Me and 7 friends are playing badminton. We play doubles (2 vs 2) on 2 courts, so 2 matches takes place at the same time. Let's say each match lasts the same amount of time. We want to play 7 matches each, in a way that for each new match, everyone plays with a new person in their team. So in total there will be 14 matches played.
For the sake of this task I have designated the players as A, B, C, D, E, F, G and H. Before we start playing every player will randomly get one of these designations, with individualized court position sequence schedules.
We also want the matches to have a maximum amount of variation between who everyone plays against, so there should be a rotation of opponents as well.
So for example if A is playing against CD, CE, CF, CF and CH that would not be an example of maximum variation of opponents because then A plays against C many times. While A playing against for example BC, DE, FG, HB, GD, FC and EH would be a good variation because A only plays against each player maximum 2 times out of the 7 matches.
(Of course A's team mate for each match should be AB, AC, AD, AE, AF, AG and AH but the order doesn't matter).
MAIN QUESTION
So for seven matches that each player is in, that have these criteria fulfilled, i.e.
- Each player teams with each other player only once
- Each player has each other player in the opposing team maximum twice.
What would be a possible sequence of court position schedule?
I can add that this is an real world situation, and we're actually looking for to find a sequence of team combinations that we can use to have a schedule ready before we start playing. However, since this is Math exchange, I understand people might be looking for more than just an answer, like to find an algorithm or maybe even the most most effective algorithm, a general solution than can be mathematical proven, and so on. I don't have anything against all that, but I just want to clarify that what we're actually looking for right now is merely the court position schedule to bring to make it easy to setup games next time we play.
With that said, I thought about it, and had some other extension ideas as well, for those who might be interested.
Extra alternative 1: To make it even simpler for each match switch, would there be possible with a solution where each player only has to move maximum one step from their current position?
The courts are next to each other and I have made an illustration what I mean with "one step". Each arrow indicates one step. Note that some steps require other steps to be valid (in order to prevent players being in the same team more than once). So all criteria from the main question must still be fulfilled.
Extra alternative 2. Instead of having "maximum one step place switch" as a criteria, another alternative for maximum variation would also be if each player plays at as many of the 8 court positions as possible. What would a solution look like that on top of the criteria from the main question also varies the positions of each player?
Again, although it would be fun to see possible solutions to extra alternative 1 and 2, the main question is what we're mainly looking for.
Thanks!