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Someone wants to organize a badminton tournament, where each match is a 2 versus 2, i.e. by teams. The idea is to have teams rotate, so that you can play with everyone.

If there are $n$ players, where $n$ is a multiple of $4$, is it possible to organize $n-1$ rounds, where each round is composed of $n/4$ simultaneous games, such that for every pair of players, they are teammates exactly once and opponents exactly twice ?

For 4 players it's obvious, for 8 players I already needed help from a SAT Solver, which found a solution, and for 12 players the SAT solver also found a solution after 11 hours and 20 minutes.

I could not find a nice combinatorial insight into this problem. It's even nicer that it's from real life. By the way this is a true story, the tournament will really happen, likely with 20 people.

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I finally found the keyword allowing to search for solutions, it's called a "Whist Tournament".

Solutions can be found here for instance: https://www.devenezia.com/downloads/round-robin/rounds.php?it=12&v=w1

However it's not so easy to build them, constructions are described e.g. in chapter 6 of the Master Thesis "Scheduling of tournaments with several opponents in one game" by Csaba Filip.

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  • $\begingroup$ What an edit! (from "clear" to "not clear") $\endgroup$
    – mathworker21
    Apr 17 at 18:51
  • $\begingroup$ Hehe yes I first thought there was a clear pattern, that was mistakenly confirmed by chatGPT, but looking more closely it's wrong. $\endgroup$
    – Denis
    Apr 17 at 18:54

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