<p>I finally found the keyword allowing to search for solutions, it's called a &quot;Whist Tournament&quot;.</p>
<p>Solutions can be found here for instance: <a href="https://www.devenezia.com/downloads/round-robin/rounds.php?it=12&amp;v=w1" rel="noreferrer">https://www.devenezia.com/downloads/round-robin/rounds.php?it=12&amp;v=w1</a></p>
<p>However it's not so easy to build them, constructions are described e.g. in chapter 6 of the Master Thesis &quot;Scheduling of tournaments with several opponents in one game&quot; by Csaba Filip.</p>

CC BY-SA 4.0
2023-04-17T18:51:08.463Z
What an edit! (from "clear" to "not clear")
mathworker21
CC BY-SA 4.0
2023-04-17T18:54:46.067Z
Hehe yes I first thought there was a clear pattern, that was mistakenly confirmed by chatGPT, but looking more closely it's wrong.
Denis
CC BY-SA 4.0
2023-04-17T18:09:41.320Z
2023-04-17T18:55:45.530Z
2023-04-17T18:55:45.530Z
Denis
<p>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.</p>
<p>If there are <span class="math-container">$n$</span> players, where <span class="math-container">$n$</span> is a multiple of <span class="math-container">$4$</span>, is it possible to organize <span class="math-container">$n-1$</span> rounds, where each round is composed of <span class="math-container">$n/4$</span> simultaneous games, such that for every pair of players, they are teammates exactly once and opponents exactly twice ?</p>
<p>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.</p>
<p>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.</p>

<co.combinatorics>
Combinatorics of a badminton tournament
Denis
# Combinatorics of a badminton tournament

(Asked by: username_0 on 2023-04-16)

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.

**Tags:** co.combinatorics


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## 1 answers

### Answer by username_0 on 2023-04-17. Score: 12

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.

> **Comments:**
> **username_1**: What an edit! (from "clear" to "not clear")  
> **username_0**: Hehe yes I first thought there was a clear pattern, that was mistakenly confirmed by chatGPT, but looking more closely it's wrong.  

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