Question 24
Suppose $k$ and $n$ are positive integers such that $k \le n \le 2k - 1$. Julian has a large pile of rectangular $k \times 1$-tiles. Merlijn picks a positive integer $m$, and receives from Julian $m$ tiles to place on an $n \times n$-board. On each tile, Julian writes whether this tile should be placed horizontally or vertically. Tiles may not overlap on the board, and they must fit entirely inside the board. What is the largest number $m$ that Merlijn can pick while still guaranteeing he can put all tiles on the board according to Julian's instructions?