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Packing Problems

Pack geometric objects into a container without overlap, making them as large or as numerous as possible. This hub indexes every packing family on the site, with deterministic verification, literature citations, and optimality status for each instance.

Families23
Sub-problems1489
Still open1374

What a packing problem is

A packing problem specifies three elements: the objects being placed (equal circles, unequal circles, rotatable squares, lines through the origin), the container (a square, disc, L-shape, cross, semicircle, sector, or triangle), and the objective to maximize (common radius, sum of radii, common side length, or minimum angle). For most parameter values, the optimal arrangement remains an open problem.

Checking a candidate layout is fast and exact, whereas finding the best layout is computationally hard. Anyone can submit coordinates; the verifier scores them in milliseconds using exact integer arithmetic, and any valid improvement over the standing record takes the lead immediately.

Packing, covering, dispersion and energy

Related geometric optimization families differ in their constraints: packing forbids overlap and maximizes size; covering forbids uncovered gaps and minimizes size; dispersion places points to maximize the minimum pairwise distance; energy problems minimize a pairwise potential sum. The site features all four categories; this page focuses on packing.

How machine verification works

Every submission consists of fixed-point decimal coordinates. The verifier converts them into exact integers and checks containment and non-overlap using BigInt arithmetic — no floating-point rounding, no square roots, and no approximate tolerances. Every certificate produces the exact same score on any machine.

External benchmarks fall into two distinct categories: a proven optimum has a mathematical proof that no better arrangement exists, whereas a best-known value is simply the strongest published construction so far. Every benchmark links directly to its source.

P02 · Equal-circle packing in a unit circle · n = 181
P10 · Equal circles in a plus sign · n = 12
P57 · Sum of radii in the unit square · n = 19

Circles

The classic circle-packing problems: place n circles inside a container. Most fix the container and maximize a shared radius; P57 allows unequal radii, while P66 optimizes the enclosing triangle.

P01 · Equal-circle packing in a unit squarePlace n non-overlapping equal circles inside the unit square to maximize their common radius.
300 sub-problems · 269 open · 31 settled · 300 with a cited value
P02 · Equal-circle packing in a unit circlePlace n non-overlapping equal circles inside the unit circle to maximize their common radius.
300 sub-problems · 285 open · 15 settled · 300 with a cited value
P08 · Equal circles in an LPlace n non-overlapping equal circles inside an L-shaped region to maximize their common radius.
15 sub-problems · 14 open · 1 settled · 15 with a cited value
P09 · Equal circles in a half-discPlace n non-overlapping equal circles inside a half-disk of radius 1 to maximize their common radius.
14 sub-problems · 14 open · 0 settled · 14 with a cited value
P10 · Equal circles in a plus signPlace n non-overlapping equal circles inside a cross-shaped region to maximize their common radius.
16 sub-problems · 13 open · 3 settled · 3 with a cited value
P11 · Equal-circle packing in a right trianglePlace n non-overlapping equal circles inside a right triangle with legs 1 and 0.75 to maximize their common radius.
11 sub-problems · 8 open · 3 settled · 3 with a cited value
P12 · Equal-circle packing in a 2:1 rectanglePlace n non-overlapping equal circles inside a 2 × 1 rectangle to maximize their common radius.
98 sub-problems · 94 open · 4 settled · 98 with a cited value
P30 · Equal circles in a quadrantPlace n equal circles in a quarter-disc of radius 1 to maximize their common radius.
14 sub-problems · 14 open · 0 settled · 14 with a cited value
P57 · Sum of radii in the unit squarePlace n non-overlapping circles of any sizes in the unit square, maximizing the sum of their radii.
30 sub-problems · 29 open · 1 settled · 2 with a cited value
P66 · Unit circles in a minimum-area triangleChoose both the triangle shape and the circle arrangement to enclose n non-overlapping unit circles in a triangle of minimum area.
47 sub-problems · 47 open · 0 settled · 47 with a cited value
P67 · Variable-radius circles in a fixed-perimeter rectangleChoose both the rectangle's aspect ratio and the circle radii to pack n non-overlapping circles into a rectangle of perimeter 4, maximizing the sum of their radii.
28 sub-problems · 21 open · 7 settled · 28 with a cited value
P92 · Sum of radii in the unit circlePlace n non-overlapping circles of arbitrary radii inside a unit disc so as to maximise the sum of their radii. In Friedman's tables, this problem is listed under maximising total perimeter (2π times the sum of radii).
50 sub-problems · 48 open · 2 settled · 50 with a cited value
P93 · Sum of radii in the equilateral trianglePlace n non-overlapping circles of arbitrary radii inside a unit equilateral triangle so as to maximise the sum of their radii. In Friedman's tables, this problem is listed under maximising total perimeter (2π times the sum of radii).
50 sub-problems · 38 open · 12 settled · 30 with a cited value
P94 · Sum of radii in the regular pentagonPlace n non-overlapping circles of arbitrary radii inside a regular pentagon of side 1 so as to maximise the sum of their radii. In Friedman's tables, this problem is listed under maximising total perimeter (2π times the sum of radii).
50 sub-problems · 42 open · 8 settled · 50 with a cited value

