Langton's Ant
An ant moves on a regular grid of squares that are coloured either black or white.
The ant is always oriented in one of the cardinal directions (left, right, up or down) and moves from square to adjacent square according to the following rules:
- if it is on a black square, it flips the colour of the square to white, rotates $90$ degrees counterclockwise and moves forward one square.
- if it is on a white square, it flips the colour of the square to black, rotates $90$ degrees clockwise and moves forward one square.
Starting with a grid that is entirely white, how many squares are black after $10^{18}$ moves of the ant?