Bakayev E.V.

In a $10 \times 10$ square, all cells of the left upper $5 \times 5$ square are painted black, and the rest of the cells are white. Into what maximum number of polygons can this square be cut (along the cell boundaries) so that in each polygon, the number of black cells is three times less than the number of white cells? (Polygons do not have to be equal or even of equal area.)
In each polygon of the partition, there must be cells of both colors. This means that there must be a black cell adjacent to a white one. However, there are only 9 such cells. See the example of cutting into 9 polygons in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_c07b2c4be0d3dcbcc4e5g-27.jpg?height=421&width=417&top_left_y=1&top_left_x=829)

## Answer

Into 9 polygons.