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
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
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
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
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
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
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
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
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
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).