Question 40
Players $A$ and $B$ play a game with $N \geq 2012$ coins and $2012$ boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least $1$ coin in each box. Then the two of them make moves in the order $B, A, B, A, \ldots$ by the following rules:
- On every move of his $B$ passes $1$ coin from every box to an adjacent box.
- On every move of hers $A$ chooses several coins that were not involved in $B$'s previous move and are in different boxes. She passes every chosen coin to an adjacent box.
Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.
- On every move of his $B$ passes $1$ coin from every box to an adjacent box.
- On every move of hers $A$ chooses several coins that were not involved in $B$'s previous move and are in different boxes. She passes every chosen coin to an adjacent box.
Player $A$'s goal is to ensure at least $1$ coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.