# Problem 287: Quadtree encoding (a simple compression algorithm)

![graphic](img287.gif)

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 ?