21 Teams 3 & 5 Byes per week

[Gavel1995]

To everyone,
Looking for two schedules for 21 Teams.
3 & 5 bye combinations.  Also the Teams that have the byes in week #1 cannot compete against each other in week #2.

Lastly, looking for two 22 Team schedules one with 4 byes and the other with 6 byes each week and the Bye teams compete against each other the next week.

thank you,
Gavel

[Ian Wakeling]

I think I can help with the 22 team schedule that has 4 byes if you don't mind the fact that the bye teams don't compete against each other in the final 4 weeks.  It works for the first 22 weeks because of the cyclic format.

( 1  8) (13 18) (20 10) ( 4  7) ( 3 11) (16 22) ( 5  6) (17 15) (12 21) ( 2  9 14 19)
( 2  9) (14 19) (21 11) ( 5  8) ( 4 12) (17  1) ( 6  7) (18 16) (13 22) ( 3 10 15 20)
( 3 10) (15 20) (22 12) ( 6  9) ( 5 13) (18  2) ( 7  8) (19 17) (14  1) ( 4 11 16 21)
( 4 11) (16 21) ( 1 13) ( 7 10) ( 6 14) (19  3) ( 8  9) (20 18) (15  2) ( 5 12 17 22)
( 5 12) (17 22) ( 2 14) ( 8 11) ( 7 15) (20  4) ( 9 10) (21 19) (16  3) ( 6 13 18  1)
( 6 13) (18  1) ( 3 15) ( 9 12) ( 8 16) (21  5) (10 11) (22 20) (17  4) ( 7 14 19  2)
( 7 14) (19  2) ( 4 16) (10 13) ( 9 17) (22  6) (11 12) ( 1 21) (18  5) ( 8 15 20  3)
( 8 15) (20  3) ( 5 17) (11 14) (10 18) ( 1  7) (12 13) ( 2 22) (19  6) ( 9 16 21  4)
( 9 16) (21  4) ( 6 18) (12 15) (11 19) ( 2  8) (13 14) ( 3  1) (20  7) (10 17 22  5)
(10 17) (22  5) ( 7 19) (13 16) (12 20) ( 3  9) (14 15) ( 4  2) (21  8) (11 18  1  6)
(11 18) ( 1  6) ( 8 20) (14 17) (13 21) ( 4 10) (15 16) ( 5  3) (22  9) (12 19  2  7)
(12 19) ( 2  7) ( 9 21) (15 18) (14 22) ( 5 11) (16 17) ( 6  4) ( 1 10) (13 20  3  8)
(13 20) ( 3  8) (10 22) (16 19) (15  1) ( 6 12) (17 18) ( 7  5) ( 2 11) (14 21  4  9)
(14 21) ( 4  9) (11  1) (17 20) (16  2) ( 7 13) (18 19) ( 8  6) ( 3 12) (15 22  5 10)
(15 22) ( 5 10) (12  2) (18 21) (17  3) ( 8 14) (19 20) ( 9  7) ( 4 13) (16  1  6 11)
(16  1) ( 6 11) (13  3) (19 22) (18  4) ( 9 15) (20 21) (10  8) ( 5 14) (17  2  7 12)
(17  2) ( 7 12) (14  4) (20  1) (19  5) (10 16) (21 22) (11  9) ( 6 15) (18  3  8 13)
(18  3) ( 8 13) (15  5) (21  2) (20  6) (11 17) (22  1) (12 10) ( 7 16) (19  4  9 14)
(19  4) ( 9 14) (16  6) (22  3) (21  7) (12 18) ( 1  2) (13 11) ( 8 17) (20  5 10 15)
(20  5) (10 15) (17  7) ( 1  4) (22  8) (13 19) ( 2  3) (14 12) ( 9 18) (21  6 11 16)
(21  6) (11 16) (18  8) ( 2  5) ( 1  9) (14 20) ( 3  4) (15 13) (10 19) (22  7 12 17)
(22  7) (12 17) (19  9) ( 3  6) ( 2 10) (15 21) ( 4  5) (16 14) (11 20) ( 1  8 13 18)

