Start - end by turning left and right alternatively, why?

[.d9n.]

Homework Statement

In a certain area there are n villages linked by a network of roads
in such a way that exactly three roads meet in each village,
and none of the roads meet between villages. Prove that if
someone starts in one village and moves round this road system
alternately turning right and left whenever he comes to a village,
then he will eventually return to the village he started from

Homework Equations

The Attempt at a Solution

 

[.d9n.]

i assume this only works if the number of villages is even. Let i=1,2,... then when we have 2i villages we have 3i roads. I can see that when you go alternatively left and right you end up at the start point, I am just not sure why/ can't prove it.