Social rectangle => squad rotation

[Nilx]

Hi and thanks for a great site.

We are merging two teams of a total of 20 players into one team. For twelve weeks we play two games a week. Our goal, in order to create a unified team spirit, is that we would like every player to play an equivalent number of games with every other player. In addition no player should play more than one game per week. By brute force I have managed to find a system that gives each player at least four games with every other, however two players still play as much as 11 games together.

Ideally every player should play together with another player five or six times. I have read through many posts and tried to adjust many of the round-robin systems here, but still I have not been able to find the "optimal" set up. Have I missed something obvious? or is there only a complex answer?  In short: two games a week (1 and 2 in the table below), 20 players to rotate internally from week to week and a total of 24 games in 12 weeks.

I would appreciate any help on this, (also if they just tell me that the answer is very obvious and points me in the right direction

Thanks for your help,
Nils

[Ian Wakeling]

I can find a solution to the original problem where all but 6 of the possible pairings of the 20 players occur together between 4 to 7 times.  However I think there are nicer solutions if we change the requirements slightly.  If we arrange it so that 10 of the 190 pairings never occur on the same team, then it's possible to have all 180 other pairings exactly 6 times each.

In the schedule below the pairs 1-2, 3-4, 5-6, ... ,19-20 never occur on the same team.

Is that a workable solution?

 w1 m1 (1 4 6 8  9 12 14 15 18 20)
 w1 m2 (2 3 5 7 10 11 13 16 17 19)
 w2 m1 (1 4 5 8  9 11 13 15 17 19)
 w2 m2 (2 3 6 7 10 12 14 16 18 20)
 w3 m1 (1 4 6 7 10 12 13 15 18 19)
 w3 m2 (2 3 5 8  9 11 14 16 17 20)
 w4 m1 (2 3 6 8 10 11 13 15 18 19)
 w4 m2 (1 4 5 7  9 12 14 16 17 20)
 w5 m1 (2 4 5 7 10 11 14 15 18 20)
 w5 m2 (1 3 6 8  9 12 13 16 17 19)
 w6 m1 (2 4 5 8 10 12 14 15 17 19)
 w6 m2 (1 3 6 7  9 11 13 16 18 20)
 w7 m1 (2 3 5 8  9 12 13 15 18 20)
 w7 m2 (1 4 6 7 10 11 14 16 17 19)
 w8 m1 (1 3 5 7 10 12 13 15 17 20)
 w8 m2 (2 4 6 8  9 11 14 16 18 19)
 w9 m1 (2 4 5 7  9 12 13 16 18 19)
 w9 m2 (1 3 6 8 10 11 14 15 17 20)
w10 m1 (2 4 6 8 10 12 13 16 17 20)
w10 m2 (1 3 5 7  9 11 14 15 18 19)
w11 m1 (2 3 6 7  9 12 14 15 17 19)
w11 m2 (1 4 5 8 10 11 13 16 18 20)
w12 m1 (1 3 5 8 10 12 14 16 18 19)
w12 m2 (2 4 6 7  9 11 13 15 17 20)

[Nilx]

Hi and thanks for your help!

This is really neat, and I will probably be able to use it in other circumstances, however I am afraid that we do need to keep everybody paired up at least some times. It is preferable to have them play each other between 4 and 7 times, and it is better to have outliers playing more often together than below average. (Come to think of it, I should probably have specified this in the first post). So, I can't use this exact solution for this particular situation, which is a shame since it really is beautiful in its symmetry. Therefore I would again be thankful if you could help me with other solutions.

Regards
Nils

[Ian Wakeling]

Nils,

Here is the alternative schedule that I mentioned.

 w1 m1 (17 12 15 11 7 9 14 1 19 5)
 w1 m2 (4 8 20 2 18 3 16 10 13 6)
 w2 m1 (20 16 19 14 6 11 3 15 4 5)
 w2 m2 (18 9 10 1 2 8 13 7 17 12)
 w3 m1 (5 2 18 4 17 6 15 1 8 11)
 w3 m2 (12 13 20 3 16 19 7 14 10 9)
 w4 m1 (16 6 12 18 11 5 10 13 9 15)
 w4 m2 (14 19 8 17 1 2 3 20 7 4)
 w5 m1 (10 4 11 12 19 7 6 2 14 18)
 w5 m2 (9 1 5 16 13 17 8 20 3 15)
 w6 m1 (6 20 14 13 12 5 9 4 1 2)
 w6 m2 (7 17 10 15 8 16 19 11 18 3)
 w7 m1 (19 5 4 8 15 20 13 12 18 7)
 w7 m2 (3 11 6 9 14 10 2 17 16 1)
 w8 m1 (8 14 13 2 15 17 16 6 12 19)
 w8 m2 (1 18 9 5 4 7 20 3 11 10)
 w9 m1 (15 2 3 7 9 12 11 8 6 20)
 w9 m2 (19 10 16 17 4 14 18 5 1 13)
w10 m1 (8 3 13 11 10 15 1 4 12 14)
w10 m2 (9 7 17 6 20 18 5 19 2 16)
w11 m1 (2 15 7 10 5 12 4 16 3 17)
w11 m2 (14 6 1 19 9 13 11 18 20 8)
w12 m1 (11 10 2 20 8 1 12 19 5 16)
w12 m2 (13 14 7 6 3 4 17 9 15 18)

3 pairs occur 3 times, 3 pairs occur 8 times each.  The others as I said before occur between 4 and 7 times.

Regards,

Ian.

[Nilx]

I have now implemented it, and this is the one we will use, it really suits our need.

And again; Thanks a lot!
regards Nils