2025 Putnam 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.

Alice 和 Bob 玩一个包含 $n$ 数字串的游戏，每个数字仅限于 $0$、$1$ 或 $2$。最初所有数字都是 $0$。合法的做法是在一位数字上加上或减去 $1$，以创建一个以前未曾出现过的新字符串。没有合法动作的玩家输，而另一个玩家赢。

爱丽丝先走，玩家轮流移动。对于每个 $n \geq 1$，确定哪个玩家有保证获胜的策略。