Three Teams at once

[KittyFAISE]

I've been looking around the internet for algorithms for scheduling a round robin tournament, and the issue is that everything I've found runs on the assumption that two teams are playing each other at once.

I need to be able to do a schedule for Three teams going at it at once.

To be more specific, I want to be able to schedule a laser tag tournament, on the system I'm working on you can have three different teams in one game, and when we do tournaments, we do that, is there an algorithm that accounts for three teams going at it at once?

[Ian Wakeling]

A good place to start might be here, that is if the number of teams is of the form 6n+3, n>=1 and you are wanting to play the tournament in rounds where each team plays once per round.  The key feature of these schedules is that all pairs of teams play against each other once.  If you have 6n teams on the other hand then there is no such nice solution and you will need to play each other twice before you can get a balanced schedule.  But perhaps you do not have a multiple of 3 teams?  Then what you looking for is a "block design", so try Googling for that term as well as "search" & "algorithm".

[KittyFAISE]

Thank you, do you mind elaborating on the block design thing? Not being completely sure what I'm looking sure I'm not sure which hits I should be going for.

[Ian Wakeling]

Lets say that you have 7 teams, then you could play 7 rounds as follows:

1 2 4
2 3 5
3 4 6
4 5 7
5 6 1
6 7 2
7 1 3

Then all 21 pairs of teams would have played together in a 3 way match exactly once.  It's very hard to write an algorithm that will construct balanced schedules such as this, specially when the number of teams gets larger - for this you really need a combinatorial approach or a library of designs taken from a book.  However algorithms for finding optimal or partially balanced incomplete block designs (PBIBD) do exist, so those are other search terms you could use.