Thank you for the comment. Glad you enjoyed it. I feel the same about the proof of the sufficient condition. It is mechanical indeed. Also, considering the fact

Thank you for the comment. Glad you enjoyed it. I feel the same about the proof of the sufficient condition. It is mechanical indeed. Also, considering the fact that there are multiple solutions possible for each n, the specific construction I use in my blog post identifies only one (actually two, because the reverse of the solution is also a valid solution) of those multiple solutions. To arrive at the particular construction in my blog post, I picked a solution each from the solutions for n = 2 and n = 3 and tried to explain the separation between the elements algebraically. Such an analysis is helped by the fact that the cases n = 3 and n = 4 have only two solutions each (where each solution in the pair is the reverse of the other). Further the length of each solution for these cases is also quite small thus making a manual analysis possible. The algebraic formulas so obtained then becomes both the proof of the sufficient condition as well as a method to construct one of the solutions.