Alon Amit's comment to Kevin Buzzard's answer (on the equivalence between the "upon STOP flip over the top card" game vs. the "upon STOP flip over the last card") reminded me of the Ants on a Meter Stick problem (which has been a favorite of mine since I first read it in the Harvey Mudd Fun Facts site):

One hundred ants are dropped on a meter stick. Each ant is traveling either to the left or the right with constant speed 1 meter per minute. When two ants meet, they bounce off each other and reverse direction. When an ant reaches an end of the stick, it falls off.

At some point all the ants will have fallen off. The time at which this happens will depend on the initial configuration of the ants.

Question: over ALL possible initial configurations, what is the longest amount of time that you would need to wait to guarantee that the stick has no more ants?

The insight lies in the fact that this is equivalent to the "Ghost Ants on a Meter Stick" problem, where the ants pass right through each other instead of bouncing off each other.