I’m an elementary grade teacher and I want to organize a speed meeting event at the start of the school year to introduce the new staff. Specifically, I want to organize 3-minute rounds during which trios of teachers would chat. Trios would then be split and reorganized for another round and so on. The idea is to organize the trios and the round so that any pair of teachers cannot be in the same trio more than once. In other words, if Arlene gets matched with Bob and Carlito in round 1, she could no longer be match with any of those two. She could still be matched with Denise, Farley or any other teacher.
In order for such a pairing system to work, there must be an odd number of attendees that is divisible by 3 (so 3, 9 or 15 people should do). Otherwise, there would either be some tables with less than 3 persons or some people left in pairs for the last round. Through trial and error, I was able to find two working configuration of pairings for a 9-persons meeting:
1st round: ABC DEF GHI
2nd round: ADG BEI CFH
3rd round: AEH BFG CDI
4th round: AFI BDH CEG
Things get way more complicated with 15 or 21 attendees (the latter being the target I have in mind for the event to be held in a couple of weeks). My wife and I tried to code something in R to find a working configuration, but we couldn’t beat this puzzle. Is anybody able to help me on that one? Can anybody point me to an algorithm (or to a solution!)?