( 1 12) ( 2 13) ( 3 14) ( 4 15) ( 7 18) ( 8 19) ( 9 20) (10 21) (11 22) ( 5  6 16 17)
( 1  5) ( 2  6) ( 3  7) ( 4  8) ( 9 13) (10 14) (11 15) (12 16) (17 21) (18 19 20 22)
( 5  9) ( 6 10) ( 7 11) ( 8 12) (13 17) (14 18) (15 19) (16 20) (22  4) ( 1  2  3 21)
( 5 16) ( 6 17) (18 22) (19  1) (20  2) (21  3)

[Gavel1995]

Mr. Wakeling,
I like what you have presented and it appears to work for one situation that I have.  I can use the first 22 weeks of what you have developed - thank you very much.  We have tie-breaker schemes that can be employed.

Can you develop a similiar schedule with 6 byes per week with the same cylic format that the bye teams compete against each other in the following week?

Any luck on the 21 Team schedules with 3 and 5 bye combinations?

Thank you for your assistance.  Where are you located, do you have access to a supercomputer?

Gavel

[Ian Wakeling]

Gavel,

Yes, 6 byes in a similar format is possible, for example:

(1 8) (4 16) (20 12) (7 10) (3 19) (6 11) (18 22) (14 15) (2 9 5 17 21 13) 
(2 9) (5 17) (21 13) (8 11) (4 20) (7 12) (19 1) (15 16) (3 10 6 18 22 14) 
(3 10) (6 18) (22 14) (9 12) (5 21) (8 13) (20 2) (16 17) (4 11 7 19 1 15) 
(4 11) (7 19) (1 15) (10 13) (6 22) (9 14) (21 3) (17 18) (5 12 8 20 2 16) 
(5 12) (8 20) (2 16) (11 14) (7 1) (10 15) (22 4) (18 19) (6 13 9 21 3 17) 
(6 13) (9 21) (3 17) (12 15) (8 2) (11 16) (1 5) (19 20) (7 14 10 22 4 18) 
(7 14) (10 22) (4 18) (13 16) (9 3) (12 17) (2 6) (20 21) (8 15 11 1 5 19) 
(8 15) (11 1) (5 19) (14 17) (10 4) (13 18) (3 7) (21 22) (9 16 12 2 6 20) 
(9 16) (12 2) (6 20) (15 18) (11 5) (14 19) (4 8) (22 1) (10 17 13 3 7 21) 
(10 17) (13 3) (7 21) (16 19) (12 6) (15 20) (5 9) (1 2) (11 18 14 4 8 22) 
(11 18) (14 4) (8 22) (17 20) (13 7) (16 21) (6 10) (2 3) (12 19 15 5 9 1) 
(12 19) (15 5) (9 1) (18 21) (14 8) (17 22) (7 11) (3 4) (13 20 16 6 10 2) 
(13 20) (16 6) (10 2) (19 22) (15 9) (18 1) (8 12) (4 5) (14 21 17 7 11 3) 
(14 21) (17 7) (11 3) (20 1) (16 10) (19 2) (9 13) (5 6) (15 22 18 8 12 4) 
(15 22) (18 8) (12 4) (21 2) (17 11) (20 3) (10 14) (6 7) (16 1 19 9 13 5) 
(16 1) (19 9) (13 5) (22 3) (18 12) (21 4) (11 15) (7 8) (17 2 20 10 14 6) 
(17 2) (20 10) (14 6) (1 4) (19 13) (22 5) (12 16) (8 9) (18 3 21 11 15 7) 
(18 3) (21 11) (15 7) (2 5) (20 14) (1 6) (13 17) (9 10) (19 4 22 12 16 8) 
(19 4) (22 12) (16 8) (3 6) (21 15) (2 7) (14 18) (10 11) (20 5 1 13 17 9) 
(20 5) (1 13) (17 9) (4 7) (22 16) (3 8) (15 19) (11 12) (21 6 2 14 18 10) 
(21 6) (2 14) (18 10) (5 8) (1 17) (4 9) (16 20) (12 13) (22 7 3 15 19 11) 
(22 7) (3 15) (19 11) (6 9) (2 18) (5 10) (17 21) (13 14) (1 8 4 16 20 12)

With the extra byes, there are now 55 possible matches that are not played.  Although I have used a computer to generate these schedules, they were essentially constructed by hand on an Excel spreadsheet.  The cyclic property means that each column contains distinct subsets of all possible matches, all with the same difference.  So the problem is reduced to finding round one as the rest of the scehdule can be generated from it.

