$\underline{\text { Folklore }}$

Two ten-digit numbers are called adjacent if they differ in only one digit in some place (for example, 1234567890 and 1234507890 are adjacent). What is the maximum number of ten-digit numbers that can be written down so that there are no adjacent numbers among them?
There are $9 \cdot 10^{9}$ ten-digit numbers. Let's divide them into $9 \cdot 10^{8}$ tens: each ten includes numbers that differ only in the last digit (for example, the first ten includes numbers from 1000000000 to 1000000009). In each ten, all numbers are adjacent, so no more than one number can be selected from it. On the other hand, in each ten, there is exactly one number whose sum of digits is divisible by 10. By selecting all these numbers (they obviously cannot be adjacent), we obtain a set of $9 \cdot 10^{8}$ numbers.

## Answer

$9 \cdot 10^{8}$.