A shape has 2012 sides. This shapes corners are all labeled with a number 1-2012 in some order (however all are only used once). A corner is called an up if it is a larger number than its adjacent corners. A number is called a down if it is smaller than its adjacent numbers. How do you prove the number of ups equals the number of downs?

I am finding much difficulty in this question. I'm not quite sure how to work it seeing as numbers are randomly placed. I have looked at other polygons but they have told me nothing as to a pattern. Any help would be useful, I'm not sure how to attack the question