13
$\begingroup$

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.

$\endgroup$

1 Answer 1

12
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ What an edit! (from "clear" to "not clear") $\endgroup$ Apr 17, 2023 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, 2023 at 18:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.