9.5. What is the smallest number of colors needed to color all the cells of a 6 by 6 square so that in each row, column, and diagonal of the square, all cells have different colors? Explanation: a diagonal of the square is understood to mean all rows of at least two cells running diagonally from one edge of the square to the other at an angle of $45^{\circ}$ or $135^{\circ}$ to the horizontal.
Answer: In 7 colors.

Solution. Let's provide an example of coloring in 7 colors that satisfies the condition of the problem. Consider a 7 by 7 square, and color it in the required way using 7 colors with a known technique: the coloring of each subsequent row is obtained from the coloring of the previous row by a cyclic shift of 2 cells. Then, we select the left lower corner 6 by 6 square from it, which will be the required example.

Let's prove the minimality of 7 colors. Each row of the square contains 6 cells, so at least 6 colors are required for a correct coloring. Suppose we managed to color the square in 6 colors in the required way, then there will be exactly 6 cells of each color, and they will all be in different verticals and horizontals of the square. Let's call the color of the left lower corner cell A of the square black, and show that the remaining 5 black cells cannot be located in the upper right 5 by 5 square in different verticals and horizontals and at the same time not lie on its main diagonal, already controlled by cell A. Otherwise, we can assume that at least three of them would be located below the main diagonal in a "staircase" figure consisting of 10 cells, made up of 4 strips of 4, 3, 2, and 1 cells respectively. However, it is directly verified that for any cell of the "staircase," cells not in the same row, column, or diagonal can correctly contain no more than one black cell. The possible positions of this second black cell are shown in the figure. Cells controlled by it are colored light gray.

Therefore, it is impossible to color the 6 by 6 square in the required way in 6 colors.

Grading criteria. Any explicit and correct example with 7 colors: 3 points. Its correctness does not need to be proven.

Noted that there should be no fewer than 6 colors: 1 point. Proven that there should be no fewer than 7 colors: 4 points.
![](https://cdn.mathpix.com/cropped/2024_05_06_992a8eb7463540e96d9cg-2.jpg?height=500&width=552&top_left_y=2192&top_left_x=1364)