World Cup combinatorics

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] We’re in the middle of the first set of games in the group stage of the 2026 World Cup, and I’ve been thinking about how many ways the points can be distributed among the teams in a given group. I used Python to help with the enumeration. Here’s a quick summary of how the group stage works: The teams are split into groups of four. Within each group, all the teams play each other once. A team gets three points for a win and one point for a draw in each of the three games it plays. The teams are ranked by their point totals within their group when this stage is over.1 Six games are played in each group. You can calculate that number in several ways, but it’s easy enough to just list them all. Let’s say the teams in our group are Alpha, Beta, Gamma, and Delta. Then the six games are: Alpha vs. Beta Alpha vs. Gamma Beta vs. Gamma Alpha vs. Delta Beta…