11.6. Each cell of a $7 \times 8$ table (7 rows and 8 columns) is painted in one of three colors: red, yellow, or green. In each row, the number of red cells is not less than the number of yellow cells and not less than the number of green cells, and in each column, the number of yellow cells is not less than the number of red cells and not less than the number of green cells. How many green cells can there be in such a table?
Answer: 8.

Solution. 1) In each row of the table, there are no fewer red cells than yellow ones, so in the entire table, there are no fewer red cells than yellow ones.

In each column of the table, there are no fewer yellow cells than red ones, so in the entire table, there are no fewer yellow cells than red ones.

Thus, in the table, there is an equal number of red

| 3 | 3 | ॠ | $\mathbf{K}$ | Ж世 | К | WK | $\mathbf{K}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| К | Ж | 3 | 3 | К | Ж | $\mathbf{K}$ | Ж |
| З | $\boldsymbol{\kappa}$ | $\mathbf{K}$ | Ж | 3 | 6 | ऊ |  |
| К | Ж | ЖК | $\mathbf{K}$ | ЗК | К | 3 | 3 |
| ЗК | $\mathbf{K}$ | $\mathbf{K}$ | Ж | K | Ж | $\mathbf{K}$ |  |
| K | ऊ币 | $\overline{\mathbf{K}}$ | Ж | $\mathbf{K}$ | ऊ | $\mathbf{K}$ | ज्ञ |
| Ж | $\mathbf{K}$ | З | $\mathbf{K}$ | ЗK | K | Ж |  |

and yellow cells.

2) Suppose that in some column, there are more yellow cells than red ones. Since in each of the other columns, there are no fewer yellow cells than red ones, then in the entire table, there would be more yellow cells than red ones, but this is not the case (see 1). Therefore, in each of the eight columns, there are an equal number of red and yellow cells.
3) Since in each column, there are no fewer yellow cells than green ones, the cases where in each column: a) 1 yellow, 1 red, 5 green cells and b) 2 yellow, 2 red, 3 green cells are excluded.

The only remaining case is when in each column there are 3 red, 3 yellow, and 1 green cell. Then, in total, there are 8 green cells in the table.

This case is possible. For example, see the table.

## Evaluation Criteria

+ A complete and well-reasoned solution is provided

$\pm$ A generally correct solution is provided, containing minor gaps or inaccuracies

干 It is proven that there can only be 8 green cells, but an example is not provided

Ғ Only the correct answer and an example are provided

- Only the answer is provided
- The problem is not solved or is solved incorrectly