Hi Funsters, Surely this is well known, but somehow not to me. How do you arrange round robin tournament pairings? I actually needed to do it for six teams, and after a little doodling I came up with a pretty symmetric- looking pairing scheme. Numbering points in cyclic order, you pair 1-2 3-4 5-6 and its (one) cyclic shift, and pair 1-4 2-6 3-5 and its (two) cyclic shifts. The first two match each person with the two sitting on either side of them, and the last three match each person with their antipode and antipode's neighbors. Is it always possible to do this? (Where "this" is a set of pairings that's invariant under cyclic shifts.) For an odd number of people, I can certainly come up with one. Number the people from -n to n, and have each k play -k, with 0 sitting out. This plus its cyclic shifts (just adding mod n) do the job, and it's easy to see why -- draw the people in order in a circle, and each round's pairings are a full family of parallel lines. For any even number of players, you can run the above scheme, with one distinguished person playing againt the guy who would have sat out above. But I don't like singling someone out like that. (Though I guess you can still draw asymmetric-looking picture of it, by having the distinguished person sitting in the middle of the circle.) But is there always a way that's as symmetric as the 6-person example? And of course there are the counting questions: in how many different ways (modulo renaming the players, of course!) can you pair people up, both in general and with the symmetry requirement. I looked in the EIS for round-robin and didn't get any answers. Okay, now you can all tell me how well-known all this is. --Michael Kleber kleber@brandeis.edu