rationale for 6n+2 rule

[nomadreid]

In several websites there is a rule of thumb to determine the number of pi bonds in a non-cyclic molecule called the "6n+2 rule", in which n = the number of non-hydrogen atoms in the molecules, and V = total number of valence electrons, so that the number of electrons involved in pi bonds is 6n+2-V (i.e., the number of pi bonds is (6n+2-V)/2. This rule was introduced in the article "Lewis Structures and the octet rule. An automatic procedure for writing canonical forms." by Lever, A.B.P. in the J.Chem Educ. 1972. 49(12), pages 819-821. Unfortunately, there is a paywall to this article, and none of the websites which are freely available explain why the rule (generally) works. I considered that the number 6 might have something to do with a maximum of three sigma bonds around an atom and the 2 to do with a hydrogen bond, but that didn't pan out. (Also, the examples tend to be organic molecules, but is this necessary?) In short, why does the rule work (when it works)? Thanks very much for any pointers.