PICKING TEAMS

Keep it fair with a strategy that relies on geometry

假如你和朋友们打算踢足球，你们8个人可分成红蓝两队，一队4人。怎样组队才公平呢？下面的文章，作者引入几何模型，介绍如何在特定场比赛中，让每名球员都与自己组队次数相同。

You and some friends are planning an afternoon of soccer scrimmages. There are eight of you, so you can play in teams of four. You want to play enough games so that each player gets to play with everyone else. And, to be fair, you would like each pair of friends to be teammates the same number of times.

How would you choose the teams and decide how many games to play?

One way to try to solve the problem is to pick the teams randomly. You could have four blue and four red slips of paper in a box. Then, at the start of a game, each player would reach in without looking and pull out a slip. Someone who picked red would join the red team.

On average, in seven games, two players would be on the same team about three times. But there’s still a small chance that you and your best buddy (or some other pair of players) never end up on the same team. Because there are eight of you, there’s actually a neat geometrical solution to the problem that guarantees fairness.

Grab a cube. It has six faces, twelve edges, and eight corners (or vertices). Let each corner (or vertex) represent a player. Picking a team of four means assigning the same color to four of the corners. If the four corners of one face of the cube represent the blue team, then the four corners of the opposite face represent the red team. There are three such sets of opposite faces. Play your first three games with your teams set up as shown below.