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        riddles >> easy >> NFL playoff eligibility
        
(Message started by: Colbey on Dec 19^th, 2006, 6:31pm)

Title: NFL playoff eligibility
Post by Colbey on Dec 19^th, 2006, 6:31pm

Given the current National Football League division alignments, playoff eligibility rules, and scheduling guidelines, what is the worst possible record that a team can have and still:

0. Have the best record in the NFL
1. Have the best record in their conference
2. Have a bye in the playoffs (i.e. have one of the two best records in the conference)
3. Win their division
4. Gain a wild card berth

Assume that no games in the entire season finish in a tie.

The below information is accurate for the NFL of 2002-2006 (and probably beyond). Those who know the NFL well do not need these guidelines to have sufficient information to solve the puzzle.

DIVISION ALIGNMENTS:
There are 32 teams in the NFL. They are divided evenly into two "conferences", and each conference has four "divisions" of four teams each.

PLAYOFF ELIGIBILITY RULES:
The best team in each division makes the playoffs. Of the remaining teams, the two best in each conference also make the playoffs as "wild card" teams. The "best" team is determined by overall win-loss record. If teams are tied, there is a complicated series of tiebreaking rules that are applied to determine which team(s) are the "best", and which would make for great puzzles, but which aren't important here. (If you're curious, see http://sports.espn.go.com/nfl/news/story?page=tiebreakers .) The only relevant things to know about the tiebreaking rules are that they can always definitively decide among all ties of any number of teams in the same conference, that they never require that additional games be played, and that they almost always work in a deterministic way based on prior games and their results (though there is the hypothetical need of a coin flip in the rare case that all other tiebreaking methods fail).

SCHEDULING GUIDELINES:
Each team plays 16 games in a season. These opponents are determined before the season begins as follows:

Each team plays 2 games against each division opponent (total: 6). Each division is matched up with a designated division from the other conference, and each team plays all other teams in that matched-up division (total: 4). Each division is also paired up with a designated division from the same conference, and each team plays all other teams in that paired-up division (total: 4).

The last two games are played against teams from the other two divisions in your conference. For example, if your team's conference contains divisions A, B, C, and D, and your team is in division A, and your team plays a game against each team in division B, then your team will also play one game against a team in C and one game against a team in D. Furthermore, the C team that you play and the D team that you play share a "B" opponent for their other game from this category (i.e. the game that isn't against your team).

Title: Re: NFL playoff eligibility
Post by flamingdragon on Dec 19^th, 2006, 10:02pm

How about riddles listed in the "easy" section should have to be less than 20 words. ;)

Title: Re: NFL playoff eligibility
Post by Colbey on Dec 20^th, 2006, 4:27am

Sorry, I'm new here. This seemed to be the rough difficulty of the other "Easy" ones, if perhaps a bit detailed for those not already familiar with the system. Is there a better place for me to have posted it? I did want to share the problem regardless of the detail, because the reasoning required is interesting and somewhat unusual, and this seemed like the best place...

Title: Re: NFL playoff eligibility
Post by SMQ on Dec 20^th, 2006, 4:45am

0. [hide]Everyone can tie at 8-8, forcing tie-breaking rules, but the best record would still be 8-8.[/hide]
1. [hide]Everyone in a conference can tie at 6-10[/hide]
2. [hide]Same as #1. Even if one team in a conference wins every in-conference game they play, there are still 6 other teams in the conference they don't play. Among those 6 teams none can do worse than 6-10 without making another one better, so the worst possible second-best record in the conference is still 6-10.[/hide]
3. [hide]If they all lose every game outside of the division, everyone in a division can tie at 3-13[/hide]
4. [hide]This is the trickiest one. Within a division, one team can win at most 6 of the 12 total in-division games, so the worst possible second-best record is 2 wins within the division. I think the four division winners (in a conference) can collectively play at most 22 of the 48 in-conference cross-division games, leaving 26 games distributed among the other 12 teams, so there will be at worst two non-division-winning teams in the conference with three in-conference cross-division wins and overall 5-11 records.[/hide]

--SMQ

Title: Re: NFL playoff eligibility
Post by SMQ on Dec 20^th, 2006, 4:51am

on 12/20/06 at 04:27:02, Colbey wrote:

Sorry, I'm new here.

Welcome! I disagree with F.D.; I think Easy is indeed an appropriate place for your riddle -- it's where I think I would have posted it. It's also possible, given the winking smiley, that F.D. was partly or mostly kidding.

--SMQ

Title: Re: NFL playoff eligibility
Post by Colbey on Dec 20^th, 2006, 6:12am

Oh, interesting. Your answer to #4 is to a slightly different problem that I didn't think to pose: "What is the worst possible record of the better wild card team in a conference?" (And I agree with your answer.) But do you get a different answer if you're looking at the second (i.e. worse) wild card team?

Title: Re: NFL playoff eligibility
Post by SMQ on Dec 20^th, 2006, 6:30am

on 12/20/06 at 06:12:36, Colbey wrote:

But do you get a different answer if you're looking at the second (i.e. worse) wild card team?

I didn't think I did, since the worst case (among the 12 non-division-winning teams in a conference) I could posit seemed to be [hide]two teams at 5-11 and 10 teams at 4-12[/hide], but in light of your comment I guess I'll revisit that line of reasoning.
;)

[edit]Actually, I'm revisiting my reasoning on #2 at the same time...[/edit]

--SMQ

Title: Re: NFL playoff eligibility
Post by Colbey on Dec 20^th, 2006, 6:33am

[hide]But maximizing the wins of the 4 division winners and one other team gives you fewer wins to distribute among the other 11 teams, which might (or might not) make the minimum number of wins for the 6th team less than 5.[/hide]

Title: Re: NFL playoff eligibility
Post by flamingdragon on Dec 20^th, 2006, 7:09am

on 12/20/06 at 04:51:23, SMQ wrote:

Of course I was joking. :)

Hmm...... F.D., has a nice ring to it. ;D

Title: Re: NFL playoff eligibility
Post by SMQ on Dec 20^th, 2006, 7:31am

Revisited:
2. [hide]My reasoning was slightly blurry, but the result still holds: the team with the best record in a conference can win at most 12 games from other conference teams, leaving 84 in-conference games divided among 15 teams so at least one of those teams must win 6 or more in-conference games.[/hide]

5. [hide]I overlooked the fact that the first wildcard team can pick up as many as 5 of the 26 in-conference cross-division games not involving the division champions, leaving 21 wins divided among the remaining 11 teams. That means the second wildcard spot can go to a team with 2 in-division wins and only 2 in-conference cross-division wins, or an overall record as bad as 4-12.[/hide]

Unfortunately for the Lions, though, even if they win their last two (not blooming likely), the rest of the NFC just isn't bad enough to make that happen.
;)

--SMQ

Title: Re: NFL playoff eligibility
Post by CapriRS302 on Dec 20^th, 2006, 9:09am

What you forgot to mention was that the stupid NFL rules committee has the sudden death overtime rule AND allows a game to end in a tie (you would think that they would get rid of at least one of those, Americans hate ties) so theoretically, every game could end in a tie, moreover a 0-0 tie, which would not only leave the best record in the league at 0-0-16, but leave the playoff spots as a coin toss.

Title: Re: NFL playoff eligibility
Post by rmsgrey on Dec 20^th, 2006, 9:55am

on 12/20/06 at 09:09:19, CapriRS302 wrote:

Though it's not clear which is the better record of 8-8-0 and 0-0-16...