A chess king starts at a position A in the top row of a standard chessboard. The number of paths of length 7 to a position B in the bottom row is a perfect square, but not a perfect cube.
The number of paths of length 7 from A to a position C in the bottom row is a perfect cube, but not a perfect square.
The number of paths of length 7 to a position D is both a perfect square and a perfect cube.

How many chess king paths of length 5 are there from B to C?