Let $k$,$n$ be positive integers, $k,n>1$, $k<n$ and a $n \times n$ grid of unit squares is
given. Ana and Maya take turns in coloring the grid in the following way: in each turn, a unit square is colored black in such a way that no two black cells have a common side or vertex. Find the smallest positive integer $n$ , such that they can obtain a configuration in which each row and column contains exactly $k$ black cells. Draw one example.
1. **Claim**: We must have \( n \geq 4k \).

2. **Proof of Bound**:
   - Consider an \( n \times n \) grid divided into \( 2 \times 2 \) squares.
   - Each \( 2 \times 2 \) square can have at most 1 black square due to the restriction that no two black cells can share a side or vertex.
   - Therefore, the total number of black squares in the grid is at most \( \frac{n^2}{4} \).
   - Since each row and column must contain exactly \( k \) black squares, the total number of black squares is \( nk \).
   - Hence, we must have \( nk \leq \frac{n^2}{4} \), which simplifies to \( k \leq \frac{n}{4} \) or \( n \geq 4k \).

3. **Construction**:
   - Consider \( n = 4k \).
   - We need to show that it is possible to color the grid such that each row and column contains exactly \( k \) black squares.
   - For \( k = 1 \), \( n = 4 \), we can construct a \( 4 \times 4 \) grid as follows:
     \[
     \begin{array}{|c|c|c|c|}
     \hline
     \blacksquare & & & \\
     \hline
     & & \blacksquare & \\
     \hline
     & \blacksquare & & \\
     \hline
     & & & \blacksquare \\
     \hline
     \end{array}
     \]
   - For general \( k \), consider a \( 4k \times 4k \) grid. Divide it into \( k \times k \) blocks of \( 4 \times 4 \) grids.
   - Each \( 4 \times 4 \) grid can be filled as shown above, ensuring that each row and column in the \( 4k \times 4k \) grid contains exactly \( k \) black squares.

4. **Verification**:
   - Each \( 4 \times 4 \) block has exactly 1 black square in each row and column.
   - Since there are \( k \) such blocks in each row and column of the \( 4k \times 4k \) grid, each row and column of the entire grid will have exactly \( k \) black squares.

Thus, the smallest positive integer \( n \) such that they can obtain a configuration in which each row and column contains exactly \( k \) black cells is \( n = 4k \).

The final answer is \( \boxed{ n = 4k } \)