World Cup camera coverage poses a moving math puzzle

Mathematicians have considered how to watch every corner of a space—but soccer adds moving players, blocked views and constant action

I can already imagine the shouts that will erupt this summer during the International Federation of Association Football (FIFA) World Cup: “That was a bad call!” “That wasn’t a foul!” “The other team should have had a penalty!”

Fortunately, video replay allows people to validate—or refute—a referee’s decision. Of course, that technology also sparks heated debate among fans. But my interest is in the mathematics that accompany video evidence and video assistants.

A dear colleague recently approached me with a seemingly harmless question: How many cameras are needed, at minimum, to cover a playing field as accurately as possible, and where is the best place to position them to guarantee that every action is recorded? As it turns out, this question is anything but easy to answer.

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In mathematics, this type of question is more familiarly encountered as the “art gallery problem.” In 1973 mathematician Václav Chvátal asked his colleague Victor Klee for an interesting geometry problem. Klee responded by challenging him to find how many guards are needed, at a minimum, to protect a gallery.

It’s a classic optimization problem that depends on the shape of the gallery. For a rectangular room with pictures hanging on the walls, assuming there are no columns or people to block one’s view, a single guard is theoretically sufficient. The guard stands in a corner and can easily oversee the entire area.

For more complex spatial shapes, finding an answer is not so easy. In 1975 Chvátal published a paper that proved that the minimum number of guards in a room with n corners is at most n ⁄ 3 , rounding down the result if it is not an integer.