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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P88
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
P80
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
P75
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
P71
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
P65
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
P63
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
P62
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
P56
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
P32
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
P31
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
P28
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
P26
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
P25
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
P24
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
P21
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
P20
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
P19
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
P17
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
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).