Problem 8.6. The numbers $13,14,15, \ldots, 25$ are painted in five colors: one black number, three red, three blue, three yellow, and three green.

It is known that:

- all four sums of three same-colored numbers are equal;
- the number 13 is red, 15 is yellow, 23 is blue.

(a) (1 point) Find the black number.

(b) (3 points) Find the three green numbers.
# Answer:

(a) (1 point) 19.

(b) (3 points) 14, 21, 22.

Solution. (a) Let's calculate the sum of the numbers from 13 to 25:

$$
13+14+\ldots+25=\frac{(13+25) \cdot 13}{2}=247
$$

The sum of all numbers except the black one should be divisible by 4. Since 247 gives a remainder of 3 when divided by 4, there are exactly three options for the black number: 15, 19, 23. Since 15 is a yellow number and 23 is a blue number, 19 must be the black number.

(b) Let's understand what we already know about the numbers:

- black number: 19
- red number: 13
- yellow number: 15
- blue number: 23
- green numbers are still unknown;
- numbers whose color is not yet determined: 14, 16, 17, 18, 20, 21, 22, 24, 25.

It is clear that the sum of the monochromatic (red/yellow/green/blue) numbers is $\frac{247-19}{4}=57$. From the condition, we know that the number 13 is red, so the sum of the remaining red numbers is $57-13=44$. There is only one way to represent 44 as the sum of two numbers whose color is not yet determined: $24+20$. Therefore, the numbers 24 and 20 are red.

Now we know the following about the numbers:

- black number: 19;
- red numbers: 13, 20, 24
- yellow number: 15
- blue number: 23
- green numbers are still unknown;
- numbers whose color is not yet determined: 14, 16, 17, 18, 21, 22, 25.

From the condition, we know that the number 15 is yellow, so the sum of the remaining yellow numbers is $57-15=42$. There is only one way to represent 42 as the sum of two numbers whose color is not yet determined: $25+17$. Therefore, the numbers 25 and 17 are yellow.

We get the following:

- black number: 19;
- red numbers: 13, 20, 24
- yellow numbers: 15, 17, 25;
- blue number: 23;
- green numbers are still unknown;
- numbers whose color is not yet determined: 14, 16, 18, 21, 22.

The number 14 is definitely not blue, because otherwise the remaining blue number would be $57-23-14=20$, so the number 14 is green. The sum of the remaining green numbers is $57-14=43$, so it is not difficult to determine them - they are 21 and 22. The remaining numbers 16 and 18 are definitely blue. Thus, we have uniquely restored the coloring of all numbers:

- black number: 19
- red numbers: 13, 20, 24;
- yellow numbers: 15, 17, 25
- blue numbers: 16, 18, 23
- green numbers: 14, 21, 22.