Sounds like a graph problem!
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Sounds like a 130 or 135 problemâŚcanât remember which was whichâŚnot numerical analysis but the other oneâŚI think it started with an E.
Edit: ah, yes, Ăperations Research, or Operations Research for those who use the modern spelling
Hamilton Circuits for the win! Guaranteed in a complete digraph (tournament) but they donât all play.
More like cyclic graphs.
If you have two disjoint cycles, then youâd have the âimpossibleâ situation twig is wondering about.
But I think the way that team schedules are set up, I think the âparityâ graph is a possibility each year.
If I understand what weâre talking about then this isnât true. I thought weâre talking about A defeats B, B defeats C, C defeats D, D defeats A. (Except with 32 teams instead of 4.)
Itâs trivial to show that it canât be done in 2017, for example. The Browns had no wins at all.
But even if every team has both a win and a loss it could still be impossible. If Team A and Team B both defeat Team C and have no other wins then it wouldnât be possible. And you can get into more complex cases where it wouldnât be possible also.
Unless you mean at the beginning of the season it would be possible for a parity graph to come into existence⌠that might be true.
But at the end of the season based on actual game results then it would not be.
The answer is obvious
Buffalo game reportedly moved to Detroit. No idea why the time on this release is from the future
ETA: they changed the time. It was 5 hours later. Maybe one of the employees sent it from Europe.
BOOOO!!!
Great Buffalo Blizzard of 2014 all over again for the Bills. Bills have a perfect record though when playing home games in Detroit.
More like time-stamped with GMT rather than a âlocalâ time.
Why not start with the beginning? But yes, I was talking about at the beginning of the season
But the answer to the âbeginning of the seasonââwhich would include paths to demonstrate parityâwill make working on this part far, far more tractable.
Ok, but the parity appears to be achieved after there are wins and losses.
The possibility of parity is what you are discussing, not the achievement of it.
Yes. First requirement is that every team has won a game and has lost a game.
Second requirement: at least one team from each conference has beaten a team from the other conference.
And I think there are more.
So, what are all (and, by extension, for logical purity, the least number) of the requirements to complete the circle?
Well youâd have to spell out a requirement that would mean this hadnât happened:
Itâs easier for me at the moment to come up with examples where it doesnât work than requirements that would ensure it does, however.
If you were trying to put the ring together youâd want to start with the teams with only 1 win or only 1 loss as those will have only one possibility for the team they defeated / team that defeated them.
Youâd want to list those and make sure that there are no duplicates.
But if A and B and C each have two wins⌠against D and E then again⌠not possible.
Or A only beat D, B only beat E, C only beat E and FâŚ
You can get into a closed loop pretty fast. Iâm not sure how to write a requirement preventing this.
I think a look at intraconference divisions
These are useful, to be sure; but they require knowledge of a complete season and donât help in looking at an active season (or looking at the question of âwhatâs the minimum number of weeks to demonstrate parity?â). But I agree that if one is looking at a completed season to see if there was parity; the starting point should be the teams with the fewest wins and the fewest losses.
But I would take a more generalized approach and look for the (theoretical) bottlenecks of the possibilities where parity can be demonstrated as a season progresses.
For example, DTNF hits one with the interconference games: if all of the xFC teams beats the yFC teams, parity cannot be achieved. So you need at least one âwinâ for each side for the given games. Furthermore, at least two of these games are required for parity to be achieved; so I would call this the first bottleneck.
I believe that the intraconference divisions wins is another bottleneck to addressâand it would be an extension of DTNFâs; but weâd be working with four such groups (rather than two); and the cycle can leverage the two (or more) interconference games to get a cyclic graph developed.
Under current scheduling, teams within a division are scheduled to play the teams of the other divisions that based on their standings within the division of the prior year. I think this provides the necessary conditions to establish parity. And intra-divisional games used to help facilitate completing a chain.
So as a season progresses, I think looking at inter-conference games to find âanchor pointsâ to start the parity graph would be the starting point. I wonder if itâs possible that a parity graph could be constructed by just looking at these sorts of games . . .
Once you meet that requirement, then start adding in inter-divisional games to see if the graph can be extended (and completed). And if not, take a quick peek at intra-divisional games to see if that helps
Or even a partially completed season to see if it has occurred yet or as of Week X.
I donât think you necessarily need that on a division level.
If every NFC East team beats every NFC South team, parity can still be achieved.
Agreed; but youâd need a path across the four groups (rather than the much simpler two). So focus on the four already made paths across them (and âsupplementâ an intra-divisional game to jump between these paths and the interconference games needed to complete parity).