Are you looking for 21 round solutions for the 21 team scenario?

Regards,

Ian.
(from Norfolk in England)

[Gavel1995]

Mr. Wakeling,
Thank you for this schedule.  It is extremely helpful for our lacrosse league.

Yes, I would greatly appreciate a 21 League schedule with 3 Bye and 5 Bye options as we may have a situation in which one of our schools may not be able to field a Team.  However, if this situation develops it is important that the teams that have the byes each week do not play against each other in the subsequent week.  I know that is opposite of what you have so graciously developed for the 22 Team schedule.

Again, Thank You.

Gavel

[Jeff]

Gavel,
Ian sent me a message that you are looking into schedules that are similiar to what I am using.  I was wondering if you have done any research into looking into PBTDs -- Partitioined Balanced Tournament Designs?  Specifically, for 18 Teams.

I have some tournament schedules that might be of interest to you for less than 20 teams if you are interested in them.

Regards,
Jeff

[Ian Wakeling]

Gavel,

Here are some more cyclic solutions for the 21 team tournaments.  They are of course incomplete as some matches are missing from both schedules.

Can I ask why the property about the byes playing or not playing each other is important to you?

Ian.

(17 12) ( 5 14) (11  9) (13 21) (20 10) ( 1 16) ( 8  7) ( 2 19) ( 6  3) (15  4 18)
(18 13) ( 6 15) (12 10) (14  1) (21 11) ( 2 17) ( 9  8) ( 3 20) ( 7  4) (16  5 19)
(19 14) ( 7 16) (13 11) (15  2) ( 1 12) ( 3 18) (10  9) ( 4 21) ( 8  5) (17  6 20)
(20 15) ( 8 17) (14 12) (16  3) ( 2 13) ( 4 19) (11 10) ( 5  1) ( 9  6) (18  7 21)
(21 16) ( 9 18) (15 13) (17  4) ( 3 14) ( 5 20) (12 11) ( 6  2) (10  7) (19  8  1)
( 1 17) (10 19) (16 14) (18  5) ( 4 15) ( 6 21) (13 12) ( 7  3) (11  8) (20  9  2)
( 2 18) (11 20) (17 15) (19  6) ( 5 16) ( 7  1) (14 13) ( 8  4) (12  9) (21 10  3)
( 3 19) (12 21) (18 16) (20  7) ( 6 17) ( 8  2) (15 14) ( 9  5) (13 10) ( 1 11  4)
( 4 20) (13  1) (19 17) (21  8) ( 7 18) ( 9  3) (16 15) (10  6) (14 11) ( 2 12  5)
( 5 21) (14  2) (20 18) ( 1  9) ( 8 19) (10  4) (17 16) (11  7) (15 12) ( 3 13  6)
( 6  1) (15  3) (21 19) ( 2 10) ( 9 20) (11  5) (18 17) (12  8) (16 13) ( 4 14  7)
( 7  2) (16  4) ( 1 20) ( 3 11) (10 21) (12  6) (19 18) (13  9) (17 14) ( 5 15  8)
( 8  3) (17  5) ( 2 21) ( 4 12) (11  1) (13  7) (20 19) (14 10) (18 15) ( 6 16  9)
( 9  4) (18  6) ( 3  1) ( 5 13) (12  2) (14  8) (21 20) (15 11) (19 16) ( 7 17 10)
(10  5) (19  7) ( 4  2) ( 6 14) (13  3) (15  9) ( 1 21) (16 12) (20 17) ( 8 18 11)
(11  6) (20  8) ( 5  3) ( 7 15) (14  4) (16 10) ( 2  1) (17 13) (21 18) ( 9 19 12)
(12  7) (21  9) ( 6  4) ( 8 16) (15  5) (17 11) ( 3  2) (18 14) ( 1 19) (10 20 13)
(13  8) ( 1 10) ( 7  5) ( 9 17) (16  6) (18 12) ( 4  3) (19 15) ( 2 20) (11 21 14)
(14  9) ( 2 11) ( 8  6) (10 18) (17  7) (19 13) ( 5  4) (20 16) ( 3 21) (12  1 15)
(15 10) ( 3 12) ( 9  7) (11 19) (18  8) (20 14) ( 6  5) (21 17) ( 4  1) (13  2 16)
(16 11) ( 4 13) (10  8) (12 20) (19  9) (21 15) ( 7  6) ( 1 18) ( 5  2) (14  3 17)


