**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.

Andy DI like how the ‘friendliness’ of two teams will vary continuously as one moves across the playing field in these games. For example, in the limit of a very large number of teams on a very large circular field, players on teams with adjacent goals will in some cases be very closely aligned (and willing to pass to each other, for example).

This subtlety could give mathematicians a crucial advantage on the playing field.

However, there’s also the more mundane worry that it might be too hard to score a goal in these games. Certainly any game with fewer goals than ordinary soccer risks being totally unwatchable. So which multi-team variants are most likely to lead to decent scores?

Gordon RoyleI’m no celebrity watcher, but even I know that her surname is Beckham, not Backham!

GK:thanks!!teageegeepeaWon’t the offense team always win and the defense team lose, unless they tie in a goal-less game?

Gil KalaiPost authorI dont think so. The offense team is happy for every goal scored and the defense team is happy for every goal not scored. The left and right teams happiness depends on the side the goal is scored. We do not assign a winner or loser among the four playing teams.

gowersI was intrigued by the hexagonal-football link but unfortunately it seems to be broken. Is it fixable?

GK:fixedGil KalaiPost authorHere is another link to the same picture http://www.onenewspage.co.uk/news/Front+Page/20110531/22770091/Three-sided-football-match-played-on-hexagonal.htm

Ehud FriedgutHere’s my main recollection from the game at the Newton Institute:

Since Gil was my Ph.D. advisor I have great faith in him, and several times while our team was attacking, and I had the ball, I passed to him. Unfortunately, the first three trials ended with Gil kicking the air. However, I didn’t lose faith, and my fourth pass finally paid off, as Gil kicked the opponent who was guarding him. I firmly believe that he was aiming for the ball, though.

Gil KalaiPost authorEhud gave indeed an accurate account of the game. I clearly remember kicking the air a few times when the ball came in my direction, and indeed once I kicked (unintentionally) Nisheeth Vishnoi (who was guarding me at the time). Nisheeth, from the famous Khot-Vishnoi theorem (and many other things) told me later that he never saw me with such a serious expression on my face as during the football game, and indeed I regard football as a very serious matter. Let me add also that I never raised my hand suggesting that the ball will be passed to me. (I try to have a similar approach regarding mathematical problems I pose.) Also, since usually some player from the other team guarded me without being fully aware of my talents and abilities, my presence did serve some purpose for the team. (If I remember correctly the devision to teams was done according to the color of the closing, although later to make the game more even some players switched teams at one point.) Among the 20 or so overall players it seems that 5-6 really knew what they were doing and this gave them much advantage over the other players. One player was clearly athletically much superior to verybody else but somehow this did not effect the game so much.