difficult situation to solve

[pitchman]

 :-/
Need help figuring out a schedule for a pitch league I run.  There can be any number of players from 12 to 48.  The league runs for 10 weeks and I want it so that every player plays the least amount of time on another player's table.  So if there are 28 players (there are 28 players this time) and there are 4 players per table each week for 10 week then no player should be on the same table with another player for the first 9 weeks and only the 10th week will a player have played on the same table.

I'm in desperate need to figure this out and have a program that can generate this depending on the number of players each session.

Thanks for all the help anyone can give.

[Ian Wakeling]

There are some schedules on the message board that may help.  For example:

28 players  This is 9 rounds, but as it is balanced, you can use any assignment to tables for round 10.

follow all the links here for other numbers

[pitchman]

This is exactly what I've ben looking for and asking everyone to solve.  Thank you so very much.  

[christiegs]

I am also in a Pitch league, and I have also been trying to find a balanced schedule.  We typically have either 28 or 32 players per 13 week session.  Our goal is to have each player play with a different partner each week, and against an evenly random distribution of the other players - i.e. not playing against the same player 5 or 6 times, while never being at the table with a some of the other players...

So - I guess I am looking for two schedules - - one for 28 players over 13 weeks, and a second for 32 players over 13 weeks...

Thank you so much in advance for your help...

Christie Getches

[Ian Wakeling]

Hi Christie,

I think the following schedule will work well for 28 players.  Each player should have 13 different partners and 26 different opponents (all but one of the 27 possible opponents exactly once).  Clearly there must be some overlap, so each player will have 12 or 13 other players with whom they play both as partners and opponents.

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

32 players and 13 weeks is a much harder problem to solve. Is there any flexibility in the number of weeks, as I think I could find a better schedule for 15 weeks rather than 13 weeks?

Ian.

[christiegs]

Hi Ian,

Thank you so much for your help.  I am thinking that if you are able to generate a schedule for 15 weeks, and if in that 15 week schedule players don't partner twice with the same player, and end up with a random distribution of playing against the remaining players, then, if we simply play 12 weeks of the 15 week schedule, we'd still be good, no?  Am I missing something?

I have been playing around with this for a while, and I am ever so grateful for your help...

Christie Getches

[Ian Wakeling]

Hi Christie,

Here is the equivalent 32 player schedule in 15 rounds, it has properties that are similar to the 28 player version above.

