Question 30
The squares of an $8\times8$ grid are coloured black or white. A colouring is called balanced if each $2\times2$ subgrid contains exactly two squares of each colour. An example of a balanced colouring of a $3\times3$ grid is shown on the left. The unbalanced colouring on the right fails this requirement since the $2\times2$ subgrid on the bottom right contains three white squares. Counting rotations and reflections of a pattern as different, how many balanced colourings of the $8\times8$ grid are there?
