2023 Putnam A6

Alice and Bob play a game in which they take turns choosing integers from $1$ to $n$. Before any integers are chosen, Bob selects a goal of ``odd'' or ``even''. On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the $n$th turn, which is forced and ends the game. Bob wins if the parity of \{k: \text{the numberk$was chosen on the$kth turn}\} matches his goal. For which values of $n$ does Bob have a winning strategy?

Alice 和 Bob 玩一个游戏，他们轮流选择从 $1$ 到 $n$ 的整数。在选择任何整数之前，鲍勃选择“奇数”或“偶数”目标。在第一回合，Alice 选择 $n$ 个整数中的一个。在第二轮，鲍勃选择剩余的整数之一。他们继续交替选择尚未选择的整数之一，直到第 $n$ 回合，这将被强制并结束游戏。如果 \{k: \text{第k$轮选择的数字$k}\} 的奇偶性与他的目标相符，Bob 获胜。对于 $n$ 的哪些值，Bob 有获胜策略？