How many rounds in a round-robin tournament consisting of 4v4 matchups?

Question

Hello, I've been fascinated with tournament systems and am a bit stuck on this seemingly simple question. For a round-robin tournament consisting of 1v1 matchups of *n* contestants, you have *(n / 2) \* (n - 1)* rounds. E.g. a round-robin tournament of 16 would need 8 \* 15 = 120 total matches to complete.  


I'm stuck on how the number of total matches changes if you increase a match size to *m* contestants per match. My application of a Mario kart tournament usually involves matches of 4, but I'm curious if there's a model for an arbitrary size *m* where *m <= n*.

The criteria for a round-robin tournament of match-size *m* would require each contestant to play every other contestant *at least once*. For example, a tournament consisting of a match-size of 4 with 8 total contestants, *a, b, c, d, e ,f, g, h,* would consist of the following matchups.

|Match 1|abcd|
|:-|:-|
|Match 2|efgh|
|Match 3|aebf|
|Match 4|cgdh|
|Match 5|agbh|
|Match 6|cedf|

Which yields 6 total matches, in this case. I've done some googling and haven't found a good answer for a general equation. Would anyone happen to know, or be able to point me to the right resource? Thank you! (giving this a discrete math flair because I don't see a combinatorics flair, apologies if that's incorrect!)

Answer 1

What I've attempted to solve this problem by trying to find a pattern in how many matches are needed for each number of contestants, on a small scale. For *m = 4*, I've tried *n = 4* through *n = 8,* but I haven't really been able to find a solution.

Answer 2

8!/(4!4!)=70 ways to split 8 players into two teams of 4. If there is no home-field advantage or other distinction between Team A and Team B, half that many matches is enough.

You are going to need 8 fairly dedicated players to get through all 35 lineups - but it's not entirely impossible.

On rereading, it appears you may have some weaker requirement than having every possible foursome play. If so, the answer is going to be quite sensitive to exactly what that requirement is. You may, for instance, want to require that you have every player as teammate 3 times and opponent 4 times, even if you don't play every possible lineup. (If one player is much weaker or much stronger than the others, playing 6 rounds rather than 7 would be a very unfair format to someone.)

Answer 3

There are n! 1v1s to do and each m player match provides m! 1v1s so you need at least n!/m! matches.