12. (12 points) There is a $6 \times 6$ square, divided into 36 $1 \times 1$ squares. Select some of these $1 \times 1$ squares and draw their diagonals, such that no two drawn diagonals have a common point. What is the maximum number of diagonals that can be drawn?
【Solution】Solution: As shown in the left figure, rows $a_{1}$ and $a_{2}$ can draw at most 7 diagonals (there are 7 points on $l_{1}$, and each diagonal must use one point).

Similarly: rows $a_{3}$ and $a_{4}$ can also draw at most 7 diagonals, and rows $a_{5}$ and $a_{6}$ can do the same.
Therefore, the maximum number of diagonals that can be drawn is $7 \times 3=21$.
The answer is 21.
Construction as shown in the right figure.