3. According to the rules of chess, the minimum number of steps required to move the king from one square to another on a grid is called the distance between these two squares. How many squares can be marked at most on a $100 \times 100$ grid so that the distance between any two marked squares is not equal to $15 ?$
3. At most $55^{2}=3025$ cells can be marked.

As shown in Figure 8, a $100 \times 100$ grid can be divided into 9 $30 \times 30$ squares, 6 $10 \times 30$ rectangles, and 1 $10 \times 10$ square.

Each $30 \times 30$ square can be divided into $15^{2}$ "quad cells," where each quad cell consists of four cells with coordinates $(a, b)$, $(a+15, b)$, $(a, b+15)$, and $(a+15, b+15)$. Thus, in each quad cell, at most one cell can be marked. Therefore, in each $30 \times 30$ square, at most $15^{2}$ cells can be marked.

Similarly, each $10 \times 30$ rectangle can be divided into "cell pairs" with a distance of 15 (for example, in a rectangle with a horizontal length of 30, such cell pairs consist of cells with coordinates $(a, b)$ and $(a, b+15)$). Thus, in each $10 \times 30$ rectangle, at most $15 \times 10$ cells can be marked. Finally, in the $10 \times 10$ square, there are only $10^{2}$ cells.
Therefore, the total number of cells that can be marked does not exceed
$$
\begin{array}{l}
9 \times 15^{2}+6 \times 15 \times 10+10^{2} \\
=(3 \times 15+10)^{2}=55^{2} .
\end{array}
$$

A specific marking method is shown in Figure 9.