(a) Suppose that each square of a 4 x 7 chessboard is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board) whose four distinct unit corner squares are all of the same color. (b) Exhibit a black-white coloring of a 4 x6 board in which the four corner squares of every rectangle, as described above, are not all of the same color.