4. Three three-digit numbers, in the notation of which all digits except zero participate, add up to 1665. In each number, the first digit was swapped with the last digit. This resulted in three new three-digit numbers. What is the sum of the new numbers
Solution: The sum of the last digits of the three original numbers is 5, 15, or 25. However, 5 and 25 are excluded, as they cannot be represented as the sum of three different digits (from 1 to 9), so 15 remains. Therefore, the sum of the middle digits is also 15, and the sum of the first digits is 15. It then becomes clear that by rearranging the digits, we will again obtain three numbers whose sum is 1665.

Finally, let's present a triplet of numbers (one of the possible ones) that

satisfies the conditions of the problem: $159, 672, 834$.

Answer: 1665.

Grading criteria: Correct answer with all necessary explanations 7 points. Presenting a triplet of numbers that satisfy the conditions of the problem is not mandatory. Their existence follows from the problem's conditions.