Amélia and Beatriz play battleship on a $2n\times2n$ board, using very peculiar rules. Amélia begins by choosing $n$ lines and $n$ columns of the board, placing her $n^2$ submarines on the cells that lie on their intersections. Next, Beatriz chooses a set of cells that will explode. Which is the least number of cells that Beatriz has to choose in order to assure that at least a submarine will explode?
To solve this problem, we need to determine the minimum number of cells Beatriz must choose to ensure that at least one of Amélia's submarines will explode. 

1. **Understanding the Board and Submarine Placement:**
   - The board is a \(2n \times 2n\) grid.
   - Amélia places \(n\) submarines in \(n\) rows and \(n\) columns, meaning there are \(n^2\) submarines in total.
   - Each submarine is placed at the intersection of a chosen row and a chosen column.

2. **Beatriz's Strategy:**
   - Beatriz needs to choose cells such that at least one of the cells contains a submarine.
   - We need to find the minimum number of cells Beatriz must choose to guarantee this.

3. **Analyzing the Problem:**
   - Consider the worst-case scenario for Beatriz, where Amélia places her submarines in such a way that they are as spread out as possible.
   - If Beatriz chooses fewer cells, there might be a configuration where no submarine is hit.

4. **Calculating the Minimum Number of Cells:**
   - Let's denote the number of exploding cells in each of the \(2n\) rows as \(x_1, x_2, \ldots, x_{2n}\).
   - We need to ensure that the sum of these cells is sufficient to cover at least one submarine in every possible configuration.

5. **Case Analysis:**
   - If \(x_{n+1} \leq 1\), then the sum of the smallest \(n\) values \(x_1 + x_2 + \cdots + x_n \leq n\). This implies there exists a set of \(n\) rows containing at most \(n\) exploding cells, leaving at least \(n\) columns without any exploding cells in those rows.
   - If \(x_{n+1} \geq 2\), then the sum of the largest \(n\) values \(x_{n+1} + x_{n+2} + \cdots + x_{2n} \geq 2n\). This implies \(x_1 + x_2 + \cdots + x_n \leq 3n - 2n = n\), again leaving at least \(n\) columns without any exploding cells in those rows.

6. **Conclusion:**
   - To ensure that at least one submarine is hit, Beatriz must choose enough cells such that the sum of the cells in any \(n\) rows is greater than \(n\).
   - This leads to the conclusion that Beatriz needs to choose at least \(3n + 1\) cells to guarantee hitting at least one submarine.

The final answer is \(\boxed{3n + 1}\).