Problem 6.1. Jerry has nine cards with digits from 1 to 9. He lays them out in a row, forming a nine-digit number. Tom writes down all 8 two-digit numbers formed by adjacent digits (for example, for the number 789456123, these numbers are $78, 89, 94, 45$, $56, 61, 12, 23$). For each two-digit number divisible by 9, Tom gives Jerry a piece of cheese. What is the maximum number of pieces of cheese Jerry can get?
Answer: 4.

Solution. Note that among two-digit numbers, only 18, 27, 36, 45, and numbers obtained by swapping their digits are divisible by 9 (there are also 90 and 99, but we do not have the digit 0 and only one digit 9). Thus, only four pairs of digits from the available ones can form a number divisible by 9. To get an example, we need to arrange all these pairs in any order:

$$
182736459
$$

## Criteria

The highest applicable criterion is used:

## 76 . Any correct solution to the problem.

3 6. It is proven that it is impossible to get 5 two-digit numbers divisible by 9.

## 3 6. An example with 4 two-digit numbers divisible by 9 is provided.

## 16 . The correct answer is given.