Consider a standard 8 x 8 chess board with white and black squares. Now remove two opposing corners to get a board of 62 squares. Suppose we have 31 dominoes (2 x 1 or 1 x 2 rectangles) which we will use to cover the board by placing each domino over exactly two squares. We will not allow overlaps or squares hanging off the edge of the board. In how many ways can we cover the modified board with dominoes?