1. Each cell of a $1000 \times 1000$ board is painted blue or white. We will call a cell balanced if among its neighbors there are an equal number of blue and white cells. Is it possible to color the board so that there are more than 600000 blue balanced cells? (Cells are considered neighboring if they share a side.)
Solution. It is possible. The figure shows how to make approximately $2 / 3$ of the cells on the board blue and balanced.

![](https://cdn.mathpix.com/cropped/2024_05_06_2b7c43278745f884d641g-1.jpg?height=301&width=309&top_left_y=522&top_left_x=903)

A more precise calculation: in each row (except the first and last), all blue cells, except for the two at the ends, are balanced. We have 998 rows, each with at least 664 balanced cells. $998 \cdot 664 > 600000$.