Circles of radius 1…n

Circle radii are fixed at 1, 2, …, n; the goal is to arrange them so the enclosing container is as small as possible.

P06 · Packing circles of radius 1,2,…,n into a circlePack n non-overlapping circles of radii 1, 2, …, n into an enclosing circle and minimize its radius.
200 sub-problems · 196 open · 4 settled · 200 with a cited value
P13 · Packing circles of radius 1,2,…,n into a squarePack n non-overlapping circles of radii 1, 2, …, n into an enclosing square and minimize its side length.
99 sub-problems · 96 open · 3 settled · 99 with a cited value

Tilted squares

Allowing each square to rotate freely expands the search space from positions alone to positions and orientations, leaving most instances open.

P05 · Tilted equal squares in a circlePlace n equal squares with arbitrary orientations inside a circle of radius 1 to maximize their common side length.
38 sub-problems · 33 open · 5 settled · 38 with a cited value
P18 · Tilted equal squares in the unit squarePlace n equal squares with arbitrary orientations inside the unit square to maximize their common side length.
28 sub-problems · 18 open · 10 settled · 28 with a cited value

High-dimensional and abstract

Packings of directions, lines, or subspaces in higher dimensions. Verification still runs in exact integer arithmetic, benchmarked against classical spherical and Grassmannian packing tables.

P60 · Line packing in real projective spaceChoose n lines through the origin of Rᵈ so that the smallest angle between any two is as large as possible.
32 sub-problems · 31 open · 1 settled · 32 with a cited value
P61 · Codebook packing in complex projective spaceChoose n directions in complex d-space so that the largest pairwise coherence is as small as possible.
25 sub-problems · 24 open · 1 settled · 25 with a cited value
P64 · The most separated family of subspacesChoose n two-dimensional subspaces (planes) of Rᵈ so that the closest pair is as far apart as possible.
16 sub-problems · 16 open · 0 settled · 16 with a cited value
P65 · The most erasure-robust measurement directionsChoose n measurement directions so that the smallest normalized volume spanned by any d survivors is as large as possible.
11 sub-problems · 7 open · 4 settled · 4 with a cited value
P68 · Spherical codes: maximize the minimum anglePlace n directions on the unit sphere so that the smallest angle between any two is as large as possible.
17 sub-problems · 17 open · 0 settled · 17 with a cited value

Latest breakthroughs

condorP05 · Tilted equal squares in a circle · n = 15
0.394765492→0.394767126
condorP12 · Equal-circle packing in a 2:1 rectangle · n = 50
0.100284574→0.100355834
condorP54 · Minimum star discrepancy in the unit square · n = 42
0.031446541485732→0.031408308023611
condorP96 · Consecutive L shapes in a square · n = 48
346→345
condorP54 · Minimum star discrepancy in the unit square · n = 31
0.041055718530329→0.041023842931929
condorP96 · Consecutive L shapes in a square · n = 50
367→366

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Public datasets

Every problem family exposes a stable JSON endpoint containing all sub-problems, authoritative scores, proof statuses, coordinates, citations, and verifier versions:

GET https://minmaxarena.com/data/square-circle-packing.json

Published under CC BY 4.0 for free use with attribution. Because live scores update as records fall, please include the generatedAt timestamp when citing.