First, consider that each row and column has a red grid. A convenient drawing method is to draw six red squares on a diagonal line (as shown in figure 1) and cross out three rows and three columns at will. It can be assumed that the principle of crossing out rows and columns is: the more red squares you cross out at a time, the better. For the diagram 1, cross out three rows, remove three red grids, and cross out three columns, and there will be no red grids.

First, consider that there are two squares in three rows (as shown in Figure 2). Obviously, there must be two squares in three rows at the same time. At this time, we can cross out three rows with two red squares, leaving three rows, with only one in each row and one in each column, so the three columns with red squares have no red grid.

In order to leave at least one red grid, just draw another red grid, and this red grid will add a row, two red grids and a column to the diagram, as shown in Figure 3.

So the conclusion is that at least 10 squares need to be painted red.