P27 · Spreading points in an annulus
Frozen since 2026-08-27 · This narrow annulus may degenerate over a wide range of n, making evenly spaced single-ring configurations provably optimal. Frozen pending a degeneracy audit.
72 verifiable
mathematical arenas
TopicPacking ProblemsThe hub for all packing families: categories, open sub-problems, verification rules, and downloadable datasets.
TopicCovering ProblemsSix families, 127 instances, cited constructions, and interactive editors in one covering hub.
DRAGGABLE
P96Packing and covering
Consecutive L shapes in a square
Pack n L shapes with short sides 1, 2, …, n into a square without overlap. Each L consists of three equal squares. Translate and rotate the pieces freely to minimise the container side.
- Instance
- n = 1–50
- Objective
- Minimize the square's side
P95Packing and covering
Consecutive dominoes in a square
Pack n dominoes with short sides 1, 2, …, n (the k-th is a k × 2k rectangle) without overlap into a square, making the square's side as small as possible. Each domino may be moved, laid flat or stood up, and turned through any angle.
- Instance
- n = 1–41
- Objective
- Minimize the square's side
P94Packing and covering
Sum of radii in the regular pentagon
Place 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).
1
- Instance
- n = 1–50
- Objective
- Maximize the sum of the radii
P93Packing and covering
Sum of radii in the equilateral triangle
Place 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).
1
- Instance
- n = 1–50
- Objective
- Maximize the sum of the radii
P92Packing and covering
Sum of radii in the unit circle
Place 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).
1
- Instance
- n = 1–50
- Objective
- Maximize the sum of the radii
P91Packing and covering
L's in a square
Pack n L-trominoes (each formed by three unit squares) into a square container with boundary contact allowed and disjoint interiors, minimising the side length of the container square.
- Instance
- n = 1–40
- Objective
- Minimize container square side
P90Packing and covering
Dominoes in a square
Pack n 1 × 2 dominoes into a square container with boundary contact allowed and disjoint interiors, minimising the side length of the container square.
- Instance
- n = 1–40
- Objective
- Minimize container square side
P89Packing and covering
Tans in a square
Pack n right isosceles triangles of leg 1 (tans) into a square container with boundary contact allowed and disjoint interiors, minimising the side length of the container square.
- Instance
- n = 1–24
- Objective
- Minimize container square side
P88Extremal configurations
The sofa in winding hallways
Design a polygonal sofa of maximum area that can move continuously through a winding hallway of width 1. Four hallway geometries are included: a zigzag (two opposite right-angle turns connected by a straight run of length d), a double U-turn around two thin walls, a three-corner staircase, and a hairpin with a lateral jog along its vertical connector.
- Instance
- n = 1–10
- Objective
- Maximize the area of the sofa
P87Packing and covering
Equal discs covering a disc
Cover a circular disc with n unit-radius discs, maximising the radius of the covered disc.
- Instance
- n = 1–90
- Objective
- Maximize covered disc radius
P86Packing and covering
Regular pentagons in a square
Pack n unit-side regular pentagons into a square container with boundary contact allowed and disjoint interiors, minimising the side length of the container square.
- Instance
- n = 1–28
- Objective
- Minimize container square side
P85Packing and covering
Squares in an equilateral triangle
Pack n unit squares into an equilateral triangle container with boundary contact allowed and disjoint interiors, minimising the side length of the container triangle.
- Instance
- n = 1–38
- Objective
- Minimize container triangle side
P84Packing and covering
Equilateral triangles in a disc
Pack n unit-side equilateral triangles into a circular disc with boundary contact allowed and disjoint interiors, minimising the radius of the container disc.
- Instance
- n = 1–35
- Objective
- Minimize container circle radius
P83Packing and covering
Equilateral triangles in a square
Pack n unit-side equilateral triangles into a square container with boundary contact allowed and disjoint interiors, minimising the side length of the container square.
- Instance
- n = 1–38
- Objective
- Minimize container square side
P82Packing and covering
Equal circles in regular polygons
Pack n equal circles into the regular m-gon with centre (1,1), circumradius 1, and first vertex (2,1) so as to maximise their common radius.
- Instance
- n = 1–248
- Objective
- Maximize circle radius
P81Packing and covering
Irregular strip nesting
Pack a batch of polygonal industrial parts into a strip of fixed height H using only translations and 90° rotations. Pieces may touch without overlapping interiors; minimise the occupied strip length L.
- Instance
- n = 1–3
- Objective
- Minimize occupied length
P80Extremal configurations
The sofa in a hairpin
Design a polygonal sofa of maximum area that can move continuously through a hairpin hallway of width 1 (two consecutive left-hand right-angle turns separated by a wall of thickness w).
