---
id: 5900f48b1000cf542c50ff9e
title: 'Problem 287: Quadtree encoding (a simple compression algorithm)'
challengeType: 5
forumTopicId: 301938
dashedName: problem-287-quadtree-encoding-a-simple-compression-algorithm
---

# --description--

The quadtree encoding allows us to describe a $2^N×2^N$ black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:

- the first bit deals with the complete $2^N×2^N$ region;
- "0" denotes a split:
  - the current $2^n×2^n$ region is divided into 4 sub-regions of dimension $2^{n - 1}×2^{n - 1}$,
  - the next bits contains the description of the top left, top right, bottom left and bottom right sub-regions - in that order;
- "10" indicates that the current region contains only black pixels;
- "11" indicates that the current region contains only white pixels.

Consider the following 4×4 image (colored marks denote places where a split can occur):

<img class="img-responsive center-block" alt="4x4 image with colored marks denoting place where split can occur" src="https://cdn.freecodecamp.org/curriculum/project-euler/quadtree-encoding-a-simple-compression-algorithm.gif" style="background-color: white; padding: 10px;">

This image can be described by several sequences, for example : "<strong><span style="color: red">0</span></strong><strong><span style="color: blue">0</span></strong>10101010<strong><span style="color: green">0</span></strong>1011111011<strong><span style="color: orange">0</span></strong>10101010", of length 30, or "<strong><span style="color: red">0</span></strong>10<strong><span style="color: green">0</span></strong>101111101110", of length 16, which is the minimal sequence for this image.

For a positive integer $N$, define $D_N$ as the $2^N×2^N$ image with the following coloring scheme:

- the pixel with coordinates $x = 0$, $y = 0$ corresponds to the bottom left pixel,
- if ${(x - 2^{N - 1})}^2 + {(y - 2^{N - 1})}^2 ≤ 2^{2N - 2}$ then the pixel is black,
- otherwise the pixel is white.

What is the length of the minimal sequence describing $D_{24}$?

# --hints--

`quadtreeEncoding()` should return `313135496`.

```js
assert.strictEqual(quadtreeEncoding(), 313135496);
```

# --seed--

## --seed-contents--

```js
function quadtreeEncoding() {

  return true;
}

quadtreeEncoding();
```

# --solutions--

```js
// solution required
```