Problem 2

A game is played by Alice and Charlie. Alice goes first. They start with a three-digit number $n$. A move consists of choosing a non-zero digit $k$ of $n$ (meaning a digit present in the number $n$) and replacing $n$ with $n - k$. This is repeated until the number 100 is written. The player who writes 100 wins.

(a) If the starting number is 125, Alice can always win. State Alice''s first move and how Alice responds to whatever move Charlie makes at each stage.

(b) Find, with proof, for which starting values Charlie has a winning strategy.