yaworsw/euler-manager

---
:id: 103
:name: 'Special subset sums: optimum'
:url: https://projecteuler.net/problem=103
: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:\n\n1. S(B) ≠ S(C); that is, sums of subsets cannot be equal.\n2.
  If B contains more elements than C then S(B) \\> S(C).\n\nIf 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\n_n_ = 1: {1}  \n_n_ = 2: {1, 2}  \n_n_ = 3:
  {2, 3, 4}  \n_n_ = 4: {3, 5, 6, 7}  \n_n_ = 5: {6, 9, 11, 12, 13}\n\nIt _seems_
  that for a given optimum set, A = {_a_<sub>1</sub>, _a_<sub>2</sub>, ... , _a_<sub>n</sub>},
  the next optimum set is of the form B = {_b_, _a_<sub>1</sub>+_b_, _a_<sub>2</sub>+_b_,
  ... ,_a_<sub>n</sub>+_b_}, where _b_ is the \"middle\" element on the previous row.\n\nBy
  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.\n\nGiven that A is an optimum special sum set for _n_ = 7,
  find its set string.\n\nNOTE: This problem is related to [Problem 105](problem=105)
  and [Problem 106](problem=106).\n\n"