Chef and Linear Chess

Chef wants to play a game of *linear chess* on a one-dimensional board ― an infinite row of squares numbered by positive integers. In this game, he has a pawn, which is initially at a square $K$. There are also $N$ other people (numbered $1$ through $N$); Chef can choose one of them to play against. For each valid $i$, the $i$-th player would play in the following way: - Take a pawn and place it o