(16 17) (13  4) ( 9 11) (12 19) (20 10) (18  2) (15  7) ( 5  1) ( 3 14  8  6 21)
(17 18) (14  5) (10 12) (13 20) (21 11) (19  3) (16  8) ( 6  2) ( 4 15  9  7  1)
(18 19) (15  6) (11 13) (14 21) ( 1 12) (20  4) (17  9) ( 7  3) ( 5 16 10  8  2)
(19 20) (16  7) (12 14) (15  1) ( 2 13) (21  5) (18 10) ( 8  4) ( 6 17 11  9  3)
(20 21) (17  8) (13 15) (16  2) ( 3 14) ( 1  6) (19 11) ( 9  5) ( 7 18 12 10  4)
(21  1) (18  9) (14 16) (17  3) ( 4 15) ( 2  7) (20 12) (10  6) ( 8 19 13 11  5)
( 1  2) (19 10) (15 17) (18  4) ( 5 16) ( 3  8) (21 13) (11  7) ( 9 20 14 12  6)
( 2  3) (20 11) (16 18) (19  5) ( 6 17) ( 4  9) ( 1 14) (12  8) (10 21 15 13  7)
( 3  4) (21 12) (17 19) (20  6) ( 7 18) ( 5 10) ( 2 15) (13  9) (11  1 16 14  8)
( 4  5) ( 1 13) (18 20) (21  7) ( 8 19) ( 6 11) ( 3 16) (14 10) (12  2 17 15  9)
( 5  6) ( 2 14) (19 21) ( 1  8) ( 9 20) ( 7 12) ( 4 17) (15 11) (13  3 18 16 10)
( 6  7) ( 3 15) (20  1) ( 2  9) (10 21) ( 8 13) ( 5 18) (16 12) (14  4 19 17 11)
( 7  8) ( 4 16) (21  2) ( 3 10) (11  1) ( 9 14) ( 6 19) (17 13) (15  5 20 18 12)
( 8  9) ( 5 17) ( 1  3) ( 4 11) (12  2) (10 15) ( 7 20) (18 14) (16  6 21 19 13)
( 9 10) ( 6 18) ( 2  4) ( 5 12) (13  3) (11 16) ( 8 21) (19 15) (17  7  1 20 14)
(10 11) ( 7 19) ( 3  5) ( 6 13) (14  4) (12 17) ( 9  1) (20 16) (18  8  2 21 15)
(11 12) ( 8 20) ( 4  6) ( 7 14) (15  5) (13 18) (10  2) (21 17) (19  9  3  1 16)
(12 13) ( 9 21) ( 5  7) ( 8 15) (16  6) (14 19) (11  3) ( 1 18) (20 10  4  2 17)
(13 14) (10  1) ( 6  8) ( 9 16) (17  7) (15 20) (12  4) ( 2 19) (21 11  5  3 18)
(14 15) (11  2) ( 7  9) (10 17) (18  8) (16 21) (13  5) ( 3 20) ( 1 12  6  4 19)
(15 16) (12  3) ( 8 10) (11 18) (19  9) (17  1) (14  6) ( 4 21) ( 2 13  7  5 20)