Problem 1820

Let $n \geq 2$ be an integer, and consider an $n \times n$ chessboard. We place $n$ rooks on this board such that each row and each column contains exactly one rook. This is defined as a peaceful configuration of rooks. The objective is to find the greatest positive integer $k$ such that, in every possible peaceful configuration of $n$ rooks, there exists a $k \times k$ sub-square on the chessboard that is completely empty of any rooks.

### Step-by-step Solution

1. Understanding the Problem:
- In a peaceful configuration, since there is exactly one rook per row and one per column, it ensures all $n$ rooks are placed in unique row-column intersections across the $n \times n$ board.

2. Identifying Empty Squares:
- We need to ensure every configuration allows for a square sub-board of size $k \times k$ which is void of rooks.

3. **Calculation of Maximum $k$:**
- If we realize a peaceful configuration where rooks are distributed such that they occupy maximum area of the available board, each row and column combination will optimally cover the board minimally.
- The goal is maximizing $k$, ensuring the largest empty $k \times k$ square still forms on any part of the board in spite of any rook configuration.

4. Using Combinatorial and Geometric Argument:
- Let’s consider placing $n-1$ rooks. In this optimal configuration, potentially every position leading to $n-1$ coverages leaves a square potentially of size up to
$\sqrt{n - 1} \times \sqrt{n - 1}$ that is free.
- For all $n$ positions to be filled, this sub-square will obviously be smaller in the maximal empty form.

5. Conclusion:
- Upon deriving these options and observance that the largest $k \times k$ square exists, due to $\lceil\frac{n}{k}\rceil$ fraction of remaining free subset, we form:
- The greatest $k$ ensuring a $k \times k$ rupe-free square is presented by the integer part:
$$k = \left\lfloor \sqrt{n - 1} \right\rfloor.$$

Thus, the greatest positive integer $k$ such that for any peaceful configuration, there exists a $k \times k$ sub-square devoid of rooks, is:
$$\boxed{\left\lfloor \sqrt{n - 1} \right\rfloor}.$$