2020 Putnam B2

Let $k$ and $n$ be integers with $1 \leq k < n$. Alice and Bob play a game with $k$ pegs in a line of $n$ holes. At the beginning of the game, the pegs occupy the $k$ leftmost holes. A legal move consists of moving a single peg

to any vacant hole that is further to the right. The players alternate moves, with Alice playing first. The game ends when the pegs are in the $k$ rightmost holes, so whoever is next to play cannot move and therefore loses. For what values

of $n$ and $k$ does Alice have a winning strategy?

令 $k$ 和 $n$ 为整数且 $1 \leq k < n$。爱丽丝和鲍勃玩一个游戏，将 $k$ 个钉子排成一排 $n$ 个孔。游戏开始时，钉子占据最左边的 $k$ 个洞。合法的移动包括移动一个钉子

到更右侧的任何空洞。玩家交替移动，爱丽丝先玩。当钉子位于最右边的 $k$ 洞中时游戏结束，因此接下来玩的人无法移动并因此失败。为了什么价值观

$n$ 和 $k$ 中 Alice 有获胜策略吗？