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(a)brandeis.edu
