Thank you, that is an interesting math aspect. That schedule would work well in some cases. However, what I was hoping to do was to include triball as part of a day of many events, with triball as a sampling of events during a main portion, and was hoping to have 3 teams playing triball 3 of the rounds while a lot of other events are going on, so that 5 teams would get to play it once, and 2 teams would play it twice (i.e., using up the 9 team spots for 3 rounds of triball). I was using letters for teams, and in respective rounds when triball was going on, the teams were ABC, DEF, GAD in my initial schedule (i.e., A and D play twice, everyone else once). I was then planning on doing a triball after-event, and if all teams wanted in, I would take the 3 rounds that had already been done, and do four more rounds, so that each had played triball three times in total, and score the 7 total rounds. However, when I look at the Fano plane solution, I can't seem to pick three initial rounds such that 5 play once and 2 play twice. To get everyone to play at least once, I seem to have to pick a combo of 3 rows such that 1 team is in all 3 rows.
I'm wondering if I want to have the 7 round property that each team plays 3 times, against different opponents, if I have to accept that in the first 3 of those rounds, to get everyone to play at least once in those 3 rounds, that 1 team will have played all 3 rounds. If so, I'll have to think about it.
