The basic idea of the weave method is that you can't have a set of 4 fair dice, unless every 3 dice subset is also fair. You start with a n=0 dice set and doing a depth first search, you add one die at a time.

  depthS(string str, int n, int s, int lastinsert) update_fairness_calc(lastinsert) if(!fair()) return
  if(s == MAX_S)
      depthN(str, n)
  for(i = lastinsert; i <= MAX_S * (n-1); ++i)
      depthS(str.insert('a' + n - 1, i), n, s + 1, i)
  depthN(string str, int n) if(n == MAX_N) print("solution = " + n) return calculate_worth_of_all_insertion_points(str) depth(str, n + 1, 0, 0)
  main() depthN("", 0)

Here's an outline of the basic idea of the simple version. 1. First build a list of all n=2 solutions. 1. When going from a n=2 set to a n=3 set, ALL n=2 subsets of the n=3 set must be part of the list generated in step 1. Keep a list of all n=2 subsets of the n=3 solutions. 1. Again, when going from a n=3 set to a n=4 set, all n=2 subsets of the n=4 set must be part of the list generated step 2.

But we can do even better. There are 3 different n=2 subsets in all n=3 sets, ba, ca, and cb. We can keep separate lists for all n=2 subsets. When building n=4 sets, we use their n=3 subsets and again filter based on the more specific 3 different n=2 lists. For example, the db relationship in dcba is part of 2 n=3 subsets, dcb and dba. In those subsets, db is equivalent to ca and cb in cba. Full relationship chart listed below.

Lists of n=2 subset counts are available on the wiki page Results.