Let $S(A)$ represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true:
It seems that for a given optimum set, $A = \\{a_1, a_2, \ldots, a_n\\}$, the next optimum set is of the form $B = \\{b, a_1 + b, a_2 + b, \ldots, a_n + b\\}$, where b is the "middle" element on the previous row.
By applying this "rule" we would expect the optimum set for $n = 6$ to be $A = \\{11, 17, 20, 22, 23, 24\\}$, with $S(A) = 117$. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for $n = 6$ is $A = \\{11, 18, 19, 20, 22, 25\\}$, with $S(A) = 115$ and corresponding set string: `111819202225`.