Covering a regular pentagon with n equal discs
Place the centres of n equal-radius discs in a regular pentagon so their union covers it completely, and minimise the shared radius.
Formal definition
- ContainerThe closed convex polygon with successive vertices (0.5,1), (0.024471742,0.654508497), (0.206107374,0.095491503), (0.793892626,0.095491503), and (0.975528258,0.654508497)
- SubmissionExactly n distinct points inside the container
- ScoreThe distance from the hardest-to-cover place to its nearest centre, determined exactly by finitely many rational Voronoi vertices
- ObjectiveMake the covering radius as small as possible
Getting a feel for it
Fivefold symmetry need not survive
The boundary has five corners, but n is usually not a multiple of five. Extra centres reshape the interior Voronoi cells; as n changes, the farthest holes and adjacency topology change as well, so there is no obvious periodic motif that can simply be repeated.
Published work settles only one small case
Liu (2022) studies the continuous p-center problem in a regular pentagon and publicly illustrates n=3–10. Only n=5 closes the upper/lower-bound gap; after two hours of computation, n=7, 8, 9, and 10 still leave gaps of 5.87%, 5.75%, 6.35%, and 10.24%. The site reconstructs n=7–10 point by point from the vector objects in Figure 9 and uses their verifier-exact radii as known bests. No publicly reproducible external configurations were found for n=11–35, so those rows carry only transparent site multi-start baselines, not invented SOTA claims.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Data and citation
Every sub-problem in this family — authoritative scores, proof status, coordinates and sources — lives at the stable address below, published under CC BY 4.0. Scores move as records fall, so cite the generatedAt timestamp the file carries.
GET https://minmaxarena.com/data/circles-covering-a-regular-pentagon.json
Cite the frozen 2026-08 edition: records move, a frozen edition never does, so the citation is still checkable years later.
GET https://minmaxarena.com/data/editions/2026-08/circles-covering-a-regular-pentagon.json
BibTeX (click to copy)
@misc{minmaxarena-circles-covering-a-regular-pentagon-2026-08,
title = {{Covering a regular pentagon with n equal discs} (P75)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-08 edition},
url = {https://minmaxarena.com/data/editions/2026-08/circles-covering-a-regular-pentagon.json},
license = {CC BY 4.0}
}Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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