Source: P. Winkler
In a circle are light bulbs numbered 0 through n-1 all initially on. At time t, you examine the bulb number t mod n, and if its on, you change the state of bulb t+1 mod n. If bulb t is off, you do nothing.
Prove that if you continue around and around the ring in this manner, eventually all the bulbs will again be on.
Update (Nov 5 2009) : Thanx vivek, the small problem updated. Bulbs numbered 0 to n-1 instead to 1 to n. Thanx. Solution in comments!!