---
id: 5900f3d61000cf542c50fee7
title: 'Problem 103: Special subset sums: optimum'
challengeType: 5
forumTopicId: 301727
dashedName: problem-103-special-subset-sums-optimum
---

# --description--

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 $S(B) > S(C)$.

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.

$$\begin{align}
  & n = 1: \\{1\\} \\\\
  & n = 2: \\{1, 2\\} \\\\
  & n = 3: \\{2, 3, 4\\} \\\\
  & n = 4: \\{3, 5, 6, 7\\} \\\\
  & n = 5: \\{6, 9, 11, 12, 13\\} \\\\
\end{align}$$

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`.

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.

# --hints--

`optimumSpecialSumSet()` should return the string `20313839404245`.

```js
assert.strictEqual(optimumSpecialSumSet(), '20313839404245');
```

# --seed--

## --seed-contents--

```js
function optimumSpecialSumSet() {

  return true;
}

optimumSpecialSumSet();
```

# --solutions--

```js
// solution required
```