4. What is the minimum number of chips that can be placed in the cells of a $99 \times 99$ table so that in each $4 \times 4$ square there are at least eight chips?
Answer: 4801.

Solution. Add a row and a column with number 100 to the table. Place a chip in each of their cells. Divide the expanded table into 625 squares $4 \times 4$. In each square, there can be no more than eight empty cells, so there are no more than 5000 empty cells in the entire table. Therefore, the total number of chips must be at least $99^{2}-5000=4801$.

Now, place chips in the cells of the expanded table where the product of the coordinates is divisible by four. In this case, each of the 625 squares $4 \times 4$ will have exactly 8 chips. Moreover, the added row and column will be completely filled with chips. Therefore, in the original table, there will be $625 \cdot 8-199=4801$ chips.