Here's a nearly optimal solution. It minimizes the total distance walked by all people. All moves are to an orthogonally adjacent cell. One person does not move at all; everyone else moves either half the time, or once more than half the time. The orderliness is very high. I'm going to reformulate the problem in terms in 2n people moving through a series of n cells arranged in a circle. The original problem can be translated into these terms by choosing a Hamiltonian cycle. Choose a nice, symmetric one to maximize orderliness. Designate one person as the "Guru", who does not move. In each round, each person moves if they did not move the previous round; and the person with the Guru also moves. (In the first round, arbitrarily choose who will move in each room other than the Guru's.) On even numbered rounds, the people who are moving move clockwise; on odd numbered rounds, those moving move counter-clockwise. That's it. Note that each person moves in the same direction each time until meeting the Guru, at which point they turn around and move the other way for the remaining time. Franklin T. Adams-Watters P.S. This works particularly well when there is an obvious choice to be the Guru. Otherwise there might be some hard feelings. -----Original Message----- From: ed@mathpuzzle.com At a convention, 128 people all need to meet with each other, 1 on 1, for a paper exchange. Suppose a room is divided into a 8x8 square, and a clock is set up to ding 127 times, once every 30 seconds. Each person is given a map of their starting square, and a map of their route over the next hour and 4 minutes. Only two people can be in each cell. Ideally, most moves should be King moves. Not needing to move during a turn is acceptable. Simplicity is judged by the amount of text needed to explain the choreography, and the orderliness of the room. Alternately, the length of the maximum walk should be minimised. Ed Pegg Jr ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.