$8 \cdot 30$ Two players, A and B, play a number-filling game on a $5 \times 5$ grid: A goes first, and the two players take turns filling in the empty cells, with A writing a one in a cell and B writing a zero. After the grid is filled, the sum of the 9 numbers in each $3 \times 3$ square is calculated, and the largest of these sums is denoted as $A$. A tries to maximize $A$, while B tries to minimize $A$. What is the maximum value of $A$ that A can achieve?
[Solution] First, divide the first 4 rows into 20 squares into 10 $1 \times 2$ rectangles (see the figure on the right). Clearly, each $3 \times 3$ square on the board contains exactly 3 of these 10 rectangles. Therefore, player B can ensure that $A \leqslant 6$ by filling one of the two squares in each $1 \times 2$ rectangle with 0. Whenever player A fills one of the squares in a $1 \times 2$ rectangle with 1, if the other square is empty, player B can fill it with 0. This ensures that each $1 \times 2$ rectangle contains at least one 0.

On the other hand, player A can make the value of $A$ equal to 6. Player A starts by filling the center square with 1. Then, player B fills a square with 0. At this point, either the other 4 squares in the 3rd row are all empty, or the 4 squares in the 3rd column are all empty, or both sets of 4 squares are empty. Assume the other 4 squares in the 3rd row are all empty. Then, player A can fill the empty square in the 3rd row and 2nd column with 1 on the second turn. Player B then fills another empty square with 0. At this point, at least one of the 1st and 4th squares in the 3rd row is empty, say the 1st square. Player A can fill this square with 1 on the third turn, and player B fills another empty square with 0. As a result, the first 3 squares in the 3rd row are all 1, and there are 3 zeros on the entire board. Considering the two $3 \times 3$ squares in the top-left and bottom-left corners, one of them must contain at most 1 zero, meaning there are at least 5 empty squares. Thus, player A can fill 3 more 1s in these squares to make the value of $A$ at least 6.

In summary, player A can make the maximum value of $A$ equal to 6.