Question 6
(HUN 7) Let $n\ge2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2^n$-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.
(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.