio de Janeiro to Pretoria to Washington D.C. to Canberra to Ottowa to Nairobi to Beijing to Wellington to Berlin to Mexico City to Cairo to Bangkok to Madrid to Seoul to Tripoli to Tokyo to Dehli to Baghdad to Paris and then back to Rio de Janeiro.

You will have realised that there are many, many more than this - but thank you to all those of you who sent in some solutions. Thomas from New York was the only one to go on to the second part of the question. Well done Thomas for persevering! He says:

For the simpler network, there are six solutions (using initials of countries): ABFEDCA, ABCDFEA, ACBFDEA, ACDEFBA, AEFDCBA, and AEDFBCA.