Similarly, I have discovered a truly marvelous proof that if x = 1 and x = 2, then 1 = 2.  (Unfortunately, the proof is too short to fit in this space.)  The proof is complete and valid, even though no x exists that meets the initial conditions.  That turns out to be fortunate, as I would not want to live in a world where 1 = 2.

But back to S.  Here is another set that works:

(a,b,c,d,e,f) = (6,6,6,6,6,6) and S = 36.  As xdog proved, 36 must be composite.