Random musings as NFL gets into the last week of the regular season and both Saints and Cardinals are at 10-5, with only one of them going to the playoffs. Since they’re not playing each other, we may see an 11-5 team not going to the playoffs this year. And only three years ago Seahawks won their division and went to playoffs at 7-9.

There’s some math laid out in here [PDF] that looks at the schedule balance of an NFL season, and while on average it appears that in most cases we do see the best teams from each conference going to the playoffs, I always wondered what is the worst case scenario.

There are two extremes here. The first one is how bad can you be and still get to the playoffs. The second one is how good can you be and still watch the post season on the TV.

The first one is simple. Each conference has four divisions, and every division is guaranteed a spot in the playoffs (aka Seahawks ’10). If you’re not familiar with how the regular season schedule is determined, here’s how it works. For the purposes of this first extreme, it’s enough to know that each team plays the three teams in its division twice (home / road), and then 10 games elsewhere in the league. What’s the absolute worst? Well, you can get into the playoffs with zero wins. That’s right, zero. How? Imagine a division with four really bad teams, and every game in that division ending at 0:0 (or any tie). And then every team in that division loses the rest of their 10 non-division games. In that case you’d eventually get to a very awkward coin-toss to determine which one of these four teams gets the “first place” in the division. Unlikely? Extremely. Possible? Of course.

Now to the other extreme. How many games can you win and still miss the playoffs? The answer is 14 (out of 16 games you’re playing) – if my math is correct of course. Let’s look at the scheduling rules more closely.

When the league expanded to 32 teams, it brought a very nice balance to the divisions themselves and to the schedule. Two conferences, four divisions each, four teams each. All hail the powers of 2! By the way, there’s additional symmetry that you get to play / host / visit each other team every 3/4/6/8 years (depending on the division association).

Back to who gets to the playoffs. Every division sends its first place, with two more spots (wildcards) left to the two best teams in the conference after division leaders are “removed” from the equation. This means you can be a really good team and still not get into those six. The conditions are quite simple really (as one of Saints / Cardinals will see this Sunday). The first one is that you have an even better team in your division that takes first place. The second one is that you have two better teams elsewhere in the conference (such as 49ers that already secured the first wildcard spot for NFC).

Let’s look at the numbers now. How can we get to the 14-2 record and still miss the playoffs?

In the following scenario we have NFC East as NE, NFC West as NW, NFC North as NN, NFC South as NS, and their counterparts in AFC as AE, AW, AS and AN. Let’s choose three random divisions in NFC, say NE, NN and NS.

A team in NE is playing 6 games in NE, 4 in NN, 4 in AW and 2 in NS/NW. A team in NN is playing 6 games in NN, 4 in NE, 4 in AN and 2 in NS/NW. A team in NS is playing 6 games in NS, 4 games in NW, 4 games in AE and 2 games in NN/NE.

In general you meet all teams in your division twice, all teams in another division in your conference once, all teams in a division in the other conference once, and then two teams from the other two divisions in your conference that finished at the same place as you last year.

What we’re trying to do is to get as many wins as possible for the #2 team in each one of our divisions (NE, NN and NS). There are only two wildcards available in each conference, and we don’t care what happens in NW or the entire AFC.

For each pair of teams in NE, NN and NS we want to maximize the number of wins while still keeping in mind that they play each other. This year each team from NE plays each team in NN twice. And each team in NS plays one team in NN and one team in NE – based on its position in the division last year.

Let’s look at NS first. Team #1 and #2 get four wins each playing #3 and #4 in their division. Then they split the wins in their two games, getting at 5-1 record for each. They then win all 4 games against NW, getting at 9-1, and all 4 games against AE, getting at 13-1. Finally, assuming that our two teams finished last year at positions that get them scheduled against NE / NN teams that will not be finishing at #1 / #2 teams this year, both teams get at 15-1 – all without taking a single win away from the four teams in NE / NN that we’ll be looking at shortly.

Now to NE / NN. We’ll look at NE, while the same logic applies to NN. Once again, teams #1 / #2 win all four games against #3 / #4 and split their own two matches, getting at 5-1 both. Now they play four games against NN. They win both games against #3 / #4 teams, getting at 7-1 each, and split the wins against #1 / #2. We need to split the wins in order not to take away “too many” wins from those two teams in NN. So we end up with 8-2. Now they win all four games against AW, getting at 12-2. Then they get one win against NW getting to 13-2. Finally, they have one game to play against NS. Applying the same selection logic, the best scenario for us is to get them scheduled against a team that is not at #1 / #2 this year (but at the same position they were last year), which gets both teams to 14-2.

And now the same goes to the first two teams in NN, getting them to 14-2 both. Which is why we need to split the NE/NN games between #1/#2 teams.

Now we have #2 team in NS at 15-1 and #2 teams in both NE and NN at 14-2 each. One of them will have to stay out of playoffs. Unlikely? Extremely. Possible? Of course.

Waving hands in the air, it feels that the first scenario is much less likely to happen due to how few ties we usually see in the league. Even though it can happen in any one of the eight divisions, and the second scenario involves three divisions in the same conference, it’s still much less likely to happen. What if we remove the ties from the equation?

A 4-team division has every team playing every other team twice. All in all you have 12 inner-division games in every division. If no game ends in a tie, the most extreme case is that all teams end up winning and losing 3 games in their division, and losing all other 10 games, with 3-13 record for each. One of them will go to the playoffs. That would also answer the question of how many games can you lose and still go to the playoffs. In the previous scenario (no wins), you have every team in the division at 0-10-6 record, so it’s “only” 10 losses. With this scenario you have a 13-loss team going to the playoffs.

It would appear that this new extreme is more likely to happen, as it only involves teams in a single division, and the other one (14-2 not going to playoffs) involves teams in three divisions.

Now two questions remain. Can we get to a 15-1 record and stay out of playoffs? And, more importantly, is there a fatal flaw in the logic outlined above?