Problem 287
Quadtree encoding (a simple compression algorithm)

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 ?

These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK