Suppose there are $n$ ants on a stick which has length 1. At any time, the ants may be facing left or right, the initial directions of the ants are arbitrary. Each ant can be modeled as a point in the stick.
At time 0, all ants start moving in the direction they're facing at speed 1.
Each time an ant collides with another ant, both ants reverse direction instantly and continue moving in the new direction.
If an ant reaches the ends of the stick, it falls down (quite a strange species of ant).
Now, you are observing this stick full of ants, starting at time 0. You are very particular of one ant, which appears in a distinct color from the other ants, so you focus your eyes just on that particular ant.
What is the maximum time you'll be watching that particular ant until it falls off the stick?