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.
(a) If initially box A contains 6 coins, show that Alice can win in 4 turns.
(b) If initially box A contains 31 coins, show that Alice cannot win in 10 turns.
(c) What is the minimum number of turns needed for Alice to win if box A initially contains 2018 coins?