Let $k$ be a positive integer. Keira and Roland play a game of reverse chess. Initially, Roland chooses a positive integer $n>\frac{k}{2023}$. Keira places $k$ kings on $k$ distinct squares of a $2023 \times n$ chess board. Then Roland places a rook on an unoccupied square of the board. Both players then take turns moving any number (possibly zero) of their pieces, with Keira starting first. Each king cannot move to a square occupied by another king, but it can capture the rook. Furthermore, if Keira chooses to move more than one king in her turn, she moves them one at a time. Roland's rook is not permitted to capture any king, nor may it pass through a square occupied by a king.

For which $k$ can Keira guarantee to capture Roland's rook, regardless of Roland's moves or choice of $n$ ?
(A king can move exactly one square in any horizontal, vertical or diagonal direction. A rook can move any number of squares in a horizontal or vertical direction. One piece (rook or king) captures another by moving to the square it occupies.)
We will show the answer is $k \geq 1012$.
We first show that Roland can avoid capture if $k \leq 1011$. Let $n$ be the number of rows. Choose $n>2027 \times 1012$, so that there always exists a row which is at least 1014 rows away from any rows occupied by a king. Call such rows safe. We will show that Roland's rook can avoid capture by moving between safe rows.

Initially, Roland places his rook in a safe row. After Keira's first turn, Roland moves the rook to the leftmost column. In each subsequent turn, Roland checks whether there is a king in the same column as his rook.
- If there is no king in the same column, then Roland can move the rook vertically to a safe row.
- If there is a king in the same column, then Roland will move the rook two squares to the right.

The second case can occur at most 1011 times, since a king can only move by at most one column per turn, so a different king is needed to block the rook's column each turn. After at most 1012 turns (including the first turn where Roland moved the rook to the leftmost column), Roland can move his rook vertically to a new safe row. Note that Keira will have taken at most 1013 turns during this, which is not enough to capture the rook which started on a safe row.

By repeating this method, Roland's rook can move between safe rows and avoid capture indefinitely if $k \leq 1011$.

We now show that $k=1012$ (and hence $k \geq 1012$ ) kings are enough to capture the rook. Keira moves the kings until there is a king in each of the topmost squares of columns $1,3, \ldots, 2023$. Label these kings with $K_{1}, \ldots, K_{1012}$, from left to right. For each $i$, assign
$K_{i}$ to guard columns $2 i-1$ and $2 i$ (with $K_{1012}$ only guarding the rightmost column). After achieving this, let us assume that the rook is not in the topmost column on Keira's move, otherwise the kings can immediately capture it.
The kings move down by one row every turn, so that each king moves to one of the two squares in its assigned double-column in the row below. If the rook is in the doublecolumn guarded by $K_{i}$, then $K_{i}$ will choose its move so that it is in the same column as the rook. As a result, the rook must always be below the row occupied by the kings. The game will eventually reach a state where, after its move, the rook is in the row directly below the kings. Keira can then capture the rook on her next move.