Let
![n \geq 2](/media/m/2/1/f/21fe2458de6d1580c44fd06e0fac11bb.png)
be an integer. Consider an
![n \times n](/media/m/9/d/8/9d8eac5b3234425afb9f970edbfe93ef.png)
chessboard consisting of
![n^2](/media/m/3/6/a/36a0dd5a7e7ff0e6bbc714e33ddb1d63.png)
unit squares. A configuration of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
rooks on this board is
peaceful if every row and every column contains exactly one rook. Find the greatest positive integer
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
such that, for each peaceful configuration of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
rooks, there is a
![k \times k](/media/m/6/f/4/6f45c34ff31ce4529a6e656d064c528f.png)
square which does not contain a rook on any of its
![k^2](/media/m/5/c/2/5c29a9cd3ceff73827290ef0ae5ed48c.png)
unit squares.
%V0
Let $n \geq 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is [i]peaceful[/i] if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.