Three given natural numbers are arranged in order of magnitude. Determine them based on the following information:

- the arithmetic mean of the given three numbers is equal to the middle one,
- the difference between some two of the given numbers is 321,
- the sum of some two of the given numbers is 777.

(L. Šimünek)
Let the middle sought natural number be $x$. For the average of all three numbers to be equal to $x$, the third number must be as much greater than $x$ as the first number is less than $x$. Therefore, the sought numbers can be denoted as

$$
x-a, \quad x, \quad x+a
$$

where $a$ is some natural number. According to the problem, the difference between some two of the sought numbers is 321. The difference between two consecutive numbers is $a$, and the difference between the first and the third number is $2a$. The number 321 is odd, and the difference $2a$ is even, so $2a$ cannot be 321, and thus $a=321$. The sought numbers are

$$
x-321, x, x+321 \text {. }
$$

According to the problem, the sum of some two of the sought numbers is 777. The sum of the first and the second is $2x-321$, the sum of the second and the third is $2x+321$, and the sum of the first and the third is $2x$. The sum $2x$ is even, so $2x$ cannot be 777. We also rule out the possibility $2x+321=777$, because then $x=228$, and thus the first sought number would be negative. Therefore, it must be $2x-321=777$, from which $x=549$. The sought numbers are

$$
228,549,870 \text {. }
$$

Evaluation. 1 point for the consideration of the average; 1 point for the consideration of the difference $321$; 2 points for the consideration of the sum $777$; 2 points for the correct conclusion.