Take a random point $Z$, i.d. in [0,1], which defines a stick.
Break the stick in two, (random i.d.).
Take the left part of the broken stick and break it again in two (i.d.)
You thus obtain three sticks.
What is the expected length of the stick where $Z$ is located?
(Prove its $5/9$. Can you come up with an intuitive reason for that?)
Generalise for n breaks of the stick in the left (i.e. always breaking the segment near the origin).
(Edit: solution was found analytically and tested on MC simulation. would be interesting to see if there is an intuitive reason though.)
// (Hope you enjoyed this series of puzzles - this was the last one!)