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>.
