Alternatively, the general statement can be formulated in terms of the total number of electrons using simple rules of divisibility by a straightforward analysis of two cases.
First, consider the case where the total number of electrons is 4n + 2:
:4n + 2 = a(4q + 2) + b(4p + 2) + c(4t) + d(4r),
where a, b, c, and d are coefficients indicating the number of each type of component.  This equation implies that one of, but not both, a or b is odd, for if a  and b are both even or both odd, then the sum of the four terms is 0 (mod 4).
The generalized statement of the Woodward–Hoffmann rules states that a + d is odd if the reaction is allowed.  Now, if a is even, then this implies that d is odd.  Since b is odd in this case, the number of antarafacial components, b + d, is even.  Likewise, if a is odd, then d is even.  Since b even in this case, the number of antarafacial components, b + d, is again even.  Thus, regardless of the initial assumption of parity for a  and b, the number of antarafacial components is even when the electron count is 4n + 2.  Contrariwise,, b + d is odd.
In the case where the total number of electrons is 4n, similar arguments (omitted here) lead to the conclusion that the number of antarafacial components b + d must be odd in the allowed case and even in the forbidden case.
Finally, to complete the argument, and show that this new criterion is truly equivalent to the original criterion, one needs to argue the converse statements as well, namely, that the number of antarafacial components b + d and the electron count (4n + 2 or 4n) implies the parity of a + d that is given by the Woodward–Hoffmann rules (odd for allowed, even for forbidden).  Another round of (somewhat tedious) case analyses will easily show this to be the case. The pericyclic selection rule states:
In this formulation, the electron count refers to the entire reacting system, rather than to individual components, as enumerated in Woodward and Hoffmann's original statement.  In practice, an even or odd number of antarafacial components usually means zero or one antarafacial components, respectively, as transition states involving two or more antarafacial components are typically disfavored by strain.  As exceptions, certain intramolecular reactions may be geometrically constrained in such a way that enforces an antarafacial trajectory for multiple components.  In addition, in some cases, e.g., the Cope rearrangement, the same (not necessarily strained) transition state geometry can be considered to contain two supra or two antara π components, depending on how one draws the connections between orbital lobes.  (This ambiguity is a consequence of the convention that overlap of either both interior or both exterior lobes of a σ component can be considered to be suprafacial.)
This alternative formulation makes the equivalence of the Woodward–Hoffmann rules to the Dewar–Zimmerman analysis (see below) clear.  An even total number of phase inversions is equivalent to an even number of antarafacial components and corresponds to Hückel topology, requiring 4n + 2 electrons for aromaticity, while an odd total number of phase inversions is equivalent to an odd number of antarafacial components and corresponds to Möbius topology, requiring 4n electrons for aromaticity.  To summarize aromatic transition state theory: Thermal pericyclic reactions proceed via (4n + 2)-electron Hückel or (4n)-electron Möbius transition states.
As a mnemonic, the above formulation can be further restated as the following: