54 lines
1.8 KiB
Markdown
54 lines
1.8 KiB
Markdown
---
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id: 5900f3d61000cf542c50fee7
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title: 'Problem 103: Special subset sums: optimum'
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challengeType: 5
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forumTopicId: 301727
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dashedName: problem-103-special-subset-sums-optimum
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---
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# --description--
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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:
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1. $S(B) ≠ S(C)$; that is, sums of subsets cannot be equal.
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2. If B contains more elements than C then $S(B) > S(C)$.
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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.
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$$\begin{align} & n = 1: \\{1\\} \\\\ & n = 2: \\{1, 2\\} \\\\ & n = 3: \\{2, 3, 4\\} \\\\ & n = 4: \\{3, 5, 6, 7\\} \\\\ & n = 5: \\{6, 9, 11, 12, 13\\} \\\\ \end{align}$$
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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.
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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`.
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Given that A is an optimum special sum set for $n = 7$, find its set string.
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**Note:** This problem is related to Problem 105 and Problem 106.
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# --hints--
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`optimumSpecialSumSet()` should return the string `20313839404245`.
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```js
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assert.strictEqual(optimumSpecialSumSet(), '20313839404245');
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```
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# --seed--
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## --seed-contents--
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```js
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function optimumSpecialSumSet() {
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return true;
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}
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optimumSpecialSumSet();
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```
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# --solutions--
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```js
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// solution required
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```
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