55 lines
1.6 KiB
Markdown
55 lines
1.6 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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S(B) ≠ S(C); that is, sums of subsets cannot be equal.
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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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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}
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It seems that for a given optimum set, A = {a1, a2, ... , an}, the next optimum set is of the form B = {b, a1+b, a2+b, ... ,an+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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`euler103()` should return 20313839404245.
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```js
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assert.strictEqual(euler103(), 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 euler103() {
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return true;
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}
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euler103();
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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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