---
id: 5900f4761000cf542c50ff88
title: 'Problem 265: Binary Circles'
challengeType: 5
forumTopicId: 301914
dashedName: problem-265-binary-circles
---

# --description--

$2^N$ binary digits can be placed in a circle so that all the $N$-digit clockwise subsequences are distinct.

For $N = 3$, two such circular arrangements are possible, ignoring rotations:

<img class="img-responsive center-block" alt="two circular arrangements for N = 3" src="https://cdn.freecodecamp.org/curriculum/project-euler/binary-circles.gif" style="background-color: white; padding: 10px;">

For the first arrangement, the 3-digit subsequences, in clockwise order, are: 000, 001, 010, 101, 011, 111, 110 and 100.

Each circular arrangement can be encoded as a number by concatenating the binary digits starting with the subsequence of all zeros as the most significant bits and proceeding clockwise. The two arrangements for $N = 3$ are thus represented as 23 and 29:

$${00010111}_2 = 23\\\\
{00011101}_2 = 29$$

Calling $S(N)$ the sum of the unique numeric representations, we can see that $S(3) = 23 + 29 = 52$.

Find $S(5)$.

# --hints--

`binaryCircles()` should return `209110240768`.

```js
assert.strictEqual(binaryCircles(), 209110240768);
```

# --seed--

## --seed-contents--

```js
function binaryCircles() {

  return true;
}

binaryCircles();
```

# --solutions--

```js
// solution required
```