This is probably my favorite math problem. The solution requires no complex mathematics, but rather several separate, elegant insights. If you can't solve it, but think you have some insights to add, post them.
There is an irregularly shaped castle wall with 12 irregularly spaced guard towers around the perimeter. The towers may be evenly spaced; they may all be clustered together; they may be somewhere in between those two options. There is a guard at each tower. Each guard patrols the castle perimeter, walking at a pace that allows him to make a complete loop around the perimeter in exactly one hour. (Thus, all 12 guards walk at the same pace as each other.) At noon, each guard starts at his own station and begins to walk either clockwise or counterclockwise (to be determined randomly). Whenever two guards meet each other, they immediately each turn around and start walking back in the direction from which they came. Their turnaround is immediate and they lose no time in switching directions.
Prove that at midnight each guard is back at his original tower.