mathematics - Maximum time for ants to fall off stick

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 interested 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?

Answer

And the answer is:

Surprisingly, it's 1, that is, it's the same as if nothing happens when the ants collide, and this doesn't depend on $n$.

Explanation:

The phrase that says you're watching a particular ant is just a distraction so that people will think that there is complexity in this riddle, while in fact it's very simple. When two ants collide and each reverses its direction, position-wise it's the same as if each ant simply continues moving in the original direction. So the maximum time is when there is an ant at on end of a stick facing the other end. Since the maximum time an ant can be in the stick is 1 (from one end to the other), therefore by time 1 all ants will already fall off the stick, including the particular ant that you're watching.

Alternative explanation:

Imagine that each ant holds a number, and when two ants collide (and reverse their directions), they will exchange the number. Now the crucial observation is: the numbers will never change its direction. So the maximum time a number can be on the stick is 1 (the numbers are always moving on the same direction with the same speed). So by the time 1, all numbers will have fallen down. Since every number is held by an ant, when there is no number, there is no ant either. Therefore by the time 1, all ants will have already fallen from the stick, including the particular ant you are watching.