Problem 7.6. Natural numbers 13, 14, 15, 16, 17, 18, 19, 20 are placed in the circles on the diagram (each number in one circle) such that the sums of the three numbers along each line are equal. What number can be in the circle $X$? List all possible options.

![](https://cdn.mathpix.com/cropped/2024_05_06_4d36ea080cd7b65cb574g-5.jpg?height=287&width=422&top_left_y=1447&top_left_x=516)
Answer: 15, 18.

Solution. Let $a$ be the number in the circle $X$. By adding the three identical sums along the lines, we get the sum of all nine numbers once, plus $a$. Therefore, $(13+14+\ldots+20)+a=132+a$ is divisible by 3. Then $a$ is divisible by 3, so $a=15$ or $a=18$.

The value $a=15$ is possible, for example, as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_4d36ea080cd7b65cb574g-6.jpg?height=279&width=418&top_left_y=355&top_left_x=518)

In each line, the sum is 49.

The value $a=18$ is possible, for example, as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_4d36ea080cd7b65cb574g-6.jpg?height=277&width=418&top_left_y=798&top_left_x=518)

In each line, the sum is 50.