## Why do the Patriots host the Broncos 3 years in a row?

### NFL Scheduling

Every once in a while, somebody will notice that a given NFL team will host another non-divisional team from the same conference three years in a row. Today the Patriots are hosting the Broncos for the third year in a row. That's so unfair! Or so it would seem. But it turns out that there are sound mathematical reasons that it works out this way.

To explain the reasons, let's consider how the NFL schedule is generated. The NFL has 32 teams, organized in two conferences, each of which has 4 divisions with 4 teams each. And the season lasts 16 games long for each team. This coincidence of numbers makes for a very elegant scheduling algorithm, which will disappear as soon as the NFL expands again.

Each team plays a mix of divisional games, intraconference games, and interconference games. For starters, each team plays a home game and an away game against each of its three division rivals.

The next consideration is the desire to have regular games against all the other teams in the league. That's neatly arranged by having all of the teams in each division play all the other teams in each other division on a rotating basis. For example, this season all of the teams in the AFC East play all of the teams in the NFC North. That rotation is on a four-year schedule, since there are four divisions in each conference.

In contrast, there is a three year rotation for interconference matchups. This year, the AFC East plays the AFC West, while the AFC North plays the AFC South. Last year the AFC East played the AFC North. Next year, they'll play the AFC South.

So each team plays 6 games in the division, 4 games against all the teams of one other division in the same conference, and 4 games against all the teams in one division intra-conference. The home-and-away matchups for division vs. division play alternates so no city will go too, too long without being able to see every other team. That means every eighth season for interconference, every sixth season for intraconference. These account for 14 of the 16 games played every season.

That leaves two slots to fill up the season. The NFL fills the last two slots by having every team play one team from each of the other two divisions in the same conference, namely, the two teams that finish in the same respective place in the standings. In recent years, the Patriots have finished in first place, and thus had to play the first place team from the other two divisions.

For many years, this led to the Patriots playing the Colts every season. One season, when the Patriots finished in second, the Colts did the same thing, so they kept playing each other. Now that Peyton Manning has moved to Denver, there is focus on that matchup. And two years ago, back when Tim Tebow was their QB, the Broncos visited Foxboro to play the Pats. Last year, they did the same thing.

And a few weeks ago they hosted the Broncos again.

Why is that? Well, let's look at how this can go. The three other first place teams last season were the Broncos, the Bengals, and the Colts. As I mentioned earlier, the Pats play all the teams of the AFC West this season. That they host the Broncos is part of the divisional schedule. Three years ago they went to Denver, and they'll go to Denver again three years from now. And if they'd gone to Denver this year, they would have hosted Denver three years ago, so they still would have hosted Denver three years in a row.

Why does it happen? It happens because the home-away switch has to interleave with the rotating division vs. division matchup. Recall the list of the first-place teams: Patriots, Colts, Bengals, and Broncos. Since the Patriots hosted the Bengals and visit Indy this season, the Broncos have to do the opposite, just to ensure all four teams have exactly 8 home games and exactly 8 road games. This makes for a "cycle": Pats host Bengals, who host the Broncos, who host the Colts, who host the Pats. A more compact notation here would be (Pats, Bengals, Broncos, Colts), where each team hosts the team after it. Or, to use the geographic names, (East, North, West, South). Last year was (East, West, North, South). And the year before that was (East, West, South, North).

If I keep East in the first slot, there are six ways to fill the other three slots. And it turns out the NFL is on a 6-year rotating schedule in this respect. Now if I keep East in the first slot, the third slot is in a regular rotation:

(East, *, South, *)

(East, *, North, *)

(East, *, West, *)

(East, *, South, *)

(East, *, North, *)

(East, *, West, *)

So, let's start with (East, West, South, North) and see what we're forced to do. The next year we have (East, *, North, *). If we want to avoid the (East, West, *, *) option, we have to use (East, South, North, West). That gives us

(East, West, South, North)

(East, South, North, West)

and, if we want to again avoid a repeat

(East, North, West, South)

So far, so good, yes? Well, at least as far as the East goes. But notice that the South hosts the North two years in a row, the West hosts the South both times it is possible, and the North hosts the West two years in a row.

So, it's possible to schedule things so the East team doesn't have any repeat hosting, but doing so would force every other divisional matchup to be constant in terms of who's on the road and who's hosting. You just cannot alternate a two-year rotation and a three-year rotation for everybody. So what the league does is use all six possibilities.

(East, West, South, North)

(East, West, North, South)

(East, North, West, South)

and those three reversed

(East, North, South, West)

(East, South, North, West)

(East, South, West, North)

In other words, East@West, East@West, division vs. division, West@East, West@East, division vs. division. And since the division vs. division matchups themselves alternate, either the third or sixth year is also going to be East@West, leaving us with three years in a row of East@West, followed by three years in a row of West@East.