$\begin{aligned} & {\left[\begin{array}{l}\text { Rebus } \\ \text { [Decimal number system] }\end{array}\right]}\end{aligned}$

From four non-zero digits, the largest and the smallest possible four-digit numbers were formed. The sum of the resulting numbers turned out to be 11990. What numbers could have been formed?
Let's denote the digits, from which the numbers are composed, in ascending order: A, B, C, and D. Then the smallest number composed of these digits is ABCD, and the largest is DCBA. We get the puzzle ABCD + DCBA = 11990 (where different letters can represent the same digits). It is immediately clear that A + D = 10. Therefore, a one has carried over from the hundreds place to the thousands place, meaning B + C > 9. Now, from the tens place, we see that B + C = 18. Therefore, B = C = 9. Consequently, D = 9, and A = 1.

## Answer

9991 and 1999.

Author: Shapovalov A.V.

In what maximum number of colors can the cells of an 8x8 chessboard be painted so that each cell borders by side with at least two cells of the same color?

## Solution

Divide the board into 16 2x2 squares and color each square with its own color. This coloring in 16 colors satisfies the condition.

A greater number of colors cannot be achieved. Indeed, if there are no more than three cells of a certain color, then only one of them can border by side with two cells of the same color (see figure).
![](https://cdn.mathpix.com/cropped/2024_05_06_46af0608f2d2cb90d801g-20.jpg?height=234&width=536&top_left_y=592&top_left_x=767)

If there are at least four cells of each color, then there are no more than sixteen different colors.

## Answer

In 16 colors.