2. Each cell of a $100 \times 100$ 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. What is the maximum number of balanced cells that can be on the board? (Cells are considered neighbors if they share a side.)
Solution. Cells lying on the border of the board but not in the corner cannot be equilibrium cells, since they have an odd number of neighbors (three). There are $4 \cdot 98=392$ such cells.

All other cells can be made equilibrium, for example, with a striped coloring (the first row is blue, the second row is white, the third row is blue, and so on). The number of these cells is $10000-392=9608$.

Answer: 9608.