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
P81Packing and covering
  • Classic
  • Applied frontier

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
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
P74Extremal configurations
  • 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
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
DRAGGABLE
P70Extremal configurations
  • 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
P68Extremal configurations
  • 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
P67Packing and covering
  • Classic
  • Applied frontier

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.

Instance
n = 1–28
Objective
Maximize sum of radii
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
P64Extremal configurations
  • 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
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
P61Extremal configurations
  • 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
P60Extremal configurations
  • 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
P59Extremal configurations
  • 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
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
P55Extremal configurations
  • 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
P54Extremal configurations
  • 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
P53Extremal configurations
  • 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

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).