Problem 10.6. In a grid table with 5 rows and 6 columns, each cell contains either a cross, a zero, or a star. It is known that:

- in each column, the number of zeros is not less than the number of crosses;
- in each column, the number of zeros is not less than the number of stars;
- in each row, the number of crosses is not less than the number of zeros;
- in each row, the number of crosses is not less than the number of stars.

How many stars can be in such a table? List all possible options.
Answer: 6.

Solution. Since in each column there are no fewer O's (noughts) than X's (crosses), then in the entire table there are no fewer O's than X's. Similarly, considering the rows, it follows that in the entire table there are no fewer X's than O's. Therefore, there are an equal number of X's and O's in the entire table, and also there are an equal number of them in each row and each column.

From the condition, it also follows that in each column there cannot be 0 or 1 O's, so there are no fewer than 2. Since the number of X's and O's in the column is equal, there cannot be 3 or more, i.e., there are exactly 2. Therefore, in each column there is exactly one star, i.e., there are a total of 6.

It is also worth noting that the described construction indeed exists. For example, in the following table all conditions are met:

| x | $\times$ | $\times$ |  |  | O |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 0 | O | 0 | $\times$ | $\times$ | $x$ |
| $*$ | * | $\times$ | $\times$ | 0 | O |
| ○ | 0 | * | $*$ | $\times$ | $x$ |
| $\times$ | $x$ | 0 | 0 | $*$ | ＊ |