1. The sum of the digits of a natural number is 2017. Moreover, any ten consecutive digits of the number are all different. Find the first 10 digits of the smallest and largest of such numbers. Justify your answer.
Solution. Among every ten consecutive digits, all digits from 0 to 9 appear exactly once, meaning the sum of every 10 consecutive digits is 45. Therefore, the first digit is the same as the eleventh, the second is the same as the twelfth, and so on, which means the number consists of repeating blocks of 10 digits (the last block may be incomplete).

Notice that $2017=44 \cdot 45+37$. This means the number consists of 44 blocks of 10 digits each, and at the end, there are a few more digits with a sum of 37. In other words, it is 8 less than 45 blocks (450 digits), which can be achieved by one (8), two (35), three (521), or four (5210) digits, but not more (since $0+1+2+3+4>8$). Therefore, the length of the number is between 446 and 449 digits.

The number will be the smallest if it consists of 446 digits, and the first 10 digits form the smallest possible number. The sum of the first six digits should be 37. The first digit can be 2, then the others are 5, 6, 7, 8, 9 (if the first digit is 1, the sum of the remaining five should be 36, which is impossible for different digits). The next four digits are 0, 1, 3, 4, forming the smallest number 0134. The smallest number starts with the digits 2567890134.

The number will be the largest if it consists of 449 digits, and the first nine digits form the largest possible number with a sum of 37 (then the tenth digit will be 8). The largest number formed by the first nine digits is 976543210.

Answer: The first ten digits of the smallest number are 2567890134, and the largest number are 9765432108.