You are given ten billiard balls numbered 1 through 10. Show that there are exactly two ways to arrange them in a circle so that the difference between any two adjacent numbers is 1 or 2. (You will observe that each of the two arrangements is simply a mirror image of the other.)
Solution
Build it. Start with 1. You can discover that as soon as you place the 2, you can make no further choices but each billiard ball's position is then determined.