MathDB

Natalie has a copy of the unit interval

[0,1]

that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value

x\in [0,1]

uniformly at random, and
(a) If

x\leq\tfrac12

she colors the interval

[x,x+\tfrac12]

with her marker.
(b) If

x>\tfrac12

she colors the intervals

[x,1]

and

[0,x-\tfrac12]

with her marker.
What is the expected value of the number of steps Natalie will need to color the entire interval black?

Problem Statement