I left a clue after I realized this may be a tad hard for those virgin to hardcore mathematics. For all intent and purpose, this problem is purely mathematical and has nothing to do with conventional Morse. This problem is solely Combinatorics and nothing else. See above for more details, there was already one answer. I am interested to find more but if no one get it by the New Year I shall post the answer. Thanks for posting though, I was wondering where everyone was on this forum. Also if you are interested in these more advanced kind of riddles let me know. If there is demand I will post.

Sincerely,
Vainstar