# Problem 103: Special subset sums: optimum
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: S(B) ≠ S(C); that is, sums of subsets
cannot be equal. If B contains more elements than C then S(B) > S(C).
If S(A) is minimised for a given n, we shall call it an optimum special
sum set. The first five optimum special sum sets are given below. n = 1:
{1}n = 2: {1, 2}n = 3: {2, 3, 4}n = 4: {3, 5, 6, 7}n = 5: {6, 9, 11, 12,
13} It seems that for a given optimum set, A = {a1, a2, ... , an}, the
next optimum set is of the form B = {b, a1+b, a2+b, ... ,an+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. Given that A is an optimum
special sum set for n = 7, find its set string. NOTE: This problem is
related to Problem 105 and Problem 106.