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
P88
  • 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
P80
  • 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
P78
  • Classic

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
DRAGGABLE
P77
  • Classic

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
DRAGGABLE
P76
  • Classic

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
DRAGGABLE
P75
  • 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
P74
  • Classic
  • Applied frontier

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
DRAGGABLE
P73
  • Classic

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.

Instance
n = 1–31
Objective
Minimize the covering radius
DRAGGABLE
P72
  • Classic
  • Weak baseline

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
DRAGGABLE
P71
  • 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
DRAGGABLE
P70
  • Classic
  • Applied frontier
  • Weak baseline

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)
DRAGGABLE
P68
  • Classic
  • Applied frontier

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
P65
  • 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
P64
  • Classic
  • Applied frontier

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
P63
  • 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
P62
  • 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
P61
  • Classic
  • Applied frontier

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
P60
  • Classic
  • Applied frontier

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
P59
  • Classic
  • Applied frontier

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.

Instance
n = 1–25
Objective
Minimize L2-star discrepancy
DRAGGABLE
P58
  • Classic

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.

Instance
n = 1–10
Objective
Maximize the smallest triangle's area
DRAGGABLE
P56
  • 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
P55
  • Classic
  • Applied frontier
  • Weak baseline

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.

Instance
n = 1–25
Objective
Minimize the average squared distance
DRAGGABLE
P54
  • Classic
  • Applied frontier
  • Weak baseline

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.

Instance
n = 1–29
Objective
Minimize the worst gap D*
DRAGGABLE
P53
  • Classic
  • Applied frontier
  • Weak baseline

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.

Instance
n = 1–13
Objective
Maximize the area
DRAGGABLE
P52
  • Classic

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.

Instance
n = 1–20
Objective
Minimize max-to-min distance ratio
DRAGGABLE
P51
  • Classic

Lighting a unit square

Place n unit-brightness point lights in the unit square to maximize the intensity at the darkest point.

Instance
n = 1–34
Objective
Maximize minimum intensity
DRAGGABLE
P34
  • Classic

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.

Instance
n = 1–16
Objective
Minimize Riesz 2-energy
DRAGGABLE
P33
  • Classic
  • Weak baseline

Riesz 2-energy in a square

Place n points in the unit square to minimize the sum of 1/distance² over all pairs.

Instance
n = 1–20
Objective
Minimize Riesz 2-energy
DRAGGABLE
P32
  • 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
P31
  • 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
P29
  • Classic

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.

Instance
n = 1–6
Objective
Maximize the smallest triangle's area
DRAGGABLE
P28
  • 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
P26
  • 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
P25
  • 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
P24
  • 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
P22
  • Classic

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.

Instance
n = 1–10
Objective
Maximize the smallest triangle's area
DRAGGABLE
P21
  • 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
P20
  • 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
P19
  • 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
P17
  • 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
P16
  • Classic

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.

Instance
n = 1–97
Objective
Maximize the smallest distance between two points
DRAGGABLE
P03
  • Classic

Heilbronn minimum triangle area

Place n points in the unit square to maximize the minimum triangle area over all point triples.

Instance
n = 1–12
Objective
Maximize minimum triangle area

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).