Ants on a Stick

The Problem

An even number of ants is placed randomly on a meter stick. One more ant named Alice is placed at the center of the stick. At a certain moment, every ant starts moving in a randomly chosen direction left or right at a constant speed of one meter per minute. When two ants meet, they instantaneously reverse directions. When an ant reaches an end of the stick, it also instantaneously reverses direction.

After one minute, all the ants stop. What is the probability (as a function of the number of ants) that Alice is in the center of the stick?

In the animation below, each circle represents an ant. When you press "Resume" they will start moving. After a logical minute (the animation speed is faster than one meter per minute), the ants will stop. Where is Alice? To add interest, the ants carry a token that they exchange with the other ants when they meet and reverse direction. Watch the token. Where does it go and where does it end up? Alice is the white ant with a black outline and starts carrying the black token.

Click "New Pattern" to create a new pattern.
Click "Start" to start the display.
Click "Pause" to pause.
Click "Resume" to resume.

The Answer

Use this answer to verify your results.