The Run is the graduation examination that the Assassins have to take undertake. They are forbidden to tell any undergraduates details of their examination. Should they fail their examination they are themselves assassinated. It consists of one of the lecturers acting in the capacity of examiner (for example Pteppic got Mericet in Pyramids), and the undergraduates then have to take a certain route to a preselected destination, find the victim, and assassinate them. Any student who manages to assassinate their examiner is considered to have passed the examination.
This worried Pteppic on his run, as he knew he could not kill anyone. He attempted instead to assassinate Mericet and failed, but was still considered to have passed.