【Analysis】Under the condition that $m$ takes the maximum value, $n$ should be as large as possible to make the value of $m+n$ the largest.
【Solution】According to the analysis, $1 \leqslant$ the number of rows for black and white cells $\leqslant 7 ; 1 \leqslant$ the number of columns $\leqslant 7$. When $m=7$, we can assume that the number of black cells in 7 columns is 3, so the total number of black cells is: $3 \times 7=21$. Then, we can place 21 black cells in rows 1 to 5 with 4 black cells in each row, 1 black cell in row 6, and none in row 7. This way, there are 5 rows where the number of black cells exceeds the number of white cells, so $n=5$, making $m+n=12$ the maximum. As shown in Figure 1: When $m=6$, we can assume that the number of black cells in 6 columns is 3, and the number of black cells in the remaining column is 7, so the total number of black cells is $3 \times 6+7=25$. Then, we can make the number of black cells in rows 1 to 6 be 4, and only 1 in the last row. This way, there are 6 rows where the number of black cells exceeds the number of white cells, so $n=6$, making $m+n=12$, as shown in Figure 2: When $m \leqslant 5$, $m+n \leqslant 12$. In summary, the maximum value of $m+n$ is 12. Therefore, the answer is: 12.
14. (15 points) A $7 \times 7$ grid is colored black and white. If the number of columns where black cells are fewer than white cells is $m$, and the number of rows where black cells are more than white cells is $n$, find the maximum value of $m+n$.