MathNet 2015-part-03 — 191 problems (EN–VI) Question 2

Let $n \in \mathbb{N}^{*}$. We place $1 \times n$ and $n \times 1$ rectangles on a $2 n \times 2 n$ chessboard in such a way that two rectangles never intersect and each rectangle covers exactly $n$ squares. A set of such rectangles is called saturated if it is impossible to add a rectangle without intersecting a rectangle already in place.
Find the smallest $k$ such that there exists a saturated set of $k$ rectangles.