Interesting that the sum of the 3 parities for eternity pieces always
equals a multiple of 4.  To explain this, note that we can split the
12 different single drafters into 4 groups of 3 (each group containing
3 drafters that are 120-degrees from each other), and the three
different ways of pairing these give the three parities.  So one is
a+b-c-d, one is a+c-b-d, and one is a+d-b-c, where a,b,c,d represent
the 4 groups.  Now, we actually take the absolute values of these
numbers, but since they are all even (guaranteed, since the total
a+b+c+d = 12, so each of these is the difference of two even or two odd
numbers), each is congruent (mod 4) to its absolute value.  Anyway,
so the sum of the three parities is congruent (mod 4) to 3a-b-c-d,
or 4a (since a+b+c+d = 12 = 0 mod 4).
--Joseph DeVincentis