13. As shown in Figure $3-17$, a regular triangular chessboard with a side length of 10 (each cell is a $1 \times$ 1 small equilateral triangle), a "rook" can control the three rows of small equilateral triangle cells it is on (the shaded part in the figure). How many "rooks" can be placed on the chessboard at most, so that no "rook" is within the control range of another "rook"?
13. Number the rows in three directions, so that the cell where the “rook” is placed can be represented by a three-dimensional coordinate $(x, y, z)$, as shown in Figure 5-10, the cell marked with 0 is $(3,2,3)$. Each coordinate is the integer part of the ratio of the distance from the center of the cell where the “rook” is placed to the corresponding side to the height of the small equilateral triangle, so $x+y+z \leqslant 9$. Suppose the chessboard can accommodate at most $k$ “rooks”, $\left(x_{i}, y_{i}, z_{i}\right)(i=1,2, \cdots, k)$, none of which control each other, then the corresponding coordinates of any two rooks are all different, therefore, on the one hand $\sum_{i=1}^{k}\left(x_{i}+y_{i}+z_{i}\right) \leqslant 9 k$, on the other hand $\sum_{i=1}^{k}\left(x_{i}+y_{i}+z_{i}\right)$ $=\sum_{i=1}^{k} x_{i}+\sum_{i=1}^{k} y_{i}+\sum_{i=1}^{k} z_{i} \geqslant 3 \sum_{i=1}^{k} i=\frac{3}{2} k(k-1)$, hence $\frac{3}{2} k(k-1) \leqslant 9 \quad k \leqslant 7.7$ The method of placing the “rooks” (in the shaded small triangles in the figure) that meets the requirements is shown in Figure 5-10.