Source: 2018 Canadian Open Math Challenge Part C Problem 2 Alice has two boxes $A$ and $B$. Initially box $a$ contains $n$ coins and box $B$ is empty. On each turn, she may either move a coin from box $a$ to box $B$, or remove $k$ coins from box $A$, where $k$ is the current number of coins in box $B$. She wins when box $A$ is empty. $\text{(a)}$ If initially box $A$ contains 6 coins, show that Alice can win in 4 turns. $\text{(b)}$ If initially box $A$ contains 31 coins, show that Alice cannot win in 10 turns. $\text{(c)}$ What is the minimum number of turns needed for Alice to win if box $A$ initially contains 2018 coins?