## Special subset sums: optimum

### Problem 103

Published on Friday, 26th August 2005, 06:00 pm; Solved by 3997Let 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 = {*a*_{1}, *a*_{2}, ... , *a*_{n}}, the next optimum set is of the form B = {*b*, *a*_{1}+*b*, *a*_{2}+*b*, ... ,*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.

Given that A is an optimum special sum set for *n* = 7, find its set string.