Round robin schedule needed - 29 teams, 6 days, 10 venues

[TimSlack]

I want to create a round robin schedule for 29 croquet teams over 6 playing days, with 10 venues available. Each day 2 or 3 teams meet at a venue and play each other, and can only use that venue on that day. Also assuming each venue is home to 3 teams it would be nice if possible to balance Home and Away venues. The constraint on venues makes it complicated! Can anyone help?

[Ian Wakeling]

I think the problem is a lot easier if you think of it as 30 teams with exactly 3 teams per venue per day.  So here is a solution to the 30 team problem:

   V1         V2        V3         V4         V5         V6       V7        V8         V9        V10
(1 12 25) (21 4  7) (18  8 19) (20 26 10) (30 22 17) (13 3 11) (2 28 23) (6 24 14) (29 15  9) (16 27 5)
(2  7 26) (22 5  8) (13  9 20) (21 27 11) (25 23 18) (14 4 12) (3 29 24) (1 19 15) (30 16 10) (17 28 6)
(3  8 27) (23 6  9) (14 10 21) (22 28 12) (26 24 13) (15 5  7) (4 30 19) (2 20 16) (25 17 11) (18 29 1)
(4  9 28) (24 1 10) (15 11 22) (23 29  7) (27 19 14) (16 6  8) (5 25 20) (3 21 17) (26 18 12) (13 30 2)
(5 10 29) (19 2 11) (16 12 23) (24 30  8) (28 20 15) (17 1  9) (6 26 21) (4 22 18) (27 13  7) (14 25 3)
(6 11 30) (20 3 12) (17  7 24) (19 25  9) (29 21 16) (18 2 10) (1 27 22) (5 23 13) (28 14  8) (15 26 4)

Then team 30 can be assigned as the non-existent 'ghost' team.  So the 6 occasions where 30 is scheduled to play are the times when only two teams meet at a venue.  The 12 teams who play the ghost, will of course only have 11 opponents in total, while the other 17 teams will have 12 opponents.

The schedule is also arranged so that no team plays more than once at a venue, or equivalently that each team plays at six different venues.  If the teams are assigned in sequence to home venues, so 1,2 & 3 are home teams at venue 1, teams 4,5 & 6 at venue 2, etc., then every team will have one meet at their home venue, and 5 meets at away venues.

I hope that is useful.

[TimSlack]

Thanks so much, I don't know how you do it!
The organising team are now wanting to see whether we can organise it around 4 teams per venue, giving six games per venue per day, which is doable in the time we have available.  Can you please help with that.
Thanks in advance. 

[Ian Wakeling]

Do you still have 10 venues?  So 40 teams in total?

[TimSlack]

No, still 30 teams, with 10 venues available but not all used for a day.

[Ian Wakeling]

This will make things much more complicated and the method I used above will not provide a full solution.  The first thing to do, is to decide how to divide the teams between venues, as 8 out of 10 venues must be used each day, and either (option 1) 7 venues have 4 teams and one venue has 2 teams, or (option 2) 6 venues have 4 teams and 2 venues have 3 teams.  I think option 1 should be rejected, since 12 teams will have 2 fewer opponents than the rest and it will therefore be the more unfair than option 2, where it can be arranged that 6 teams have 1 fewer opponent than the rest.

So here is a solution similar to the one above, but for option 2.

(30 13 20 11) (3 14 26  8) (18 23 25  9) (12 17 5 19) (27 21 10 2) (1 24 15 29) (22 6 16) (4 28  7)
(25 14 21 12) (4 15 27  9) (13 24 26 10) ( 7 18 6 20) (28 22 11 3) (2 19 16 30) (23 1 17) (5 29  8)
(26 15 22  7) (5 16 28 10) (14 19 27 11) ( 8 13 1 21) (29 23 12 4) (3 20 17 25) (24 2 18) (6 30  9)
(27 16 23  8) (6 17 29 11) (15 20 28 12) ( 9 14 2 22) (30 24  7 5) (4 21 18 26) (19 3 13) (1 25 10)
(28 17 24  9) (1 18 30 12) (16 21 29  7) (10 15 3 23) (25 19  8 6) (5 22 13 27) (20 4 14) (2 26 11)
(29 18 19 10) (2 13 25  7) (17 22 30  8) (11 16 4 24) (26 20  9 1) (6 23 14 28) (21 5 15) (3 27 12)


The problem is that you will have to make your own assignments of the 8 groups for each day to the 10 venues, as my software can not handle this.  And once you have done this you can then decide on the home teams at each venue.  However the basic structure is in the plan above, teams 1 to 6 should have 16 different opponents, while teams 7 to 30 shouold have 17 different opponents.

[TimSlack]

Thanks so much, that is a great help. Now our work begins.