We have all heard the old problem about forming a triangle from breaking a stick into three pieces, with the breaks randomly distributed. Some variations make the second break contingent on the first in some way. I present a new variation (not original to me).
Problem: Take a stick and break it at a location selected with uniform density along its length. Throw away the left-hand piece and break the right-hand one at a location selected with uniform density along its length. Continue forever. What is the probability that one of the discarded left-hand pieces is more than half as long as the original stick?
I do not have a particularly elegant to solution to this one, and was wondering if one exists. Of course, any answers that solve the problem are welcome.