HMMT November 2024 T7

A weird checkerboard is a coloring of an $8 \times 8$ grid constructed by making some (possibly none or all) of the following 14 cuts:

• the 7 vertical cuts along a gridline through the entire height of the board,
• and the 7 horizontal cuts along a gridline through the entire width of the board.

The divided rectangles are then colored black and white such that the bottom left corner of the grid is black, and no two rectangles adjacent by an edge share a color. Compute the number of weird checkerboards that have an equal amount of area colored black and white.