Packing Problems
Fit objects into a container without overlap, and make them as large or as many as possible. This page indexes every packing family on the site: every score recomputed exactly by a deterministic verifier, every known value carrying its source and whether anyone has proved it optimal.
What a packing problem is
A packing problem fixes three things: what is being placed (equal circles, circles of assorted sizes, squares free to tilt, lines through the origin), what it goes into (a square, a circle, an L, a plus sign, a half-disc, a quadrant, a right triangle), and what is being maximized (the common radius, the sum of radii, the common side, the smallest angle). Once all three are fixed, what remains is the arrangement itself — and for most parameters, nobody knows the optimal one.
That asymmetry — checking an arrangement is easy, finding the best one is hard — is why packing problems stayed alive into the computer era, and it is what this site is built on. Anyone can submit coordinates; the verifier scores them exactly in integer arithmetic within milliseconds; anything better than the standing record takes it, publicly and immediately.
Packing, covering, dispersion and energy
Packing forbids overlap and maximizes size. Covering forbids gaps and minimizes size. Dispersion places no bodies at all, only points, and maximizes the smallest distance between them. Energy problems minimize a potential summed over every pair. The four are routinely conflated, but their objectives differ and so do their optima. The site carries all four; this page lists the packings.
What machine-verified means here
Every submission is a set of coordinates written to nine decimal places. The verifier reads them as integers and decides containment and overlap in BigInt arithmetic — no floats, no square roots, no approximate comparisons. The same certificate therefore scores the same on every machine, and a record never depends on whose hardware was faster.
Known values come in two kinds and the pages never blur them. A proven optimum means somebody has shown nothing beats it, and reaching it finishes the sub-problem. A best-known value means only that nobody has done better yet: matching it draws level, and beating it is a real result. Every cited value links to where it came from.
Circles
The classic family: fit n circles into a fixed container and maximize the common radius. P57 drops the equality and maximizes the sum of the radii instead.
Circles of radius 1…n
The circle sizes are fixed by the problem at 1, 2, …, n; the container is what moves, and the goal is to shrink it.
Tilted squares
Every square may turn to any angle, which widens the search from positions to positions-and-poses — and is why so many parameters in this family remain unsolved.
High-dimensional and abstract
What is packed is no longer a shape in the plane but a direction, a line, a plane. The scores are still exact integer arithmetic; the frontier is Sloane's packing tables.
Latest breakthroughs
Taking the data
Every family has a stable JSON address carrying all of its sub-problems: authoritative scores, proof status, coordinates, sources and the verifier version.
GET https://minmaxarena.com/data/square-circle-packing.json
The data is published under CC BY 4.0: use it freely, with attribution and a link back. Scores move as records fall, so cite the generatedAt timestamp the file carries alongside them.