Sixteen golfers play in foursomes on six different days.

combinatorics/golf.foursomes New

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Problem

Sixteen golfers play in foursomes on six different days.  How can the
schedule be set up so that each golfer plays with every other golfer
once (it will also be necessary for each golfer to play with three
golfers twice)?

[ Editor: problem statement condensed from a request posted by Chris
  Cole for Edward & Sandra Fabian, December 1997, who were scheduling
  a golf tournament.  The foursomes are implied by the solutions. ]

Show solution

[ Editor: the original request: ]

> I am helping schedule a golf tournament in late February. There are 16
> golfers participating in rounds on six different days. We are
> attempting to set up a schedule whereby each golfer will play with
> every other golfer once. It will also be necessary for each golfer to
> play with three golfers twice.
>
> I am able to compute numbers of combinations (astronomical!), but am
> unable to figure what the ideal combinations would be.

I'm not sure how you want to measure ideal.
For 5 days, you could use a schedule like:

 0         5          10        15
        6 9        12  13
1 4        7 8
2 3                 11 14


 1         6          11        15
        7 5        13  14
2 0        8 9
3 4                 12 10


 2         7          12        15
        8 6        14  10
3 1        9 5
4 0                 13 11


 3         8          13        15
        9 7        10  11
4 2        5 6
0 1                 14 12

 4         9          14        15
        5 8        11  12
0 3        6 7
1 2                 10 13

You'd have to repeat partners on the 6th day, I don't know what
kind of repetition would be less ideal than another.

You might also check the articles on bridge club scheduling that appear
in Mathematics Magazine Vol. 70, No. 4 Oct 1997

-- Dean Inada <dmi@questrel.com>

First Day:
ABCD
EFGH
IJKL
MNOP

Second Day:
AEIM
BFJN
CGKO
DHLP

Third Day:
AFKP
BELO
CHIN
DGJM

Fourth Day:
AGLN
BHKM
CEJP
DFIO

Fifth Day:
AHJO
BGIP
CFLM
DEKN

Sixth Day:
Whatever you want, as everybody has played with everybody
else.  I'd suggest matching best scores, with head-to-head
and then coin toss as tie breaker.

Solved this earlier this year--I think they gave
credit to the newsgroup on the tournament schedule.  Thanks.

-- RM Mentock <mentock@mindspring.com>