ILL 1971 Problem 27

Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$ - gon such that the following conditions are satisfied:

(1) only digits $1$ and $2$ are used;

(2) each number consists of exactly $n$ digits;

(3) different numbers are assigned to different vertices;

(4) the numbers assigned to two neighboring vertices differ at exactly one digit.