Consider the following one-person game: A player starts with score $0$ and writes the number $20$ on an
empty whiteboard. At each step, she may erase any one integer (call it a) and writes two positive integers
(call them $b$ and $c$) such that $b + c = a$. The player then adds $b\times c$ to her score. She repeats the step
several times until she ends up with all $1$'s on the whiteboard. Then the game is over, and the final score is
calculated. Let $M, m$ be the maximum and minimum final score that can be possibly obtained respectively.
Find $M-m$.