- Instance
- n = 1–8
- Objective
- Maximize the area of the sofa
P78Extremal configurations
Covering a disc with n equilateral triangles
Cover the unit disc with n congruent equilateral triangles, each independently translated, rotated, and flipped, to minimise their common side.
- Instance
- n = 1–12
- Objective
- Minimize the common side
P77Extremal configurations
Covering an equilateral triangle with n equilateral triangles
Cover the unit equilateral triangle with n congruent equilateral triangles, each independently translated, rotated, and flipped, to minimise their common side.
- Instance
- n = 1–10
- Objective
- Minimize the common side
P76Extremal configurations
Covering a square with n equilateral triangles
Cover the unit square with n congruent equilateral triangles. Each piece may be translated, rotated, and flipped; overlap and protrusion are allowed. Minimise the common side.
- Instance
- n = 1–11
- Objective
- Minimize the common side
P75Extremal configurations
Covering a regular pentagon with n equal discs
Place n centres in a regular pentagon to minimise the common radius required to cover the entire container.
- Instance
- n = 1–29
- Objective
- Minimize the covering radius
P74Extremal configurations
Covering an equilateral triangle with n equal discs
Place n centres in a unit equilateral triangle and minimise the common radius needed to cover every point of it.
- Instance
- n = 1–34
- Objective
- Minimize the covering radius
P73Extremal configurations
Covering a square with n equal discs
Place n circle centres in the unit square so as to minimize the shared radius needed for the n equal discs to cover the entire square.
12345
- Instance
- n = 1–31
- Objective
- Minimize the covering radius
P72Extremal configurations
Heilbronn minimum tetrahedron volume
Place n points in the unit cube and maximize the smallest tetrahedron volume among all quadruples.
- Instance
- n = 1–12
- Objective
- Maximize minimum tetrahedron volume
P71Extremal configurations
Optimal quantization in an L-shaped region
Place n representative points in an L-shaped region, minimizing the integral of squared distance to the nearest representative over the region.
12345
- Instance
- n = 1–20
- Objective
- Minimize squared-distance integral
P70Extremal configurations
Four-phase radar probe codebook
Design five length-N four-phase (QPSK) sequences for a five-channel MIMO radar so as to lexicographically minimize the peak squared aperiodic correlation sidelobe, its multiplicity, and the total sidelobe energy.
- Instance
- n = 1–9
- Objective
- Minimize peak² / ties / total energy (lexicographic)
P68Extremal configurations
Spherical codes: maximize the minimum angle
Place n directions on the unit sphere so that the smallest angle between any two is as large as possible.
- Instance
- n = 1–17
- Objective
- Minimize largest inner product
P67Packing and covering
Variable-radius circles in a fixed-perimeter rectangle
Choose 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.
123
- Instance
- n = 1–28
- Objective
- Maximize sum of radii
P66Packing and covering
Unit circles in a minimum-area triangle
Choose both the triangle shape and the circle arrangement to enclose n non-overlapping unit circles in a triangle of minimum area.
- Instance
- n = 1–47
- Objective
- Minimize the triangle area
P65Extremal configurations
The most erasure-robust measurement directions
Choose n measurement directions so that the smallest normalized volume spanned by any d survivors is as large as possible.
- Instance
- n = 1–11
- Objective
- Maximize the worst subset's normalized volume
P64Extremal configurations
The most separated family of subspaces
Choose n two-dimensional subspaces (planes) of Rᵈ so that the closest pair is as far apart as possible.
- Instance
- n = 1–16
- Objective
- Maximize the smallest squared chordal distance
P63Extremal configurations
Optimal quadrature points on the torus
Place n equal-weight sample points on the d-dimensional torus, minimizing the worst-case integration error over a class of periodic functions.
12345678
- Instance
- n = 1–12
- Objective
- Minimize the worst-case integration error
P62Extremal configurations
A sampling design uniform in every pair of columns
Design an n-row, d-column sampling table so that every two-column scatter plot is as uniform as possible.
x1 × x2WORST
x1 × x3
x1 × x4
x2 × x3
x2 × x4
x3 × x4
- Instance
- n = 1–9
- Objective
- Minimize the least-uniform pair's discrepancy
P61Extremal configurations
Codebook packing in complex projective space
Choose n directions in complex d-space so that the largest pairwise coherence is as small as possible.
- Instance
- n = 1–25
- Objective
- Minimize the largest coherence μ
P60Extremal configurations
Line packing in real projective space
Choose n lines through the origin of Rᵈ so that the smallest angle between any two is as large as possible.
