, 2025

We say that the positive integer $n \ge 3$ is happy if it is possible to arrange $n$ different positive integers in a circle such that two conditions are satisfied:

(a) If integers $u$ and $v$ in the circle are neighbours, then either $u$ divides $v$ or $v$ divides $u$;

(b) If different integers $u$ and $v$ are in the circle but are not neighbours, then neither divides the other.

Determine, with proof, which positive integers $n$ in the range $3 \le n \le 12$ are happy.