Putnam 2025 A3

Alice and Bob play a game with a string of $n$ digits, each of which is restricted to be $0$, $1$, or $2$. Initially all the digits are $0$. A legal move is to add or subtract $1$ from one digit to create a new string that has not appeared before. A player with no legal move loses, and the other player wins. Alice goes first, and the players alternate moves. For each $n \geq 1$, determine which player has a strategy that guarantees winning.