- Instance
- n = 1–32
- Objective
- Minimize the largest coherence μ
P59Extremal configurations
Minimum L2-star discrepancy in the unit hypercube
Place n sample points in the d-dimensional unit hypercube, making the mean-squared distribution error over all origin-anchored boxes as small as possible.
12345678
- Instance
- n = 1–25
- Objective
- Minimize L2-star discrepancy
P58Extremal configurations
The smallest triangle in an equilateral triangle
Place n points in an equilateral triangle of side 1 so the smallest triangle any three of them make is as large as possible.
12345
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
P57Packing and covering
Sum of radii in the unit square
Place n non-overlapping circles of any sizes in the unit square, maximizing the sum of their radii.
1
- Instance
- n = 1–30
- Objective
- Maximize sum of the radii
P56Extremal configurations
The most uniform sampling mesh in the unit square
Place n points in the unit square; the red circle is the largest uncovered hole and the blue line the closest pair. Make the ratio of hole radius to pair spacing as small as you can.
12345
- Instance
- n = 1–36
- Objective
- Minimize the uniformity M
P55Extremal configurations
Optimal quantization in the unit square
Place n points in the unit square to minimize the mean squared distance from a uniformly chosen point in the square to its nearest submitted point.
123456
- Instance
- n = 1–25
- Objective
- Minimize the average squared distance
P54Extremal configurations
Minimum star discrepancy in the unit square
Place n points in the unit square to minimize D*, the maximum discrepancy between the fraction of points in an origin-anchored axis-aligned rectangle and its area.
12345678910111213141516171819202122
- Instance
- n = 1–29
- Objective
- Minimize the worst gap D*
P53Extremal configurations
The biggest little polygon
Choose n points in the plane with pairwise distances at most 1 to maximize the area of the convex n-gon they form.
123456
- Instance
- n = 1–13
- Objective
- Maximize the area
P52Extremal configurations
Smallest ratio of largest to smallest distance
Place n points in the plane to minimize the ratio of the maximum pairwise distance to the minimum pairwise distance.
123456789
- Instance
- n = 1–20
- Objective
- Minimize max-to-min distance ratio
P51Extremal configurations
Lighting a unit square
Place n unit-brightness point lights in the unit square to maximize the intensity at the darkest point.
12
- Instance
- n = 1–34
- Objective
- Maximize minimum intensity
P34Extremal configurations
Riesz 2-energy in a disc
Place n points in a disc of radius 1 to minimize the sum of 1/distance² over all pairs.
123456789
- Instance
- n = 1–16
- Objective
- Minimize Riesz 2-energy
P33Extremal configurations
Riesz 2-energy in a square
Place n points in the unit square to minimize the sum of 1/distance² over all pairs.
12345
- Instance
- n = 1–20
- Objective
- Minimize Riesz 2-energy
P32Extremal configurations
The smallest triangle in a quadrant
Place n points in a quarter-disc of radius 1 to maximize the minimum triangle area formed by any three points.
12345
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
P31Extremal configurations
Spreading points in a quadrant
Place n points in a quarter-disc of radius 1 to maximize the minimum distance between any two points.
1234
- Instance
- n = 1–15
- Objective
- Maximize the smallest distance between two points
P30Packing and covering
Equal circles in a quadrant
Place n equal circles in a quarter-disc of radius 1 to maximize their common radius.
123
- Instance
- n = 1–14
- Objective
- Maximize common radius
P29Extremal configurations
Heilbronn's problem in a triangle
Place n points in the right triangle with vertices (0, 0), (1, 0), and (0, 1) to maximize the minimum triangle area formed by any three points.
12345
- Instance
- n = 1–6
- Objective
- Maximize the smallest triangle's area
P28Extremal configurations
The smallest triangle in an annulus
Place n points inside an annulus with outer radius 1 and inner radius 0.5 so that the smallest triangle formed by any three points has the largest possible area.
12345
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
P26Extremal configurations
The smallest triangle in a half-disc
Place n points inside a semicircle of radius 1 so that the smallest triangle formed by any three points has the largest possible area.
12345
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
P25Extremal configurations
The smallest triangle in a plus sign
Place n points inside a cross-shaped region so that the smallest triangle formed by any three points has the largest possible area.
12345
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
P24Extremal configurations
The smallest triangle in an L
Place n points inside an L-shaped region so that the smallest triangle formed by any three points has the largest possible area.
12345
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
P22Extremal configurations
The smallest triangle in a disc
Place n points in a disk of radius 1 to maximize the minimum triangle area over all point triples.
