The angle of Victoria Beckham’s hat (here in a picture from a recent wedding) is closely related to our previous post on football
One of the highlights of the recent Newton Institute conference on discrete harmonic analysis was a football game which was organized by Frank Barthe and initiated independently by Barthe and Prasad Tetali. There were two teams of 10 players (more or less), I was the oldest player on the field, and it was quite exciting. No spontaneous improvement of my football skills has occurred since my youth.
All the lectures at the conference were videotaped and can be found here. (The football game itself was not videotaped.) Let me mention an idea for a new version of football which occurred to me while playing. For an early suggested football revolution and some subsequent theoretical discussions see this post on football and the intermediate value theorem.
The New Football Game
There are four teams. Team L (the left team), Team R (the right team), Team D (the defense team) and team O (the offense team.) The left team protects the left goal and tries to score the right goal, the right team protects the right goal and tries to score the left goal, the defense team tries to prevent goals on both sides of the field and the offense team tries to score as much as possible goals on both sides of the field.
In formal terms, define X to be the number of goals scored to the right and Y the number of goals scored to the left. Then the four teams try to maximize X-Y, Y-X, -X-Y and X+Y, respectively. (I do not assume teams A and B change position at halftime, but the formula can easily be adjusted.)
We can start with five players on each team (but this is negotiable.)
Another version: Circular field, 3- and 6- teams football and root systems
The advantage of the above version is that it can be implement easily. There are some variations which are harder to implement but have some further nice aspects. For example you can have a circular court with 3 goals (see the above picture). If X, Y, and Z are the number of goals scored we can have three teams which try to maximize X+Y-2Z, X+Z-2Y and Y+Z-2X, respectively. We can even have 3 more teams maximizing 2X-Y-Z,2Y-Z-X,2Z-X-Y. Savvy football fans will recognize the similarity of the first version of this post with the root system and of the last version with the root system .
Update: A three teams footmall where each team has one side and aim to minimize the number of goals is already in place. Click on the heagon below. It is an equivalent game to our three-teams game.