The old man is a character seen by Tsukiko drawing on the ground in chalk what looks like complex mathematical equations as she is passing by on a bus.
As the detectives exit the hospital and drive away, the old man's writing is visible. It is revealed that when Tsuki had passed by the man, she was passing by the hospital, where she eventually ended up.
He appears during the opening sequence and after the end credits in a tuxedo laughing and giving a prophecy related to the next episode. After he leaves in the beginning of episode 12, he is replaced in the end prophecy by a new white haired man in a tuxedo.