12345
- Instance
- n = 1–10
- Objective
- Maximize the smallest triangle's area
P21Extremal configurations
Spreading points in a plus sign
Place n points in a cross-shaped region to maximize the minimum pairwise Euclidean distance.
1234
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
P20Extremal configurations
Spreading points in a half-disc
Place n points in a half-disk of radius 1 to maximize the minimum pairwise Euclidean distance.
1234
- Instance
- n = 1–15
- Objective
- Maximize the smallest distance between two points
P19Extremal configurations
Spreading points in an L
Place n points in an L-shaped region to maximize the minimum pairwise Euclidean distance.
1234
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
P18Packing and covering
Tilted equal squares in the unit square
Place n equal squares with arbitrary orientations inside the unit square to maximize their common side length.
- Instance
- n = 1–28
- Objective
- Maximize the smallest side
P17Extremal configurations
Spreading points in a rectangle
Place n points in a 2 × 1 rectangle to maximize the minimum pairwise Euclidean distance.
1234
- Instance
- n = 1–17
- Objective
- Maximize the smallest distance between two points
P16Extremal configurations
Spreading points in a right triangle
Place n points in a right isosceles triangle with legs of length 1 to maximize the minimum pairwise Euclidean distance.
1234
- Instance
- n = 1–97
- Objective
- Maximize the smallest distance between two points
P13Packing and covering
Packing circles of radius 1,2,…,n into a square
Pack n non-overlapping circles of radii 1, 2, …, n into an enclosing square and minimize its side length.
12
- Instance
- n = 1–99
- Objective
- Minimize square side
P12Packing and covering
Equal-circle packing in a 2:1 rectangle
Place n non-overlapping equal circles inside a 2 × 1 rectangle to maximize their common radius.
123
- Instance
- n = 1–98
- Objective
- Maximize common radius
P11Packing and covering
Equal-circle packing in a right triangle
Place n non-overlapping equal circles inside a right triangle with legs 1 and 0.75 to maximize their common radius.
12
- Instance
- n = 1–11
- Objective
- Maximize common radius
P10Packing and covering
Equal circles in a plus sign
Place n non-overlapping equal circles inside a cross-shaped region to maximize their common radius.
123
- Instance
- n = 1–16
- Objective
- Maximize common radius
P09Packing and covering
Equal circles in a half-disc
Place n non-overlapping equal circles inside a half-disk of radius 1 to maximize their common radius.
123
- Instance
- n = 1–14
- Objective
- Maximize common radius
P08Packing and covering
Equal circles in an L
Place n non-overlapping equal circles inside an L-shaped region to maximize their common radius.
1234
- Instance
- n = 1–15
- Objective
- Maximize common radius
P06Packing and covering
Packing circles of radius 1,2,…,n into a circle
Pack n non-overlapping circles of radii 1, 2, …, n into an enclosing circle and minimize its radius.
1
- Instance
- n = 1–200
- Objective
- Minimize container radius
P05Packing and covering
Tilted equal squares in a circle
Place n equal squares with arbitrary orientations inside a circle of radius 1 to maximize their common side length.
- Instance
- n = 1–38
- Objective
- Maximize the smallest side
P03Extremal configurations
Heilbronn minimum triangle area
Place n points in the unit square to maximize the minimum triangle area over all point triples.
12345
- Instance
- n = 1–12
- Objective
- Maximize minimum triangle area
P02Packing and covering
Equal-circle packing in a unit circle
Place n non-overlapping equal circles inside the unit circle to maximize their common radius.
1
- Instance
- n = 1–300
- Objective
- Maximize common radius
P01Packing and covering
Equal-circle packing in a unit square
Place n non-overlapping equal circles inside the unit square to maximize their common radius.
1
- Instance
- n = 1–300
- Objective
- Maximize common radius
Frozen problems
These problems are closed to new submissions for the reasons noted below. Their records and attributions stay as history and, since 2026-10-04, earn no contribution points.
P07 · Spreading points in a circle
Frozen since 2026-08-28 · Mathematically equivalent to P02 (equal circles in a disc): the packing radius r and minimum point separation d are related by the strictly monotone bijection d = 2r/(1 − r), so any solution to one converts directly into a solution to the other. The catalogue retains P02, the standard formulation in the literature.
P15 · Spreading points in the unit square
Frozen since 2026-08-28 · Mathematically equivalent to P01 (equal circles in the unit square) under d = 2r/(1 − 2r); Specht's reference table lists both quantities as columns of the same table. The catalogue retains the circle-packing formulation (P01).