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.
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P96
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
P95
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
P94
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
P93
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
P92
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
P91
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
P90
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
P89
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
P87
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
P86
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
P85
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
P84
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
P83
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
P82
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
P81
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
P67
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
P66
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
P57
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
P30
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
P18
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
P13
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
P12
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
P11
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
P10
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
P09
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
P08
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
P06
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
P05
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
P02
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
P01
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).