---
id: 5900f4361000cf542c50ff48
title: 'Problem 201: Subsets with a unique sum'
challengeType: 5
forumTopicId: 301841
dashedName: problem-201-subsets-with-a-unique-sum
---

# --description--

For any set $A$ of numbers, let $sum(A)$ be the sum of the elements of $A$.

Consider the set $B = \\{1,3,6,8,10,11\\}$. There are 20 subsets of $B$ containing three elements, and their sums are:

$$\begin{align}
  & sum(\\{1,3,6\\}) = 10 \\\\
  & sum(\\{1,3,8\\}) = 12 \\\\
  & sum(\\{1,3,10\\}) = 14 \\\\
  & sum(\\{1,3,11\\}) = 15 \\\\
  & sum(\\{1,6,8\\}) = 15 \\\\
  & sum(\\{1,6,10\\}) = 17 \\\\
  & sum(\\{1,6,11\\}) = 18 \\\\
  & sum(\\{1,8,10\\}) = 19 \\\\
  & sum(\\{1,8,11\\}) = 20 \\\\
  & sum(\\{1,10,11\\}) = 22 \\\\
  & sum(\\{3,6,8\\}) = 17 \\\\
  & sum(\\{3,6,10\\}) = 19 \\\\
  & sum(\\{3,6,11\\}) = 20 \\\\
  & sum(\\{3,8,10\\}) = 21 \\\\
  & sum(\\{3,8,11\\}) = 22 \\\\
  & sum(\\{3,10,11\\}) = 24 \\\\
  & sum(\\{6,8,10\\}) = 24 \\\\
  & sum(\\{6,8,11\\}) = 25 \\\\
  & sum(\\{6,10,11\\}) = 27 \\\\
  & sum(\\{8,10,11\\}) = 29
\\end{align}$$

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 $sum(U(B,3)) = 156$.

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.

Determine the sum of all integers which are the sum of exactly one of the $50$-element subsets of $S$, i.e. find $sum(U(S,50))$.

# --hints--

`uniqueSubsetsSum()` should return `115039000`.

```js
assert.strictEqual(uniqueSubsetsSum(), 115039000);
```

# --seed--

## --seed-contents--

```js
function uniqueSubsetsSum() {

  return true;
}

uniqueSubsetsSum();
```

# --solutions--

```js
// solution required
```