$13 \cdot 86$ On a plot of land, there are 10,000 trees planted, with 100 trees per row, totaling 100 rows, forming a neat grid. Try to determine the maximum number of trees that can be cut down while ensuring that when standing on any tree stump, no other tree stump can be seen behind it. Here, it can be assumed that the trees are thin enough.
[Solution] Number the rows and columns as $1,2, \cdots, 100$. Cut down all the trees where both the row and column numbers are odd. In this case, a total of 2500 trees are cut down. It is easy to prove that this method meets the requirements of the problem, so the maximum value we are looking for is no less than 2500. On the other hand, these trees are precisely located at the nodes of a $99 \times 99$ grid. If we extract the squares where both the row and column numbers are odd, there are 2500 such squares (see the right figure). Their 10000 vertices do not overlap, and they are exactly the 10000 nodes of the $99 \times 99$ grid. If 2501 trees are cut down, which means selecting 2501 nodes, by the pigeonhole principle, there will always be two points that are the vertices of a selected square. Cutting down both of these trees does not meet the requirements.

In summary, the maximum number of trees that can be cut down is 2500.