Listing every pairing is not a fixture list
The obvious way to build a league is to write down every pair and share them out across the weeks. It gives you every fixture exactly once and it looks finished.
Then somebody notices they are down to play twice on the same Tuesday. You shuffle it. Now another team has three games in a fortnight and then nothing for a month. That is an afternoon gone, and the result still is not right.
The problem is not effort, it is that "every pair once" and "nobody twice in a round" are two different constraints, and satisfying the first tells you nothing about the second.
The circle method solves it in one pass
It is about a hundred years old and it is almost embarrassingly simple.
Fix one team in place. Stand everybody else in a circle around them. Pair off across the circle, first against last, second against second-last, and so on. Then rotate everyone except the fixed team by one position and do it again.
Every pair meets exactly once. Nobody plays twice in a round. It finishes in the minimum number of rounds. No searching, no backtracking, no repairs afterwards.
The minimum is not a claim, it is arithmetic
With six teams, each one needs five games and can only play one per round. So five rounds is the floor, and the circle method hits it.
With an odd number it goes up by one. Seven teams still means six games each, but somebody has to sit out every round, so you need seven rounds to hand out the seven rests. That extra round is not inefficiency, it is unavoidable.
The byes are the bit that gets fixtures rewritten
Add a ghost team to make an odd number even, and whoever draws the ghost has the week off.
Because the circle rotates evenly, each team draws the ghost exactly once. Every side gets one rest and no side gets two. That sounds obvious, and hand-made fixture lists get it wrong constantly, because it is invisible until week nine when somebody adds it up.
Home and away is not automatic
If you pair the circle the same way every round, the team that stays fixed plays every game at home. Or every game away, depending which side of the pairing it sits.
Nothing on the fixture list looks wrong. It shows up when somebody totals the mileage in March. So the sides alternate round by round.
For a real season, turn on home and away. It plays the whole schedule twice with the sides reversed, so every pair meets once at each ground and every team has exactly as many home games as away ones.
Check the numbers before you print it
The games-per-team list is there so you can see the schedule is even rather than take it on trust. Every team should show the same total, and with home and away on, the same split.
That is the check worth doing on any fixture list, including one somebody else made you.
Common questions
How many rounds does a round robin take?
One fewer than the number of teams, if that number is even. Six teams need five rounds. That is provably the fewest possible: every team needs five games and can only play one per round, so nothing can beat it. An odd number of teams takes as many rounds as there are teams, because somebody has to sit out each week.
What happens with an odd number of teams?
A ghost team is added to make the count even, and whoever is drawn against it has that round off. The useful part is that each team rests exactly once, which is the thing hand-written fixture lists get wrong. The commonest complaint in an amateur league is one side resting twice while another never does.
Why not just list every pairing and share them out?
Because it gives you every fixture exactly once, which looks finished, and then somebody spots a team playing twice in the same week. Sorting that out by hand takes an afternoon and usually still leaves one team playing three times in a fortnight and then nothing for a month.
What is the circle method?
A hundred-year-old trick. Fix one team in place, stand the rest in a circle, rotate everyone by one position each round, and pair off across the circle. Every pair meets exactly once, nobody plays twice in a round, and it finishes in the minimum number of rounds. No searching, no backtracking, no fixing it up afterwards.
Does it balance home and away?
It alternates which side is at home from round to round, which matters more than it sounds. Pair the circle the same way every time and the fixed team ends up at home every single week, or away every single week. It is invisible on the fixture list and painfully obvious once somebody adds up the travel. For a properly balanced season turn on home and away, which plays the whole thing twice with the sides swapped so every pair gets one of each.
How many games in total?
The number of pairs, which is n times n minus one, divided by two. Six teams is fifteen games. Ten teams is forty-five. Twenty teams is a hundred and ninety, which is where people usually realise they want two divisions.
Can I reproduce the draw?
Yes. The shuffle is seeded and the seed is shown, so the same seed always produces the same fixture list. If anybody asks why their team got the hardest opening three weeks, you can show them the draw was not arranged.