---
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 2N×2N 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 2N×2N region;

"0" denotes a split:

the current 2n×2n region is divided into 4 sub-regions of dimension 2n-1×2n-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):

This image can be described by several sequences, for example : "001010101001011111011010101010", of length 30, or "0100101111101110", of length 16, which is the minimal sequence for this image.

For a positive integer N, define DN as the 2N×2N image with the following coloring scheme: the pixel with coordinates x = 0, y = 0 corresponds to the bottom left pixel, if (x - 2N-1)2 + (y - 2N-1)2 ≤ 22N-2 then the pixel is black, otherwise the pixel is white.What is the length of the minimal sequence describing D24 ?

# --hints--

`euler287()` should return 313135496.

```js
assert.strictEqual(euler287(), 313135496);
```

# --seed--

## --seed-contents--

```js
function euler287() {

  return true;
}

euler287();
```

# --solutions--

```js
// solution required
```