Tuesday, December 24, 2019

8 teams, 84 games - Part 2

Lay out the series

I use a spreadsheet for setting out the series.  It provides a nice structure for gridding things and simple math functions can be used to take care of some of the work like duplicating series and adding +1 to team numbers to create subsequent matchups.

I defined a series with a triplet of numbers.  From left to right the numbers in the triplet are:

  1. number of games in series
  2. away team
  3. home team.

The number of games I underline to make it stand out visually when working in the spreadsheet.  Here I will show the info with a fixed-width font rather than copying spreadsheet views.

So this represents a 3-game series of team 1 playing at team 2:
 3 1 2
That is one series, and we need to do all the series.

One team...

Start with one team's intradvision games.  Team # 1 is always a good starting point.  Lay out all of its division matchups.  From the previous part we said play each other team in three series both away and home with one of the series both away and home being 4-game.

3 1 2
3 1 2
4 1 2
3 2 1
3 2 1
4 2 1
3 1 3
3 1 3
4 1 3
3 3 1
3 3 1
4 3 1
3 1 4
3 1 4
4 1 4
3 4 1
3 4 1
4 4 1

One division...

Next we fill in the rest of the division games for the division.  That is pretty trivial for a 4-team division because there is only one other matchup to define.

3 1 2 3 3 4
3 1 2 3 3 4
4 1 2 4 3 4
3 2 1 3 4 3
3 2 1 3 4 3
4 2 1 4 4 3
3 1 3 3 4 2
3 1 3 3 4 2
4 1 3 4 4 2
3 3 1 3 2 4
3 3 1 3 2 4
4 3 1 4 2 4
3 1 4 3 2 3
3 1 4 3 2 3
4 1 4 4 2 3
3 4 1 3 3 2
3 4 1 3 3 2
4 4 1 4 3 2

Rest of division series...

Now make the other divisions intradivision games as a mirror image of the first.  That means put the corresponding teams (i.e., 5:1, 6:2, 7:3, 8:4) into the matchup slots, and flip the home/away of each matchup.  E.g., from a 3-game series of 1 at 2, make a 3-game series of 6 at 5.  The flip of the home/away is to preserve the symmetry that appears when making interdivision games.

3 1 2 3 3 4 3 6 5 3 8 7
3 1 2 3 3 4 3 6 5 3 8 7
4 1 2 4 3 4 4 6 5 4 8 7
3 2 1 3 4 3 3 5 6 3 7 8
3 2 1 3 4 3 3 5 6 3 7 8
4 2 1 4 4 3 4 5 6 4 7 8
3 1 3 3 4 2 3 7 5 3 6 8
3 1 3 3 4 2 3 7 5 3 6 8
4 1 3 4 4 2 4 7 5 4 6 8
3 3 1 3 2 4 3 5 7 3 8 6
3 3 1 3 2 4 3 5 7 3 8 6
4 3 1 4 2 4 4 5 7 4 8 6
3 1 4 3 2 3 3 8 5 3 7 6
3 1 4 3 2 3 3 8 5 3 7 6
4 1 4 4 2 3 4 8 5 4 7 6
3 4 1 3 3 2 3 5 8 3 6 7
3 4 1 3 3 2 3 5 8 3 6 7
4 4 1 4 3 2 4 5 8 4 6 7

Interdivision...

Now lay out the interdivision games.  The starting case is easy for aligning the divisions with their corresponding position in the other division.

3 1 5 3 2 6 3 3 7 3 4 8
3 5 1 3 6 2 3 7 3 3 8 4

One way to fill out the rest of the matchups is to simply rotate the second division around the first.  E.g., after 1v5, 2v6, 3v7, 4v8, we could do 1v6, 2v7, 3v8, 4v5.  But here I will use the more symmetric approach involving team pairings.  E.g., we have 1v5 & 2v6, so flip those to 1v6 & 2v5.

3 1 5 3 2 6 3 3 7 3 4 8
3 5 1 3 6 2 3 7 3 3 8 4
3 1 6 3 2 5 3 3 8 3 4 7
3 6 1 3 5 2 3 8 3 3 7 4
3 1 7 3 2 8 3 3 5 3 4 6
3 7 1 3 8 2 3 5 3 3 6 4
3 1 8 3 2 7 3 3 6 3 4 5
3 8 1 3 7 2 3 6 3 3 5 4

All together...

Combine the intradivision and the interdivision, and we have our full set of 26 half-week series sets.

3 1 2 3 3 4 3 6 5 3 8 7
3 1 2 3 3 4 3 6 5 3 8 7
4 1 2 4 3 4 4 6 5 4 8 7
3 2 1 3 4 3 3 5 6 3 7 8
3 2 1 3 4 3 3 5 6 3 7 8
4 2 1 4 4 3 4 5 6 4 7 8
3 1 3 3 4 2 3 7 5 3 6 8
3 1 3 3 4 2 3 7 5 3 6 8
4 1 3 4 4 2 4 7 5 4 6 8
3 3 1 3 2 4 3 5 7 3 8 6
3 3 1 3 2 4 3 5 7 3 8 6
4 3 1 4 2 4 4 5 7 4 8 6
3 1 4 3 2 3 3 8 5 3 7 6
3 1 4 3 2 3 3 8 5 3 7 6
4 1 4 4 2 3 4 8 5 4 7 6
3 4 1 3 3 2 3 5 8 3 6 7
3 4 1 3 3 2 3 5 8 3 6 7
4 4 1 4 3 2 4 5 8 4 6 7
3 1 5 3 2 6 3 3 7 3 4 8
3 5 1 3 6 2 3 7 3 3 8 4
3 1 6 3 2 5 3 3 8 3 4 7
3 6 1 3 5 2 3 8 3 3 7 4
3 1 7 3 2 8 3 3 5 3 4 6
3 7 1 3 8 2 3 5 3 3 6 4
3 1 8 3 2 7 3 3 6 3 4 5
3 8 1 3 7 2 3 6 3 3 5 4

Next steps...

That layout is nice and pretty but obviously we do not want to play the series in that order.  We want to spread the games around.  How so?

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