7.5. Each cell of a $5 \times 5$ table is painted in one of several colors. Lada shuffled the rows of this table so that no row remained in its original position. Then Lera shuffled the columns so that no column remained in its original position. To their surprise, the girls noticed that the resulting table was the same as the original one. What is the maximum number of different colors this table can be painted with?
# Answer: 7.

Solution: Let's renumber the colors and reason about numbers instead. Both columns and rows could have been cyclically permuted or divided into a pair and a triplet. If a cyclic permutation of columns was used, then all columns consist of the same set of numbers, i.e., no more than five different numbers. The same situation applies to a cyclic permutation of rows.

If, however, both columns and rows were divided into a pair and a triplet, the table can be conditionally divided into squares $2 \times 2$ and $3 \times 3$, as well as rectangles $2 \times 3$ and $3 \times 2$, in each of which both columns and rows are cyclically permuted. In the squares, there can be no more than two and three different numbers, respectively (because the set of numbers in each column is the same), and in the rectangles, no more than one (since both in columns and rows, the set of numbers must be the same).

![](https://cdn.mathpix.com/cropped/2024_05_06_da44c9888fa01160fc84g-2.jpg?height=512&width=465&top_left_y=1720&top_left_x=1458)

Example for seven numbers see in the figure.

Criteria: Only the estimate - 3 points

Only the example - 3 points.