6. (7 points) Petya showed Vasya 37 identical-looking cards laid out in a row. He said that on the hidden sides of the cards, all numbers from 1 to 37 are written (each exactly once) such that the number on any card starting from the second is a divisor of the sum of the numbers written on all preceding cards. Then Petya showed Vasya that the number on the first card is 37, and on the second card is 1. Vasya said that he then knows what number is written on the third card. What is it?
Answer: 2.

Solution. The sum of all numbers except the last one is divisible by the last number, which means the sum of all numbers is also divisible by the last number. The sum of all numbers from 1 to 37 is $19 \cdot 37$. Therefore, the last number is 1, 19, or 37. Since 1 and 37 are in the first and second positions, the last number is 19. The third number is a divisor of $37+1=38$, which means it is 1, 2, or 19. We know that the numbers 1 and 19 are not in the third position, so the number on the third card is 2. Note. It is not necessary to provide an example of how the numbers are arranged on the other cards (or to prove its existence).

## Grading Criteria.

- Full correct solution - 7 points.
- It is stated that the number on the third card is 2 or 19, but there is no further progress - 1 point.

Maximum score for all completed tasks - 42.