1. At the vertices of a regular dodecagon, natural numbers from 1 to 12 (each exactly once) are arranged in some order. Could it happen that the sums of all pairs of adjacent numbers are prime and the sums of all pairs of numbers with exactly two numbers between them are also prime? (20 points)
Solution: Each number at the vertex participates in exactly four sums. Note that to obtain a simple sum, to the numbers 6 and 12, only $1,5,7$ and 11 can be added. This means that for the vertices where the numbers 6 and 12 are located, the sets of adjacent numbers and numbers standing two vertices away must coincide. However, for each vertex, these sets are different, so at least one of the sums will not be a prime number.

## Answer: could not