Ants on a Stick

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?

Presentation Suggestions:
You might give this at the end of lecture one day and present the answer the following lecture.

The Math Behind the Fact:
The answer is 1 minute! While ants bouncing off each other seems difficult to keep track of, one key idea (fun fact!) makes it quite simple: two ants bouncing off each other is equivalent to two ants that pass through each other, in the sense that the positions of ants in each case are identical. So, you might as well think of all ants acting with independent motions. Viewed in this way, all ants fall off after traversing the length of the stick, i.e., the longest that you would need to wait to ensure that all ants are off is 1 minute.

Seeking alternate ways to look at a problem can offer useful insights!

How to Cite this Page:
Su, Francis E., et al. "Ants on a Stick." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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