← Problem catalogCOVERING PROBLEMS

Covering Problems

Cover a target container completely with congruent shapes without leaving any uncovered gaps, while minimizing their shared size. This hub collects all continuous-covering families on the site.

Families6

Sub-problems127

Still open122

With cited constructions92

Covering vs. packing

Packing forbids overlap and protrusion while maximizing size; covering allows both overlap and protrusion, requires every point of the container to be covered by at least one piece, and minimizes shared size. Boundaries, corners, and tiny gaps between pieces typically determine the score.

The verifier never relies on pixel sampling. For disc coverings, it enumerates the finite set of Voronoi vertices and boundary intersections that can realize the maximum distance; for polygon coverings, it constructs the exact planar arrangement of edges and intersections. A finite coordinate certificate therefore rigorously proves continuous coverage.

Six families, six independent frontiers

P73Covering a square with n equal discs

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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.

31 sub-problems · 29 open · 26 with external sources

P74Covering 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.

34 sub-problems · 32 open · 30 with external sources

P75Covering 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.

29 sub-problems · 29 open · 4 with external sources

P76Covering 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.

11 sub-problems · 11 open · 11 with external sources

P77Covering 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.

10 sub-problems · 9 open · 9 with external sources

P78Covering 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.

12 sub-problems · 12 open · 12 with external sources