Point of view switcheroo
Rolling marbles Here's a puzzle: Suppose you have N identical marbles rolling along a one dimensional table-top. Each marble is randomly rolling to the left or to the right, all with the same speed. Collisions are elastic, which means the marbles just change direction. What is the maximum amount of time before all the marbles have rolled off the table? 8 marbles with speed 1 on a table of length 1 Answering this question is really difficult if you simply pick an individual marble and try to work out how long it might stay on the table as it bounces back and forth. But there's a simpler way to look at it. Prior to each collision you have one marble rolling to the left and one to the right, and afterwards you still have one rolling to the left and one to the right. If we swap labels following each collision then the labels never change direction. Now it's easy to see that the answer is the same whether there are 100 marbles or just one. And ...