These 2012 numbers will make a sequence of increasing and decreasing numbers. If you disregard the vertices which are neither up/down, you will find that you have an alternating sequence of up/down/up/down etc. The 2012 numbers form a cycle (think of it as a continuous zigzag) and you will see that the number of ups equals the number of downs.

Sorry if my solution isn't rigorous enough...there's probably a similar solution that is better explained, or involves some other monovariant.