For each 'bad' path "P", define a new path "P"′ by reflecting the part of "P" up to the first point it touches the line across it. "P"′ is a path from (−1, 1) to ("p", "q"). The same operation applied again restores the original "P". This produces a one-to-one correspondence between the 'bad' paths and the paths from (−1, 1) to ("p", "q"). The number of these paths is formula_38 and so that is the number of 'bad' sequences. This leaves the number of 'good' sequences as