70 lines
1.8 KiB
Markdown
70 lines
1.8 KiB
Markdown
---
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id: 5900f4361000cf542c50ff48
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title: 'Problem 201: Subsets with a unique sum'
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challengeType: 5
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forumTopicId: 301841
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dashedName: problem-201-subsets-with-a-unique-sum
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---
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# --description--
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For any set $A$ of numbers, let $sum(A)$ be the sum of the elements of $A$.
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Consider the set $B = \\{1,3,6,8,10,11\\}$. There are 20 subsets of $B$ containing three elements, and their sums are:
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$$\begin{align}
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& sum(\\{1,3,6\\}) = 10 \\\\
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& sum(\\{1,3,8\\}) = 12 \\\\
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& sum(\\{1,3,10\\}) = 14 \\\\
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& sum(\\{1,3,11\\}) = 15 \\\\
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& sum(\\{1,6,8\\}) = 15 \\\\
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& sum(\\{1,6,10\\}) = 17 \\\\
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& sum(\\{1,6,11\\}) = 18 \\\\
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& sum(\\{1,8,10\\}) = 19 \\\\
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& sum(\\{1,8,11\\}) = 20 \\\\
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& sum(\\{1,10,11\\}) = 22 \\\\
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& sum(\\{3,6,8\\}) = 17 \\\\
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& sum(\\{3,6,10\\}) = 19 \\\\
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& sum(\\{3,6,11\\}) = 20 \\\\
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& sum(\\{3,8,10\\}) = 21 \\\\
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& sum(\\{3,8,11\\}) = 22 \\\\
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& sum(\\{3,10,11\\}) = 24 \\\\
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& sum(\\{6,8,10\\}) = 24 \\\\
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& sum(\\{6,8,11\\}) = 25 \\\\
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& sum(\\{6,10,11\\}) = 27 \\\\
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& sum(\\{8,10,11\\}) = 29
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\\end{align}$$
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Some of these sums occur more than once, others are unique. For a set $A$, let $U(A,k)$ be the set of unique sums of $k$-element subsets of $A$, in our example we find $U(B,3) = \\{10,12,14,18,21,25,27,29\\}$ and $sum(U(B,3)) = 156$.
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Now consider the $100$-element set $S = \\{1^2, 2^2, \ldots , {100}^2\\}$. $S$ has $100\\,891\\,344\\,545\\,564\\,193\\,334\\,812\\,497\\,256\\;$ $50$-element subsets.
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Determine the sum of all integers which are the sum of exactly one of the $50$-element subsets of $S$, i.e. find $sum(U(S,50))$.
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# --hints--
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`uniqueSubsetsSum()` should return `115039000`.
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```js
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assert.strictEqual(uniqueSubsetsSum(), 115039000);
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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 uniqueSubsetsSum() {
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return true;
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}
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uniqueSubsetsSum();
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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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