Let each integer number from $1$ to $2n$ be ascribed to a vertex or a side of an $n$-gon (one number per vertex and per side). Let the weight of a side be the sum of three numbers "connected" to it (i.e. the number ascribed to the side and two numbers ascribed to its end points).
Find all possible combinations (disregarding reflections and rotations) of numbers (from 1 to 12) for hexagon such that the weights of all sides are equal. Please do not use computer.
I had a lot of fun solving the problem, and wonder if some simple recipes for finding the solutions for larger $n$ exist.
PS. The original problem was to find a single combination, but it was so funny that I decided to find them all.