Create a head-to-head schedule for 7 teams over 10 weeks?

September 21, 2010 3:13 PM

How do I go about creating a schedule for head-to-head matchups for a league with seven teams over ten weeks?

I inherited "commissioner" duty for this sports league that previously had 8 teams. One team dropped out last minute. We played the first week of games last week (giving one team a bye).

Been busy at work and figured I'd take care of it next week and just get the first week of scheduled games out of the way.

It is now time to take care of it and I still don't have the time to deal with this. Someone else started dealing with it but said he was running into problems with some teams having 9 games while others had 10, one team playing another 3 times, etc.

Is it possible for seven teams to play the same number of games over ten weeks in a relatively balanced manner (such that we can have a playoff for the top 4 teams that is relatively fair).

We have the field for four hours and each game is an hour, so if necessary we could have an extra game where one (or more) teams ended up playing twice in the same week. But this is not ideal. Also we could condence the playoff to a single week and thus add an 11th week to the season if that would help?

Anyway, I haven't given this any thought, but it seemed like either a straightforward/easy problem or an interesting problem. Haa I doubt this ends up being the case and I would not be surprised to see this question deleted. I searched google and askme but have no idea what term to use to describe this type of problem...

posted by lulz to sports, hobbies, & recreation (8 answers total)

It's hard to tell exactly what kind of format you're seeking out. A round-robin with byes, which will give you seven weeks and then time for a preferred playoff spot (perhaps double-elimination or similar)? A double round-robin and playoff isn't going to fit into 10 or 11 weeks easily, even with one team playing twice per week.

I'm not a fan of unbalanced schedules, so I'll leave others to come up with options. Perhaps just turn the spare hour into a pickup game?
posted by holgate at 3:33 PM on September 21, 2010

Each week you will have three games and one team with a bye. So in a ten week season there will be ten teams with byes, which unfortunately means 3 teams will have two bye weeks (as your friend discovered). The only ways to fix this are a 7-week season, a 14-week season, or the addition of an 8th team to eliminate the need for bye weeks.
posted by bgrebs at 3:48 PM on September 21, 2010

Seven week season. Each team gets one bye week. Three week playoff for all teams, seeded by season record. (Top team gets bye into second round.)

Total time = 10 weeks. Sounds like you were scheduled for 12, including playoffs? Make week 11 into a "pro bowl" and week 12 is a party!
posted by m@f at 4:16 PM on September 21, 2010

You're going to have the shorten the season. The math also works for making it longer, but you shouldn't count on people being available for a longer time frame. Unless you know you can count on that, then make it longer. A 7 week schedule also means you could have time to expand the playoffs to more teams or give teams a longer break between playoff games. I'm in favor of the former.

If you keep the 10 week season then you'll end up with teams having more byes, and conversely other teams having more games. On the other hand, you could take the 7 week schedule and expand it to the 10 weeks. This means teams will have more than 1 bye week but I don't see a problem with that.

As a side note, that 8th team dropping out might have been a blessing in disguise. Having 4 hour long games scheduled during the 4 hour period you have to field could have lead to some timing disasters.
posted by theichibun at 4:28 PM on September 21, 2010

Each week you will have three games and one team with a bye. So in a ten week season there will be ten teams with byes, which unfortunately means 3 teams will have two bye weeks (as your friend discovered). The only ways to fix this are a 7-week season, a 14-week season, or the addition of an 8th team to eliminate the need for bye weeks.

There's another way to fix this: some weeks, more than one team gets a bye. (This was in fact the solution used by the NFL from 1999-2001, when there were an odd number of teams in the league, so there had to be an odd number of teams receiving byes each week; every week, at least one team got a bye, but some weeks three teams got byes so each team had one bye during the season.)

In your case, there have to be at least ten byes (one each week), but with seven teams, for each team to have the same number of games, the total number of byes has to be a multiple of seven, so that makes 14 byes. Each team plays eight games and has two bye weeks; given that each week must have an odd number of teams with a bye, one way to do this would be to have eight weeks with three games (and one team receiving a bye) and two weeks with two games (three teams receive a bye).

I don't see that expanding to 11 weeks helps: then you have an odd number of weeks, and and odd number of teams having byes each week, which sum to an odd number of total byes. So assuming you want each team to play the same number of games, you have to go to 21 total byes (3 per team) and each team still plays 8 regular-season games.

Here's a possible 10-week schedule (bye teams in brackets):
1. G-A C-D E-F [B]
2. A-B E-C F-G [D]
3. A-F B-G D-E [C]
4. E-A B-C F-D [G]
5. C-A G-E [B, D, F]
6. F-B D-G [A, C, E]
7. B-A D-C F-E [G]
8. A-D B-E G-C [F]
9. C-B E-D G-F [A]
10. A-G D-B C-F [E]

This schedule has the following nice properties:
* Each team plays eight games (two of the other six twice, and the other four once each)
* For the pairs of teams that play each other twice (AB, BC, CD, DE, EF, FG, and GA) , the meetings between the two teams are at least three weeks apart
* Each team has two byes, at least three weeks apart
* I don't know if it makes sense to think of some "home field advantage" (even if it's say, just calling a coin toss) in your league, but if there is, take the second team listed in each pairing as the home team, and each team has four home games; furthermore, for the pairs of teams that play each other twice, each team is the home team once in their two meetings.
posted by DevilsAdvocate at 6:35 PM on September 21, 2010 [1 favorite]

wow, thank you... you're not in the league, are you?
posted by lulz at 6:41 PM on September 21, 2010

Nope, I just love problems like this.
posted by DevilsAdvocate at 6:43 PM on September 21, 2010

One aspect I didn't consider in my previous post was the order the games were played within each week—if you played the games each week in the order I've listed them above, team A would play the first game every week it played.

I've switched around the order the teams play within each week (without making any other changes) to even that out somewhat, if that's a concern:

1. G-A C-D E-F [B]
2. E-C F-G A-B [D]
3. D-E A-F B-G [C]
4. F-D B-C E-A [G]
5. G-E C-A [B,D,F]
6. F-B D-G [A,C,E]
7. F-E B-A D-C [G]
8. G-C B-E A-D [F]
9. C-B E-D G-F [A]
10. A-G C-F D-B [E]
posted by DevilsAdvocate at 8:22 PM on September 21, 2010