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
P81
  • 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
P67
  • 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

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).