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. Over all the initial configurations, what is the longest time you have to wait for all the ants to fall off?
Hint (highlight to see): The answer is the same if there are only 10 ants.
When two ants bump off each other it is equivalent to them crossing over. Therefore, the longest amount of time you have to wait for all the ants to fall off is the time it takes for one ant to travel the distance of the stick, 1 minute. If an ant is placed at each end, facing away from the edge, regardless of where the 98 other ants are placed, it will take exactly 1 minute for all the ants to fall off.