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From: Butch Malahide on 1 Aug 2010 19:25
On Aug 1, 12:13 pm, Danny73 <fasttrac...(a)att.net> wrote:
> On Aug 1, 5:31 am, Butch Malahide <fred.gal...(a)gmail.com> wrote:
>
>
>
>
>
> > On Aug 1, 4:09 am, Butch Malahide <fred.gal...(a)gmail.com> wrote:
>
> > > On Jul 31, 8:51 pm, Brent Hugh <brentdh...(a)gmail.com> wrote:
>
> > > > 36 people total, meeting in groups of 6. After 5 minutes, the groups
> > > > shuffle into completely new groups.  Repeat this every five minutes.
>
> > > > Is it possible to arrange the meetings in such a way that after 7 sets
> > > > of meetings, each person has been in a group with each of the other 36
> > > > people exactly once?
>
> > > No. Suppose that could be done. So you've got 36 guys and 42 groups.
> > > Now get 7 new guys, and make those 7 guys into a group (called "the
> > > line at infinity"). So now you've got 43 guys and 43 groups; one group
> > > has 7 members the rest have only 6. Now add one more member to each of
> > > those 6-member groups, as follows. Number the new guys from 1 to 7,
> > > add new guy #1 to each of the 6 groups from the first meeting, add new
> > > guy #2 to each of the 6 groups from the second meeting, and so on.
>
> > > OK, now you've got 43 guys and 43 groups; each group has 7 members;
> > > each guy is in 7 groups; any two groups have exactly one member in
> > > common; and two guys are together in exactly one group. If you call
> > > the guys "points" and the groups "lines", what you have is a structure
> > > called a "projective plane of order 6". A projective plane of order n
> > > is the same except that you have n + 1 points on a line and n + 1
> > > lines through a point; the number of lines and the number of points
> > > are both equal to n^2 + n + 1. Your question, then, is tantamount to
> > > asking whether there is a projective plane of order 6.
>
> > > It is known that a projective plane of order n exists if n is a prime
> > > number, or more generally if n is a (positive) power of a prime
> > > number; i.e., they exist for n = 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17,
> > > 19, 23, 25, 27, 29, . . . . (In other words, a projective plane of
> > > order n exists whenever there is an n-element field.) It is
> > > conjectured that they do not exist for any other value of n. I believe
> > > that many values of n have been ruled out (as possible orders of
> > > projective planes), but don't ask me what they are, I never studied
> > > that stuff. All I know is that projective planes of order 6 do not
> > > exist, and that was established a long time ago.
>
> > > Search terms: "finite projective plane", "orthogonal latin squares",
> > > "36 officers".
>
> > P.S. I looked at that "Metafilter" page, and I see they are also
> > asking how many rounds you can go without two people meeting a second
> > time. The answer is 3 rounds. Trying to do it for 4 rounds is
> > equivalent to Euler's problem of the 36 officers, and was proved
> > impossible by a guy named Tarry something like a hundred years ago.
> > Look it up on Wikipedia.
>
> I know what a round is but how many discrete meetings are possible?
> Is it more than 18 meetings?

Good question. I don't know.
From: Gerry Myerson on 1 Aug 2010 19:37
In article
<0dfa91bd-3c07-4dc4-8be3-9612cd6a9901(a)a30g2000vba.googlegroups.com>,
Butch Malahide <fred.galvin(a)gmail.com> wrote:

> On Aug 1, 1:39�am, Gerry <ge...(a)math.mq.edu.au> wrote:
> >
> > > 36 people total, meeting in groups of 6. After 5 minutes, the groups
> > > shuffle into completely new groups. �Repeat this every five minutes.
> >
> > > Is it possible to arrange the meetings in such a way that after 7 sets
> > > of meetings, each person has been in a group with each of the other 36
> > > people exactly once?
> >
> > > Interestingly there is a pretty simple solution for groups of 4, 9,
> > > 16, 25, 49 and all other squares of primes (see the link above for
> > > details).
> >
> > 16 is not the square of a prime.
>
> But it is the square of a prime power.
>
> > > But how about squares of non-prime numbers, like groups of 36?
> >
> > > Is a solution possible? �Is there some general method for finding it?
> > > Any thoughts in general? �(I find it hard to believe that this type of
> > > problem hasn't been studied already . . . )
> >
> > There is a literature about these things. Sometimes
> > it's called the social golfer problem, sometimes it
> > relates to whist tournaments (but in those cases we
> > usually want groups of 4, not 6). Maybe resolvable
> > block designs is the keyphrase.
>
> Isn't the OP is just asking for a projective plane (or 5 mutually
> orthogonal Latin squares) of order 6? And just getting through four
> sets of meetings without two people being together twice, isn't that
> just Euler's problem of the 36 officers, which was proved impossible
> by Tarry? Or am I just confused?

I had a feeling it reduced to those problems, but I couldn't quite
make the connection. I see you've worked it out in another post.
Good!

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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