7.5. A square box 3 by 3 is divided into 9 cells. It is allowed to place balls in some cells (possibly a different number in different cells). What is the minimum number of balls that need to be placed in the box so that each row and each column of the box contains a different number of balls?
Answer: 8.

Solution: Add up the number of balls in all rows and columns. Since these are 6 different non-negative numbers, this sum is at least $0+1+\ldots+5=15$.

Now notice that the sum of the numbers in the rows is equal to the sum of the numbers in the columns, since these sums are equal to the total number of balls in the box. Therefore, the sum of the six numbers is twice the sum of the numbers in the rows, which means it is even. Thus, twice the number of balls in the box is greater than 15, so the number of balls is greater than 7. An example with 8 is shown alongside.

Criteria: only the answer - 0 points, only the example - 2 points, only noted that the sum of 6 numbers is not less than $15-1$ point. Only the correct estimate - 3 points.