You're playing pinball. When you first shoot a ball it randomly comes down through 1 of 3 gates. When you go through an unlit gate, it lights up. Similarly, a lit gate will go out. What is the expected number of balls you have to throw for all 3 gates to light up?
For example, ball A could go through gate 2, B through gate 3, and C through gate 1. This scenario took 3 rolls and has probability
I've put serious thought into this question twice over the last couple of years but my answer gets more and more complicated until my brain explodes.
Douglas hit the nail on the head. For kicks, here's the Python script I used as a reality check for both the 2 and 3 gate cases.
from random import randint def pinball(gates): trials =  for trial in range(10000): state = [False for g in range(gates)] balls = 0 while not all(state): gate = randint(0, len(state) - 1) state[gate] = not state[gate] balls += 1 trials.append(1.0 * balls) print sum(trials) / len(trials) pinball(2) pinball(3)