data/problems/105.yml
---
:id: 105
:name: 'Special subset sums: testing'
:url: https://projecteuler.net/problem=105
:content: |+
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:
1. S(B) ≠ S(C); that is, sums of subsets cannot be equal.
2. If B contains more elements than C then S(B) \> S(C).
For example, {81, 88, 75, 42, 87, 84, 86, 65} is not a special sum set because 65 + 87 + 88 = 75 + 81 + 84, whereas {157, 150, 164, 119, 79, 159, 161, 139, 158} satisfies both rules for all possible subset pair combinations and S(A) = 1286.
Using [sets.txt](project/resources/p105_sets.txt) (right click and "Save Link/Target As..."), a 4K text file with one-hundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, A<sub>1</sub>, A<sub>2</sub>, ..., A<sub><i>k</i></sub>, and find the value of S(A<sub>1</sub>) + S(A<sub>2</sub>) + ... + S(A<sub><i>k</i></sub>).
NOTE: This problem is related to [Problem 103](problem=103) and [Problem 106](problem=106).