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.
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 $15 ?$