[math-fun] Choreography of large complete graphs

At a convention, 120 people all need to meet with each other, 1 on 1. Suppose a room is divided into a 8x8 square, and a clock is set up to ring 60 times, once per minute. Each person is given a map of their starting square, and a map of their route over the next hour. What is the simplest set of maps? What is the length of the minimal walks people need to take? Ed Pegg Jr