soccer league - 16 players 4 players per team

[tonyg]

I coach soccer with a bunch of 12yo kids
I want to schedule a "dutch masters" soccer league /tournament..
players achieve cumulative points scores as an individual and an individual total giving them each indidual rankings (not team rankings) at the completion of the rounds
players play in teams of 4 in each round.
each individual player earns points per round (3 win, 1 draw, 0 loss) depending how his team went in that round.
players change teams after every round
each player should play in a team with every other player once
each player should play against each other player once
assume 16 players but would be great if I could get an algorithm for more or less players depending on how many turn up each week
sorry in advance if this question has already been asked ...
its too tough for me to work out - cheers

[Ian Wakeling]

I don't think there are any fair schedules for your scenario.  The way I read it your tournament is played on one day only, so the number of rounds will be sufficiently limited that no balanced schedule is possible.  If you think of the tournament from the perspective of one kid, then each round they play in, they will have 3 partners on their team, and 4 opponents on the opposite team, the fact that there are more opponents than partners, excludes the possibility of having each player once as both partner and opponent.  There are some schedules that will work but in general they will have as many rounds as there are players.  The only one I have at the moment is the one below for 17 players and 17 rounds with one bye per round.  Here players have all possible partners exactly 3 times, and all opponents exactly 4 times.

  (14 17  5  2 v 10 16  9  3) (13  6 15  4 v  7  8 11 12)   ( 1)
  (15  1  6  3 v 11 17 10  4) (14  7 16  5 v  8  9 12 13)   ( 2)
  (16  2  7  4 v 12  1 11  5) (15  8 17  6 v  9 10 13 14)   ( 3)
  (17  3  8  5 v 13  2 12  6) (16  9  1  7 v 10 11 14 15)   ( 4)
  ( 1  4  9  6 v 14  3 13  7) (17 10  2  8 v 11 12 15 16)   ( 5)
  ( 2  5 10  7 v 15  4 14  8) ( 1 11  3  9 v 12 13 16 17)   ( 6)
  ( 3  6 11  8 v 16  5 15  9) ( 2 12  4 10 v 13 14 17  1)   ( 7)
  ( 4  7 12  9 v 17  6 16 10) ( 3 13  5 11 v 14 15  1  2)   ( 8)
  ( 5  8 13 10 v  1  7 17 11) ( 4 14  6 12 v 15 16  2  3)   ( 9)
  ( 6  9 14 11 v  2  8  1 12) ( 5 15  7 13 v 16 17  3  4)   (10)
  ( 7 10 15 12 v  3  9  2 13) ( 6 16  8 14 v 17  1  4  5)   (11)
  ( 8 11 16 13 v  4 10  3 14) ( 7 17  9 15 v  1  2  5  6)   (12)
  ( 9 12 17 14 v  5 11  4 15) ( 8  1 10 16 v  2  3  6  7)   (13)
  (10 13  1 15 v  6 12  5 16) ( 9  2 11 17 v  3  4  7  8)   (14)
  (11 14  2 16 v  7 13  6 17) (10  3 12  1 v  4  5  8  9)   (15)
  (12 15  3 17 v  8 14  7  1) (11  4 13  2 v  5  6  9 10)   (16)
  (13 16  4  1 v  9 15  8  2) (12  5 14  3 v  6  7 10 11)   (17)

[tonyg]

thanks Ian

yes I was hoping to play this tournament in one day so probably no more than 5-6 rounds or games.  but also thought it could be like a league that continued over several days at each training session if we needed more rounds to make the maths work.
I guess I was also a bit hopeful that there would be an algorithm where a player could play with each player once and then also play against each player once but on reflection I would be happy with simply each player playing at least once in the same team as every other player.  I have seen it done before with another coach at an academy where he had 16 players and each one got to play with every other player in a team of 4 over about 5-6 rounds .. so I know its possible...   im just really stumped on how to work it out

cheers
Tony    

[Ian Wakeling]

There is no simple formula that will solve all these scheduling problems - it is more a case of building solutions for specific problems.

It's not a problem to balance the 16 players so that they play with each other on the same team exactly once - the issue is that you can't also balance the opposition.  For example this is about the best that can be done:

  (15 10 13  3 v 1  2 16 11) (12  8  6  5 v 14  7  9  4)
  ( 2  8 10 14 v 4  5 13 16) (11 12  7  3 v  1  9 15  6)
  (12 13  1 14 v 4  6 11 10) ( 2  5  3  9 v 16 15  7  8)
  (14 15  5 11 v 6 13  7  2) ( 8  1  3  4 v 10 16  9 12)
  (12  4  2 15 v 6 14  3 16) ( 5 10  7  1 v 11  8  9 13)

within teams the balance is perfect, however any player has 3 other players who they never oppose, and 2 other players who they oppose 3 times.  For example player 1 never opposes 2, 5 or 14, but opposes 10 & 11 three times.  So the kid who never opposes one of the best players and opposes a weaker player three times, has an unfair advantage.