MinMax Arena
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PROBLEM CATALOG

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
P88Extremal configurations
  • Formed here

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
DRAGGABLE
P80Extremal configurations
  • Formed here

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
DRAGGABLE
P75Extremal configurations
  • Formed here
  • Applied frontier

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
DRAGGABLE
P71Extremal configurations
  • Formed here
  • Applied frontier
  • Weak baseline

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.

Instance
n = 1–20
Objective
Minimize squared-distance integral
P65Extremal configurations
  • Formed here
  • Applied frontier

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
P63Extremal configurations
  • Formed here
  • Applied frontier

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.

Instance
n = 1–12
Objective
Minimize the worst-case integration error
P62Extremal configurations
  • Formed here
  • Applied frontier

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
DRAGGABLE
P56Extremal configurations
  • Formed here
  • Applied frontier
  • Weak baseline

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.

Instance
n = 1–36
Objective
Minimize the uniformity M
DRAGGABLE
P32Extremal configurations
  • Formed here

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.

Instance
n = 1–10
Objective
Maximize the smallest triangle's area
DRAGGABLE
P31Extremal configurations
  • Formed here

Spreading points in a quadrant

Place n points in a quarter-disc of radius 1 to maximize the minimum distance between any two points.

Instance
n = 1–15
Objective
Maximize the smallest distance between two points
DRAGGABLE
P28Extremal configurations
  • Formed here

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.

Instance
n = 1–10
Objective
Maximize the smallest triangle's area
DRAGGABLE
P26Extremal configurations
  • Formed here

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.

Instance
n = 1–10
Objective
Maximize the smallest triangle's area
DRAGGABLE
P25Extremal configurations
  • Formed here

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.

Instance
n = 1–10
Objective
Maximize the smallest triangle's area
DRAGGABLE
P24Extremal configurations
  • Formed here

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.

Instance
n = 1–10
Objective
Maximize the smallest triangle's area
DRAGGABLE
P21Extremal configurations
  • Formed here

Spreading points in a plus sign

Place n points in a cross-shaped region to maximize the minimum pairwise Euclidean distance.

Instance
n = 1–17
Objective
Maximize the smallest distance between two points
DRAGGABLE
P20Extremal configurations
  • Formed here

Spreading points in a half-disc

Place n points in a half-disk of radius 1 to maximize the minimum pairwise Euclidean distance.

Instance
n = 1–15
Objective
Maximize the smallest distance between two points
DRAGGABLE
P19Extremal configurations
  • Formed here

Spreading points in an L

Place n points in an L-shaped region to maximize the minimum pairwise Euclidean distance.

Instance
n = 1–17
Objective
Maximize the smallest distance between two points
DRAGGABLE
P17Extremal configurations
  • Formed here

Spreading points in a rectangle

Place n points in a 2 × 1 rectangle to maximize the minimum pairwise Euclidean distance.

Instance
n = 1–17
Objective
Maximize the smallest distance between two points
DRAGGABLE
P10Packing and covering
  • Formed here

Equal circles in a plus sign

Place n non-overlapping equal circles inside a cross-shaped region to maximize their common radius.

Instance
n = 1–16
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.

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.

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).