5. For a $20 \times 20$ grid, each cell is colored either black or white, and the entire grid is divided into several rectangles with boundaries along the grid lines. Given a coloring method $P$, consider all partitions where no rectangle contains more than two black cells, and let $n(P)$ denote the minimum number of rectangles in these partitions that contain at most one black cell. Find the maximum value of $n(P)$ when $P$ ranges over all possible coloring methods.
5. $\max _{P}\{n(P)\}=20$.

First, explain $n(P) \leqslant 20$.
In the grid, select any row and divide this row into several small rectangles such that each rectangle contains exactly two black cells, with the last rectangle possibly containing one black cell.
Since there are 20 rows in total, 
$n(P) \leqslant 20$.
Next, define the coloring $Q$ as shown in Figure 3: each cell containing a letter is colored black (the rest are white), and a cell containing the letter $\mathrm{X}$ is called an X-cell, and a cell containing the letter Y is called a Y-cell.

Take any partition such that each rectangle contains at most two black cells. Then, a rectangle cannot contain two X-cells or two Y-cells. Therefore, any rectangle containing two black cells must contain exactly one X-cell and one Y-cell.

Since there are 38 X-cells and 18 Y-cells, there can be at most 18 rectangles containing two black cells. For the remaining at least 20 X-cells, each is contained in a rectangle with only one black cell.

Thus, for any partition, there are at least 20 rectangles containing one black cell, i.e., $n(Q)=20$.