Katee,
In an
earlier message, Richard gave a schedule that he calls a social square
*.
From this you should see that the everyone-plays-all-the-others-equally property can only be done if the number of rounds is a multiple of 5. So the best you can do for 12 rounds is to use two copies of this schedule with any two additional rounds from a third copy.
Of course if you just reuse the same plan for rounds 6-10 as you did for rounds 1-5 then the players will no doubt complain that they are playing in the same foursomes as they did several days previously. So it is prudent to randomize the second copy of the plan a little.
It turns out that you can do better than just avoid having the same foursomes come up again, it is also possible to ensure that no three players play within a foursome more than once.
For example the triads ABC, ABD, ACD and BCD all occur in the very first foursome in the plan below, however these triads will never crop up elsewhere in the schedule of 12 rounds.
(ABCD) (EFGH) (IJKL) (MNOP)
(AEIM) (BFJN) (CGKO) (DHLP)
(AFKP) (BELO) (CHIN) (DGJM)
(AGLN) (BHKM) (CEJP) (DFIO)
(AHJO) (BGIP) (CFLM) (DEKN)
(ILOP) (BDFK) (EHMN) (ACGJ)
(ABEP) (FHJL) (CDMO) (GIKN)
(FGMP) (BCLN) (EJKO) (ADHI)
(DJNP) (AKLM) (BGHO) (CEFI)
(CHKP) (DEGL) (AFNO) (BIJM)
(AEFL) (DHNO) (BJKP) (CGIM)
(BEIN) (CDKL) (FGJO) (AHMP)
In the schedule above everyone would play with everyone else exactly twice if you stopped after the 10th round. If you use all 12 rounds then all the pairs within the last two fousomes (e.g AE, AF, AL etc…) will all occur three times. When you use the plan then it would be a good idea to randomize the tee-off order both of the foursomes themselves and the players within a foursome.
Regards,
Ian.
Monday, September 5 2005, 06:18 amThanks for another great answer Ian -- Richard*In mathematics this layout is known as an affine geometry or a resolvable balanced incomplete block design.
