Problem 7.8. A set of 28 different dominoes looks like this:
![](https://cdn.mathpix.com/cropped/2024_05_06_99dc865d189ae298307eg-5.jpg?height=440&width=548&top_left_y=947&top_left_x=452)

All 28 dominoes were arranged such that the number of dots on the touching halves of the dominoes is the same. On some halves, the number of dots was completely erased. As a result, the following structure was obtained (empty halves could have been empty initially or could have contained some number of dots).

![](https://cdn.mathpix.com/cropped/2024_05_06_99dc865d189ae298307eg-6.jpg?height=587&width=763&top_left_y=102&top_left_x=349)

How many dots are on each half of the yellow tile?

Dots on half $A: \ldots$

Dots on half $B: \ldots$
Answer: $A=2, B=5$.

Solution. First, note that any number must appear exactly 8 times on the halves of the dominoes (on 6 dominoes paired with other numbers and on one domino twice). This means each number appears an even number of times.

Consider the highlighted red areas consisting of 1 or 3 cells. According to the condition, the numbers on adjacent halves are equal, so the same numbers are placed within each highlighted area. All other halves of the dominoes can be paired with adjacent halves, where the same numbers must also be placed.

![](https://cdn.mathpix.com/cropped/2024_05_06_99dc865d189ae298307eg-6.jpg?height=440&width=555&top_left_y=1367&top_left_x=447)

We get that outside the highlighted areas, each number appears an even number of times. Therefore, for a number to appear an even number of times in the entire domino layout, the number of highlighted red areas with each number must be even. We know the numbers in all highlighted areas except for the areas with halves $A$ and $B$. Now, note that only the numbers 2 and 5 appear an odd number of times (in one highlighted area), so the areas with halves $A$ and $B$ must contain these numbers.

It remains to determine where 2 and 5 are placed. Suppose that 5 is placed in half $A$, then all halves to the left of this one are uniquely determined, and the two highlighted dominoes on the picture become identical, which is impossible.

![](https://cdn.mathpix.com/cropped/2024_05_06_99dc865d189ae298307eg-7.jpg?height=436&width=589&top_left_y=533&top_left_x=432)

Thus, the only possible option is $A=2$ and $B=5$.

Remark. One of the possible configurations of the dominoes is shown in the figure below.

| 5 | 1 | 1 | 2 | 2 | 25 | 5 | 年 | 5 | 5 | 6 | 6 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  |  |  | 2 |  | 3 |  |  |  |  | 6 |
|  |  |  |  | 2 |  | 3 |  |  |  |  | 6 |
|  |  |  |  | 6 |  | 2 |  |  |  |  | 3 |
| 4 | 4 | 4 | 4 | 6 |  | 2 | 0 | 0 | 0 | 0 | 3 |
| 5 |  |  |  |  |  | 2 |  |  |  |  | 3 |
| 5 |  |  |  |  |  | 4 |  |  |  |  | 3 |
| 0 |  |  |  |  |  | 4 |  |  |  |  | 3 |
| 0 |  |  |  |  |  | 0 |  |  |  |  | 4 |
| 1 | 1 | 1 | 1 | 3 |  | 0 | 6 | 6 | 1 | 1 | 4 |

Note: The character "年" in the table is not a number and seems to be a placeholder or error in the original text. It should be replaced with the correct number if known.