7.5. On a 5 by 5 grid, several cells are marked in such a way that every 3 by 3 square of cells contains exactly one marked cell. How many cells can be marked
Answer. Any number from 1 to 4.

Solution. It is clear that there must be at least one cell. On the other hand, four corner 3x3 squares completely cover the 5x5 square, and each of them contains exactly 1 marked cell, so the total number of marked cells is no more than 4. Examples for quantities from 1 to 4 are shown in the figure.