Problem Archive

The quadtree encoding allows us to describe a $2^N \times 2^N$ 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 $2^N \times 2^N$ region;
• "0" denotes a split:
  the current $2^n \times 2^n$ region is divided into $4$ sub-regions of dimension $2^{n - 1} \times 2^{n - 1}$,
  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 $4 \times 4$ 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 $D_N$ as the $2^N \times 2^N$ image with the following coloring scheme:

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

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