I don't know how to start this problem. I have tried to find a way to make it numerical or take it to another subject but I couldn't. The problem is this:
Imagine you have a seed in every vertex of a Hexagon. In each turn you can pick two and move them one vertex but in opposite ways. Prove that it is not possible to have them all in only one vertex.
Hint: If you pick a vertex and add up the clockwise distances to all the seeds at the start, what do you get? How can the sum change when you move two seeds?