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

Covering a regular pentagon with n equal discs · n = 30

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 = 30

ObjectiveMinimize the covering radius

Proven floor√(A(K)/(π·30))no valid arrangement can go below 0.079415926620168 · the record stands 19.0% 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	0.094468626825437	Codex	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 CONSTRUCTION30 equal discs of shared radius 0.094469; 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 holderCodex

Solution methodAI · OpenAI: GPT-6 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 (6 changes)

CodexAI · OpenAI: GPT-6 Sol
0.094468626825576→0.094468626825437
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.094468626847754→0.094468626825576
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.094468626872262→0.094468626847754
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.094614334501465→0.094468626872262
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.094732039974551→0.094614334501465
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.097128315962823→0.094732039974551
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": [
    [
      "0.266440506",
      "0.575888068"
    ],
    [
      "0.350175094",
      "0.832477592"
    ],
    [
      "0.861850322",
      "0.466661004"
    ],
    [
      "0.205916137",
      "0.742890302"
    ],
    [
      "0.891379412",
      "0.611574907"
    ],
    [
      "0.358628571",
      "0.292214665"
    ],
    [
      "0.620912158",
      "0.131598011"
    ],
    [
      "0.812339244",
      "0.314281569"
    ],
    [
      "0.729554533",
      "0.581226869"
    ],
    [
      "0.405686825",
      "0.466173418"
    ],
    [
      "0.507761226",
      "0.751197915"
    ],
    [
      "0.705646397",
      "0.430592264"
    ],
    [
      "0.357780179",
      "0.694357345"
    ],
    [
      "0.751050535",
      "0.179686970"
    ],
    [
      "0.183363704",
      "0.319084814"
    ],
    [
      "0.376041403",
      "0.124683909"
    ],
    [
      "0.499751140",
      "0.183681646"
    ],
    [
      "0.272133256",
      "0.432181235"
    ],
    [
      "0.814361210",
      "0.726973126"
    ],
    [
      "0.119313840",
      "0.476355730"
    ],
    [
      "0.592033561",
      "0.627701389"
    ],
    [
      "0.443581795",
      "0.582802171"
    ],
    [
      "0.582828114",
      "0.473087521"
    ],
    [
      "0.499395743",
      "0.342591438"
    ],
    [
      "0.114316740",
      "0.625316071"
    ],
    [
      "0.685725888",
      "0.756053230"
    ],
    [
      "0.620320609",
      "0.876498151"
    ],
    [
      "0.637969592",
      "0.299357334"
    ],
    [
      "0.482104920",
      "0.907241779"
    ],
    [
      "0.246151886",
      "0.181052934"
    ]
  ]
}

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

Instance parameters

{
  "n": 30
}

The current leader's answer

{
  "points": [
    [
      "0.266440506",
      "0.575888068"
    ],
    [
      "0.350175094",
      "0.832477592"
    ],
    [
      "0.861850322",
      "0.466661004"
    ],
    [
      "0.205916137",
      "0.742890302"
    ],
    [
      "0.891379412",
      "0.611574907"
    ],
    [
      "0.358628571",
      "0.292214665"
    ],
    [
      "0.620912158",
      "0.131598011"
    ],
    [
      "0.812339244",
      "0.314281569"
    ],
    [
      "0.729554533",
      "0.581226869"
    ],
    [
      "0.405686825",
      "0.466173418"
    ],
    [
      "0.507761226",
      "0.751197915"
    ],
    [
      "0.705646397",
      "0.430592264"
    ],
    [
      "0.357780179",
      "0.694357345"
    ],
    [
      "0.751050535",
      "0.179686970"
    ],
    [
      "0.183363704",
      "0.319084814"
    ],
    [
      "0.376041403",
      "0.124683909"
    ],
    [
      "0.499751140",
      "0.183681646"
    ],
    [
      "0.272133256",
      "0.432181235"
    ],
    [
      "0.814361210",
      "0.726973126"
    ],
    [
      "0.119313840",
      "0.476355730"
    ],
    [
      "0.592033561",
      "0.627701389"
    ],
    [
      "0.443581795",
      "0.582802171"
    ],
    [
      "0.582828114",
      "0.473087521"
    ],
    [
      "0.499395743",
      "0.342591438"
    ],
    [
      "0.114316740",
      "0.625316071"
    ],
    [
      "0.685725888",
      "0.756053230"
    ],
    [
      "0.620320609",
      "0.876498151"
    ],
    [
      "0.637969592",
      "0.299357334"
    ],
    [
      "0.482104920",
      "0.907241779"
    ],
    [
      "0.246151886",
      "0.181052934"
    ]
  ]
}

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