Australian AMC 2010 - Intermediate Question 30

There are many towns on the island of Tetra, all connected by roads. Each town has three roads leading to three other different towns: one red road, one yellow road and one blue road, where no two roads meet other than at towns. If you start from any town and travel along red and yellow roads alternately (RYRY...) you will get back to your starting town after having travelled over six different roads. In fact RYRYRY will always get you back to where you started. In the same way, going along yellow and blue roads alternately will always get you back to the starting point after travelling along six different roads (YBYBYB). On the other hand, going along red and blue roads alternately will always get you back to the starting point after travelling along four different roads (RBRB).

How many towns are there on Tetra?

**Source annotation note:** The coloured sketches in the supplied page margins are handwritten annotations, not a printed official map. No official diagram is required or supplied for this question; the solution does not use those sketches.