The 1973 "art gallery problem" — a classic geometry question about the minimum number of guards needed to watch every corner of a polygonal room — gives a clean bound for static rooms, but moving players break the theorem, and the upcoming World Cup is testing a question math…
When FIFA's broadcast trucks roll into a World Cup stadium this summer, they will bring dozens of cameras to cover 22 moving players, a referee, and a ball that never stops. The question of where to place those cameras, and how few can still see everything, turns out to be a live branch of a 53-year-old problem in computational geometry.
In 1973, mathematician Václav Chvátal asked his colleague Victor Klee for an interesting geometry problem: what is the minimum number of guards needed to watch every corner of a polygonal art gallery? In 1975, Václav Chvátal proved that ⌊n/3⌋ guards always suffice for a polygon with n vertices, a result that became one of the founding theorems of an area now called the art gallery problem (O'Rourke survey, UC Davis). The proof works by triangulating the floor plan, then assigning guards to one of three color classes; that class is guaranteed to see the whole interior.
That clean result is not the whole story. Finding the actual minimum number of guards for an arbitrary polygon is NP-complete, and the ⌊n/3⌋ bound is not always tight. The theorem tells you that a certain number will work, not that no smaller number could.
A soccer pitch, on its face, should be easy. A standard field is a rectangle, and a single 90-degree camera in one corner would see the entire playing surface. The complication is that 22 players, two goalkeepers, and three officials are opaque, moving, and arranged in formations that constantly block one another from any fixed viewpoint. The rectangle is no longer empty.
In 2009, Hemanshu Kaul and YoungJu Jo, then at the Illinois Institute of Technology, proved that 10 cameras placed in the upper stands are enough to cover a soccer field containing 22 static "holes" representing the players (Kaul & Jo, "Long Art Gallery Problem" talk slides, IIT Math). The bound adapts the Chvátal technique to a rectangular polygon, with players treated as opaque disks whose centers must remain visible to at least one camera.
That proof freezes the players in place. The moment they start running, the geometry stops being a single art gallery problem and becomes a continuous family of them, one for every instant of the match, with occlusions shifting between frames. The minimum number of cameras needed to keep the ball and every player in view across a full ninety minutes is not known, and no published closed-form bound exists for the dynamic case (Scientific American: "World Cup camera coverage poses a moving math puzzle").
This is the gap the upcoming 2026 World Cup is operating inside. Broadcasters do not solve the continuous problem on a whiteboard; they stack cameras, assign redundant coverage zones, and trust operators and replay systems to catch what a single fixed geometry could not. Mathematicians, for their part, have a clean static bound (ten cameras for twenty-two frozen players) and a classical theorem that is more than fifty years old, but no published certificate that a specific number of cameras, no matter how cleverly placed, can guarantee unbroken coverage of a live match.
The honest implication, as the math journalist Manon Bischoff put it in the original German-language explainer adapted for Scientific American, is that piling on more cameras is not the answer. Placement and geometry are. The art gallery theorem gives broadcasters a lower bound to reason about and an upper bound to plan around. What it does not yet give them, and what the World Cup will not pause to find, is the exact number a perfectly designed stadium would need.
That answer, for now, is the open problem the 2026 tournament will be playing on, even if no one in the broadcast compound is keeping score.