8.4. A three-digit number, which does not contain any zeros, is written on the board. Petya wrote down in his notebook all different numbers that can be obtained from the original number by rearranging its digits (including the number written on the board). The sum of all numbers written by Petya in his notebook turned out to be 444. What number could have been initially written on the board
Answer: $112,121,211,444$.

Solution. If all the digits in the original three-digit number were different, Petya would have written down 6 different three-digit numbers, and their sum would be no less than 600. Therefore, there are two possible cases. 1) If all the digits were the same, Petya wrote down only one original number, and it is exactly 444. 2) In the original number, there were 2 identical digits. Let's say the number had two digits $a$ and one digit $b$. Then Petya wrote down the numbers $\overline{a a b}=100 a+10 a+b, \overline{a b a}=100 a+10 b+a$ and $\overline{b a a}=100 b+$ $10 a+a$. Their sum is $222 a+111 b=111(2 a+b)$. Thus, $2 a+b=$ $444 / 111=4$. From this, we get the only solution: $a=1, b=2$.

## Criteria.

1 point. Only the answer.

5 points. The case where all digits were the same is not considered (thus, the number 444 is missing in the answer).

Minus 2 points. The impossibility of the case where all 3 digits were different is not proven (or such a case is not considered at all).

Comment. If everything else written is correct, give 5 points; if the case with all three identical digits is also not considered, then with the previous criterion, it results in 3 points.