Golf foursomes

Harold Druss · science forum beginner Joined: 15 Feb 2006 Posts: 2

Hi All
I have been asked by a group of golfers (20) if it would be possible
for every golfer to play at least once with every other golfer on a 7 day
golf trip.
They will play in foursomes.
If not 7 days, then how many days would it take?
Thanks
Harold

Proginoskes · science forum Guru Joined: 29 Apr 2005 Posts: 2593

Harold Druss wrote:

Rob · science forum beginner Joined: 20 Jun 2005 Posts: 44

(Note first that time has no relavence on the math: you would need to
determine how many games can be played in a day, and then work out an
ideal schedule for the correct number of games...I am concerned here
only with determining the number of games necessary)

Idealy, you are looking for a S(2,4,20) steiner system. A S(l,k,v) is
a collection of k-tuples of a set with v elements such that each
l-tuple appears in exactly 1 of the k-tuples.

EG: S(2,3,7) is a collection of triples of a set of 7 elements (say 1,
2, 3, ..., 7), such that each pair occures in exactly one triple:
1,2,4
2,3,5
3,4,6
4,5,7
5,6,1
6,7,2
7,1,3

However there can't be a S(2,4,20), since by a well known theorem on
general block designs (see Design Theory: Volume 1 by Thomas Beth, H
Lenz, D Jungnickel, pg11, corollary 2.11, 2.11.a), it must be that

v-1 \equiv 0 (mod k-1)

and here that would mean that 20-1 \equiv 0 (mod 4-1), or 19 \equiv 0
(mod 3), which is not true.

Thus, there is no possible way that everyone could play each other
exactly once (assuming you stick to only foursoms)--to answer this
question, you will need to allow some people to play others more than
once (for example, Peter, Paul, Jane, and Wendy may play one game, and
later, it may be necessary for Peter, James, Shaun, and Wendy to play
each other (Peter and Wendy repeat).

There is a maximum number of games you need: just (20 choose 4) = 4845
games. This would cover every possible foursome. The question
remains...how many of those games do we not have to play because
everyone in them has played everyone else at least once...

(It was recently showed that for 25 people, it would work out
perfectly: see http://www.maths.gla.ac.uk/~es/steiner.html, and I think
16 people works too, though I cant find a reference at this moment)

Thats about all I can post right now timewise... Maybe I'll post more
later.

Rob

Harold Druss · science forum beginner Joined: 15 Feb 2006 Posts: 2

Hi Rob
Thanks for your reply.

Dana DeLouis · science forum beginner Joined: 06 Mar 2006 Posts: 37

Thanks for this info. I've never understood the additional info at this
link...
http://mathworld.wolfram.com/SocialGolferProblem.html
I've tried in the past to write a program, but never had any luck. :>(
--
Dana

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