A square area of size $100 \times 100$ is paved with $1 \times 1$ square tiles of four colors: white, red, black, and gray - such that no two tiles of the same color touch each other (that is, they do not share a side or a vertex). How many red tiles can there be?
We will divide the area into 2,500 squares of $2 \times 2$, each of which consists of four tiles sharing a common vertex. Therefore, each of these squares can contain no more than one red tile, and thus the number of red tiles cannot exceed 2,500. The same is true for tiles of other colors, meaning there are exactly 2,500 tiles of each color.

## Oтвет

2,500 tiles.