Thirty-three ladybugs are sitting on a one meter stick. Suddenly all ladybugs start crawling either to the left or to the right with a constant speed of one meter per minute. When two ladybugs meet, they reverse directions immediately. If one arrives at the end of the stick, it falls off. Considering all possible initial configurations, what is the longest time it can take until all ladybugs have fallen off?

Let the bugs be b1 through b33. Suppose one of the bugs (b1) initially moves in a direction where if it doesn't meet another bug, it would travel a distance d before falling off. If b1 meets say b5, then (since all ladybugs look alike) exchange their names. By continually exchanging names each time two meet, b1 will travel a distance d before falling.  This would follow for every bug and so as long as one of the bugs starts at one end of the stick moving toward the other end, the longest time for all bugs to fall would be one minute.

Posted by Dennis on 2006-12-05 15:36:54