9.5. On a 10 by 10 grid, several cells are marked such that each 3 by 3 square contains exactly one marked cell. How many cells can be marked?
Answer. Any number from 9 to 16.

Solution. Divide the board with horizontal lines at the 3rd, 6th, and 9th positions and vertical lines at the 3rd, 6th, and 9th positions into nine 3x3 squares, three vertical strips 1x3 in the rightmost column, three horizontal strips in the top row, and the top right corner cell. In each of the 3x3 squares, there must be one marked cell, so there are at least 9 in total. On the other hand, each of the 16 elements of the partition can be covered by one 3x3 square, so it cannot contain more than one marked cell, hence there are no more than 16.

Let's construct examples for each number of cells. Highlight the 3rd, 6th, and 9th columns. Each 3x3 square on the board intersects with exactly one of the highlighted columns in a 3x1 strip. Therefore, if we mark every third cell, starting from a certain one, in these columns, we will get the desired example. If we start from the first cell, there will be four marked cells in the column; if from the second or third, then three each. By varying the choice in the columns, we can get all the numbers of marked cells from 9 to 12. Now choose the 1st, 4th, 7th, and 10th columns and do the same. We will get all the numbers of selected cells from 12 to 16.