Thanks to Isabella, Y12, for the original version of this problem.

Freddy the fly has recently acquired a metallic purple marker pen which he keeps with him at all times. He has just landed on a black square on the top row of an 8x8 chessboard and decides to play a game (not chess!) He marks his starting square with a small purple cross Then, he can move to another black square in the row below, but this must be diagonal to the starting square. He repeats this process until he reaches the bottom row of the chess board, marking each square that he goes to with a purple cross each time. How many purple crosses must he mark and how many different arrangements could they be in once he finishes the game?