  (30 21 v 22 26) (10 19 v 25 17) ( 1  3 v 24 11) ( 6 28 v 16 31) ( 2 32 v  8 20) (29  9 v  5 12) (14 15 v 13 27) ( 7  4 v 18 23)
  (16 22 v 23 27) (11 20 v 26 18) ( 2  4 v 25 12) ( 7 29 v 17 31) ( 3 32 v  9 21) (30 10 v  6 13) (15  1 v 14 28) ( 8  5 v 19 24)
  (17 23 v 24 28) (12 21 v 27 19) ( 3  5 v 26 13) ( 8 30 v 18 31) ( 4 32 v 10 22) (16 11 v  7 14) ( 1  2 v 15 29) ( 9  6 v 20 25)
  (18 24 v 25 29) (13 22 v 28 20) ( 4  6 v 27 14) ( 9 16 v 19 31) ( 5 32 v 11 23) (17 12 v  8 15) ( 2  3 v  1 30) (10  7 v 21 26)
  (19 25 v 26 30) (14 23 v 29 21) ( 5  7 v 28 15) (10 17 v 20 31) ( 6 32 v 12 24) (18 13 v  9  1) ( 3  4 v  2 16) (11  8 v 22 27)
  (20 26 v 27 16) (15 24 v 30 22) ( 6  8 v 29  1) (11 18 v 21 31) ( 7 32 v 13 25) (19 14 v 10  2) ( 4  5 v  3 17) (12  9 v 23 28)
  (21 27 v 28 17) ( 1 25 v 16 23) ( 7  9 v 30  2) (12 19 v 22 31) ( 8 32 v 14 26) (20 15 v 11  3) ( 5  6 v  4 18) (13 10 v 24 29)
  (22 28 v 29 18) ( 2 26 v 17 24) ( 8 10 v 16  3) (13 20 v 23 31) ( 9 32 v 15 27) (21  1 v 12  4) ( 6  7 v  5 19) (14 11 v 25 30)
  (23 29 v 30 19) ( 3 27 v 18 25) ( 9 11 v 17  4) (14 21 v 24 31) (10 32 v  1 28) (22  2 v 13  5) ( 7  8 v  6 20) (15 12 v 26 16)
  (24 30 v 16 20) ( 4 28 v 19 26) (10 12 v 18  5) (15 22 v 25 31) (11 32 v  2 29) (23  3 v 14  6) ( 8  9 v  7 21) ( 1 13 v 27 17)
  (25 16 v 17 21) ( 5 29 v 20 27) (11 13 v 19  6) ( 1 23 v 26 31) (12 32 v  3 30) (24  4 v 15  7) ( 9 10 v  8 22) ( 2 14 v 28 18)
  (26 17 v 18 22) ( 6 30 v 21 28) (12 14 v 20  7) ( 2 24 v 27 31) (13 32 v  4 16) (25  5 v  1  8) (10 11 v  9 23) ( 3 15 v 29 19)
  (27 18 v 19 23) ( 7 16 v 22 29) (13 15 v 21  8) ( 3 25 v 28 31) (14 32 v  5 17) (26  6 v  2  9) (11 12 v 10 24) ( 4  1 v 30 20)
  (28 19 v 20 24) ( 8 17 v 23 30) (14  1 v 22  9) ( 4 26 v 29 31) (15 32 v  6 18) (27  7 v  3 10) (12 13 v 11 25) ( 5  2 v 16 21)
  (29 20 v 21 25) ( 9 18 v 24 16) (15  2 v 23 10) ( 5 27 v 30 31) ( 1 32 v  7 19) (28  8 v  4 11) (13 14 v 12 26) ( 6  3 v 17 22)

What you say is essentially correct, things will still be good if you start removing rounds from the 15 round schedule.  However the program I used to find the schedule also tries to balance the total number of times that players play together in the same match, regardless of whether they are partners or opponents.  This social balance will be slightly upset by removing some rounds above, so it might be possible to do a little better.

Ian.

[christiegs]

Thanks, Ian.  This is fabulous!!!

[christiegs]

Hi Ian,

I gave your 28 and 32 player rotations to the card league organizers last night and they were THRILLED!!!  Thanks so much again.  Their one comment was that, occasionally, there are 36 players.  Is there any chance of your creating one more schedule with 36 players?  I am told that we never go more than 12 weeks - I thought it was 13 - but any length of 12 weeks or more would be great, as we can just not use all of the weeks if need be.

Thank you, thank you, THANK YOU for what you have already done, and hopefully, if possible for the 36 player schedule...

Christie Getches

[pitchman]

Hi Ian,

So I want to say you're a genius setting up these schedules.  You say you have a program that does it.  Is there any way I can get the program so when I change the number of players it will generate a schedule for me as you did before.

Thanks again for all your great help.

[Ian Wakeling]

The program has no interface, and I am changing the code each time I want to run it,  so it's very much a prototype algorithm that I might build into a schedule generator at a later stage.  As it stands there will be many situations where the algorithm will fail to find a good schedule and I think the 36 player/12 week schedule is a good example,  a 14 week schedule that could be cut back to 12 weeks might be possible, but I think this will not be as good as taking the 11 week schedule that can be found here and then add on to it a 12th week.  Each match there is presented as (a b c d), but assume that this means (a b vs c d), then every player will have 11 partners and 22 opponents all of whom are different from each other.  Then construct the 12th round in such a way that no partnership from the first 11 rounds is repeated, that should be relatively easy, but if possible also try to cover some of the player pairs that have not yet occured either as partners or opponents.  There are exactly two of these pairs for each player and because of the way the schedule is constructed the first match for the 12th round is easy - this is (1 34 vs 35 36).  It will be a little tedious, but if you now take player 2 and work out who they have not played with, you should be able to make assignments for the second match and so on.

[christiegs]

Ian,

Thanks so much - we will give it a go to create the schedule for week 12.  Many, many, many thanks for all of your help.

Christie