One hundred ladybugs are dropped on a yard stick. Each ladybug is traveling either to the left or the right with constant speed equal to 1 m/min. When two ladybugs meet, they bounce off each other and turn around while continuing their constant speed of 1m/min. When a ladybug reaches an end of the stick, it falls off.
At some point all the bugs will have fallen off. The time at which this happens will depend on the initial configuration of the ladybugs.
Question: For all possible configurations, what is the longest amount of time that you would need to wait to guarantee that the stick has no more ladybugs?
The Math Behind the Fact:
The answer is 1 minute! While ladybugs bouncing off each other seems
difficult to keep track of, one key idea makes it quite simple: two ladybugs
bouncing off each other is equal to two ladybugs that pass through
each other, in the idea that the positions of ladybugs in each case are
identical. So, you can think of all ladybugs acting with independent
motions. Viewed in this way, all ladybugs fall off after traveling the
length of the stick, i.e., the longest that you would need to wait to
ensure that all ladybugs are off is 1 minute and this happens if one ladybug
started on the end of a stick facing the other end of the stick.
Some other interesting websites to find math problems:
http://mathforum.org/pow
http://www.thewizardofodds.com/math/
http://school.discovery.com/brainboosters/
http://www.flooble.com/perplexus/