I was at a bar once and the question of “What’s the worst NFL record you can have and still make the playoffs?” came up. We figured out pretty quickly, you can theoretically win your division (and host a home game!) without ever winning a game. If you lose all non-division games and tie all your division games you would go 0-11-6 and host like an 11 win team on wild card weekend. I pray in vain for this every year.

What’s the World Cup version of this? With the expanded version of the World Cup that now features some third place group stage finishers advancing, naturally the question is “How bad can you be an still advance?”

So, first thing’s first. How many unique final standings are there for a group? There are 6 games and each game has 3 possible outcomes. So 3^6 = 729 possible final tables, but for this question I only care about unique sets of points. That is if teams A, B, C, and D finish with 9,6,3,0 or teams D, C,B, and A finish with 9,6,3,0 those are the same for my purposes.

So how many unique point totals are there? I found 40. (I wrote code to do this. Anyone know how to do the math to solve this?)

Seven different tables when one team wins all their games:

Next there are 14 tables possible with the top team getting 2 wins and a tie:

Seven different tables where the top teams gets two wins and a loss.

9 different tables are possible where the winner get’s a single win and two ties. Finally there are two tables possible where the top team gets one win and a tie and exactly one possible table where the group winner has 1 win and 2 losses (all teams win exactly one game, my personal favorite).

But we don’t care about that for the purposes of this question. We care about what’s the worst that you can do and still finish 3rd? So let’s look at the distribution of points for the third place team:

In the forty unique tables the third place team finished with 5 or 6 points exactly once each. There are 13 ways to get to 4 points and 15 ways to get to 3 points and eight ways to get to 2 points. This leaves exactly two tables with the third place team on 1 point: 7,7,1,1 and 9,6,1,1. In both these cases the bottom two teams both lose all their games against the top 2 teams in the groups and then they tie each other. Ultimately what this means is that you can theoretically show up to the world cup and if every single group finishes with 7, 7, 1, 1, or 9, 6, 1, 1, multiple teams that went 0-2-1 in the group stage will advance! I love it so much.

The other natural question is “What’s the most points you can score and not advance?” The most points you can score and finish fourth is 4 points (everyone goes 1-1-1 in the group). So you could score 4 points and not even have a chance to advance by finishing 4th. But there is one even worse. There is a scenario where you finish 3rd with 6 full points (the bottom team loses all three games and the top three all split with each other). If this happened in every group multiple teams with TWO WINS wouldn’t advance.

In summary, the theoretically most points you can score and not advance is 6 and the fewest points you can score and theoretically advance is 1. My absolute favorite thing in sports to root for now is absurd outcomes like this. Both of these extremes are fairly unlikely in reality, but a boy can dream, can’t he?

Cheers.