Covering a regular pentagon with n equal discs
Place the centres of n equal-radius discs in a regular pentagon so their union covers the entire polygon, 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 maximum distance from any point in the container to its nearest centre, determined exactly by finitely many rational Voronoi vertices
- ObjectiveMinimise the covering radius r(P)
Getting a feel for it
Pentagonal boundary and asymmetric partitions
The boundary has five vertices, whereas n is generally not a multiple of five. Additional centres break fivefold symmetry and reorganise the interior Voronoi cells; as n varies, both the farthest uncovered corners and the adjacency topology change.
Known results and baselines
Liu (2022) studied the continuous p-center problem in a regular pentagon and reported configurations for n = 3–10. Only n = 5 closes the gap between upper and lower bounds; after two hours of computation, n = 7, 8, 9, and 10 leave relative gaps of 5.87%, 5.75%, 6.35%, and 10.24%. This catalogue reconstructs n = 7–10 from the vector graphics in Figure 9 and records their exact verifier radii as known bests. For n = 11–35, where no reproducible external coordinates are available, the table provides multi-start search baselines.
SourceCurrent best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (1) ↓Data and citation
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/circles-covering-a-regular-pentagon.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/circles-covering-a-regular-pentagon.json
BibTeX (click to copy)
@misc{minmaxarena-circles-covering-a-regular-pentagon-2026-10,
title = {{Covering a regular pentagon with n equal discs} (P75)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/circles-covering-a-regular-pentagon.json},
license = {CC BY 4.0}
}Discussion
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Grouped by player and sub-problem, showing the latest note first. Earlier versions and replies are preserved; click any note to read or reply. Notes can be published alongside record-breaking submissions.