Random ponderance about 4-team group stages

Shwafta

MLS Cup Attendee
Elite Donor
Jan 16, 2018
7,406
10,397
303
26
Long Island
So the Euros got me wondering
1) What is the highest number of points a 3rd place team could get? Along with that, what's the lowest number of points a team in 2nd needs to get to confirm advancement
2) What's the highest number of points a 4th place team can get? As well, what's the lowest number of points a 3rd place team needs to confirm potential qualification?
3) What are all the permutations that a group stage can have?

As such, I created a small javascript program that ran and here are the results I got, filtered for duplicates:
[[0,3,6,9],[1,1,6,9],[3,3,3,9],[1,3,4,9],[0,4,4,9],[1,2,4,9],[2,2,2,9],[3,3,6,6],[0,6,6,6],[1,4,6,6],[1,3,6,7],[3,4,4,6],[0,4,6,7],[1,4,5,6],[2,4,4,6],[1,2,6,7],[2,2,5,6],[3,3,4,7],[0,3,7,7],[1,3,5,7],[2,3,4,7],[0,4,5,7],[1,3,4,7],[1,4,4,7],[2,2,4,7],[1,1,7,7],[1,2,5,7],[2,2,3,7],[3,4,4,5],[2,3,5,5],[3,4,4,4],[4,4,4,4],[2,4,4,5],[1,4,5,5],[2,3,4,5],[0,5,5,5],[1,3,5,5],[2,2,5,5],[2,3,3,5],[3,3,3,3]]

From this, I found the following:
1. The highest number of points for a 3rd place team is 6, meaning that to guarantee at least second place, the second place team has to get 7 points, aka two wins and a draw. There is a possibility that even after winning two games, a team is NOT guaranteed second place!
2. The highest number of points the 4th place team is... 4! to confirm not last, 3rd place must get at least 5 points.
3. Here are all the permutations. Seeing as it's impossible to get 8 points, the only way to guarantee first place in a group is to win all three games. Most likely two wins and a draw can seal the deal on 7 points, but there are a few cases where two teams end up on 7 points, so it's not guaranteed.

I've always wondered what the possible permutations are, and this is fascinating to me so I hope some of you learned anything fun.

As an added bonus, I'm going to leave two more things:
1. The full set of possibilities, with team names connected to the points:
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2. Here is the algorithm I used to find the solutions, if anyone is interested: https://jsfiddle.net/#&togetherjs=g9lJnsbJrI
 
Good stuff. I like that.

Another one. If the second place team has 7 points, then so does the first place team.

The scenario where a 3rd place team has 6 points has always been interesting to me. Basically, this means that three teams tie at 6 points and the fourth gets zero. The three teams each beat the fourth and then one of the others - rock, paper, scissors style. It nearly happened in the 2018 World Cup, but S. Korea upset Germany in the final game. If Germany had prevailed, it would have been on 6 points with Sweden and Mexico, and Mexico would have been the odd team out based on goal differential. Imagine not advancing with 2 wins, one of them over Germany!
 
Good stuff. I like that.

Another one. If the second place team has 7 points, then so does the first place team.

The scenario where a 3rd place team has 6 points has always been interesting to me. Basically, this means that three teams tie at 6 points and the fourth gets zero. The three teams each beat the fourth and then one of the others - rock, paper, scissors style. It nearly happened in the 2018 World Cup, but S. Korea upset Germany in the final game. If Germany had prevailed, it would have been on 6 points with Sweden and Mexico, and Mexico would have been the odd team out based on goal differential. Imagine not advancing with 2 wins, one of them over Germany!
Yeah, that scenario is always nuts! I forgot that happened at the world cup, probably mostly because Germany lost.

I want to see if there's any scenario of the above that have never happened in a group stage, but that's not something I feel like actively trying to research right now lol.