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.