Reflections...

I would have spent more time preparing this note, but I thought it was more important to spend the time preparing the final for the class I'm teaching. Thanks!

And in case you are ever bored, here is a problem that you can try solving. It's fun!

Initially, n balls are placed along a circumference at arbitrary positions. They all start moving at the same rate: some in the clockwise direction, others in the counterclockwise direction. Will the balls ever return to their exact initial positions while also moving in the same direction as they were moving at the beginning?

(When two balls approach each other, they collide elastically: after the collision, each remains moving at the same speed, but in the opposite direction.)