Solution. Let's look at the moment when there are no moves left. Suppose we have $k$ colors left, then we have repainted at least $100-k$ cells. Note that if nothing can be repainted, then there are at least 4 cells of each color. In total, there are at least 25 colors left after this process. Therefore, we will repaint at least 75 cells. Thus, $N \geqslant 75$. The example is constructed simply by $2 \times 2$ squares. That is, we divide the board into $2 \times 2$ squares and repaint each square to the color of its lower left corner.
3. Andrey placed chips of a hundred different colors in the cells of a $10 \times 10$ board. Each minute, one of the chips changes color, and only a chip that was unique (i.e., differed in color from all others) in its row or column before this operation can change color. After $N$ minutes, it turned out that no chip could change color anymore. What is the smallest value $N$ could have been?