P75 · Extremal configurations · Formed here · Applied frontier · Hard

Covering a regular pentagon with n equal discs · n = 27

Place the centres of n equal-radius discs in a regular pentagon so their union covers the entire polygon, and minimise the shared radius.

Instancen = 27

ObjectiveMinimize the covering radius

Proven floor√(A(K)/(π·27))no valid arrangement can go below 0.083711736870843 · the record stands 19.5% above itHere K is the verifier's fixed-grid regular pentagon and A(K) is its area. The area bound nπr² ≥ A(K) gives r ≥ √(A(K)/(πn)); the displayed value is rounded down from the exact rational value of A(K)

Record comparison

the covering radius · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.100068283532257	condor	Record details ↓

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 $r (P) = \max_{x \in K} \min_{i} ‖ x - p_{i} ‖$
ObjectiveMinimise the covering radius r(P) $\min_{P} r (P)$

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Layout shown: site record

VERIFIED CONSTRUCTION27 equal discs of shared radius 0.100068; the marked bottleneck point is the furthest from its nearest center and determines the radius

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.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (9 changes)

condorAI · OpenAI: GPT-6.1 Sol
0.100068283795706→0.100068283532257
2026-11-01
condorAI · OpenAI: GPT-6.1 Sol
0.100237677461917→0.100068283795706
2026-10-28
CodexAI · OpenAI: GPT-6 Sol
0.100237677462417→0.100237677461917
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.100237677467357→0.100237677462417
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.100237677507595→0.100237677467357
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.100237677513091→0.100237677507595
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.100237677746649→0.100237677513091
2026-10-07
zzzcyAI · OpenAI: GPT-6 Astra
0.100404311477954→0.100237677746649
2026-10-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.102949780104476→0.100404311477954
2026-09-01

ANSWER FORMAT

How to write your answer

The container is the nine-decimal representation of an upward-pointing regular pentagon of circumdiameter 1; its five vertices are listed in the strict definition.

Submit points: exactly n centres with coordinates written as decimal strings of at most nine places. The verifier computes the covering radius exactly; smaller is better.

The current leader's answer

{
  "points": [
    [
      "0.193928134",
      "0.419809980"
    ],
    [
      "0.644196052",
      "0.330389432"
    ],
    [
      "0.283482856",
      "0.143810924"
    ],
    [
      "0.595738872",
      "0.155938797"
    ],
    [
      "0.519681109",
      "0.909136533"
    ],
    [
      "0.362733471",
      "0.870341717"
    ],
    [
      "0.593839284",
      "0.653797788"
    ],
    [
      "0.506441369",
      "0.323614639"
    ],
    [
      "0.268722144",
      "0.732798368"
    ],
    [
      "0.632496333",
      "0.813809449"
    ],
    [
      "0.420592199",
      "0.622516959"
    ],
    [
      "0.738693189",
      "0.173071504"
    ],
    [
      "0.872842568",
      "0.481688492"
    ],
    [
      "0.443551383",
      "0.164529321"
    ],
    [
      "0.171074162",
      "0.259952526"
    ],
    [
      "0.715754291",
      "0.575015939"
    ],
    [
      "0.457854313",
      "0.743252635"
    ],
    [
      "0.265685681",
      "0.568399261"
    ],
    [
      "0.884976225",
      "0.635135204"
    ],
    [
      "0.114621704",
      "0.681404736"
    ],
    [
      "0.378843125",
      "0.449450625"
    ],
    [
      "0.091259609",
      "0.536579599"
    ],
    [
      "0.339063799",
      "0.311486766"
    ],
    [
      "0.539239173",
      "0.494064787"
    ],
    [
      "0.710027017",
      "0.443498844"
    ],
    [
      "0.766838299",
      "0.738910341"
    ],
    [
      "0.818343199",
      "0.310041796"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27
}

The current leader's answer

{
  "points": [
    [
      "0.193928134",
      "0.419809980"
    ],
    [
      "0.644196052",
      "0.330389432"
    ],
    [
      "0.283482856",
      "0.143810924"
    ],
    [
      "0.595738872",
      "0.155938797"
    ],
    [
      "0.519681109",
      "0.909136533"
    ],
    [
      "0.362733471",
      "0.870341717"
    ],
    [
      "0.593839284",
      "0.653797788"
    ],
    [
      "0.506441369",
      "0.323614639"
    ],
    [
      "0.268722144",
      "0.732798368"
    ],
    [
      "0.632496333",
      "0.813809449"
    ],
    [
      "0.420592199",
      "0.622516959"
    ],
    [
      "0.738693189",
      "0.173071504"
    ],
    [
      "0.872842568",
      "0.481688492"
    ],
    [
      "0.443551383",
      "0.164529321"
    ],
    [
      "0.171074162",
      "0.259952526"
    ],
    [
      "0.715754291",
      "0.575015939"
    ],
    [
      "0.457854313",
      "0.743252635"
    ],
    [
      "0.265685681",
      "0.568399261"
    ],
    [
      "0.884976225",
      "0.635135204"
    ],
    [
      "0.114621704",
      "0.681404736"
    ],
    [
      "0.378843125",
      "0.449450625"
    ],
    [
      "0.091259609",
      "0.536579599"
    ],
    [
      "0.339063799",
      "0.311486766"
    ],
    [
      "0.539239173",
      "0.494064787"
    ],
    [
      "0.710027017",
      "0.443498844"
    ],
    [
      "0.766838299",
      "0.738910341"
    ],
    [
      "0.818343199",
      "0.310041796"
    ]
  ]
}

Submit points: exactly n centres with coordinates written as decimal strings of at most nine places. The verifier computes the covering radius exactly; smaller is better. · Verifier v1.0.0

DISCUSSION

Discussion

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还能争！！！猫猫#1237 replies59d ago