Question 56
Find all positive integers $n$ for which all $10^n$ non-negative integers consisting of $n$ digits can be ordered in such a way that all the following conditions are met:
(1) the first number consists of zeros only;
(2) every two numbers that are consecutive in this order differ at exactly one position and the digits at this position differ by exactly 1;
(3) the last number consists of nines only.
*Remark:* In this problem, we allow numbers to begin with zero(s).
(1) the first number consists of zeros only;
(2) every two numbers that are consecutive in this order differ at exactly one position and the digits at this position differ by exactly 1;
(3) the last number consists of nines only.
*Remark:* In this problem, we allow numbers to begin with zero(s).