An old man is walking with a stick with initial length 1. Every time he steps the stick breaks with uniform distribution along its (remaining) length. He holds the stick at the very end and holds on to the remaining piece. What is the expected number of steps before the stick has length less than $\epsilon$, for some $0 < \epsilon < 1$?
I've done some computation and I believe the answer is $1-\log(\epsilon)$ but I'm not sure how to go about proving it.