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

Covering a regular pentagon with n equal discs · n = 32

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 = 32

ObjectiveMinimize the covering radius

Proven floor√(A(K)/(π·32))no valid arrangement can go below 0.076894140305885 · the record stands 19.3% 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	0.091739469134229	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 CONSTRUCTION32 equal discs of shared radius 0.091739; 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 (4 changes)

CodexAI · OpenAI: GPT-6 Sol
0.091739469169885→0.091739469134229
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.091739469176014→0.091739469169885
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.091760480029818→0.091739469176014
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.094614485368856→0.091760480029818
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": [
    [
      "0.505300839",
      "0.744714227"
    ],
    [
      "0.521801902",
      "0.417178863"
    ],
    [
      "0.666063246",
      "0.330734234"
    ],
    [
      "0.661065127",
      "0.483423681"
    ],
    [
      "0.378427126",
      "0.327086190"
    ],
    [
      "0.517139621",
      "0.271370171"
    ],
    [
      "0.760386863",
      "0.578063907"
    ],
    [
      "0.265888091",
      "0.582187613"
    ],
    [
      "0.403426194",
      "0.489439029"
    ],
    [
      "0.390254040",
      "0.653643776"
    ],
    [
      "0.528817606",
      "0.577120625"
    ],
    [
      "0.634475990",
      "0.655916448"
    ],
    [
      "0.806027119",
      "0.411960518"
    ],
    [
      "0.136308695",
      "0.375639914"
    ],
    [
      "0.336384502",
      "0.791915378"
    ],
    [
      "0.642318256",
      "0.819313036"
    ],
    [
      "0.533162815",
      "0.914464289"
    ],
    [
      "0.767046704",
      "0.745100254"
    ],
    [
      "0.229831722",
      "0.727124630"
    ],
    [
      "0.812621923",
      "0.291643242"
    ],
    [
      "0.234043261",
      "0.263192887"
    ],
    [
      "0.374675280",
      "0.175184532"
    ],
    [
      "0.643969220",
      "0.175162184"
    ],
    [
      "0.104997125",
      "0.652152438"
    ],
    [
      "0.893064374",
      "0.654428422"
    ],
    [
      "0.905354158",
      "0.506445996"
    ],
    [
      "0.129937517",
      "0.536836165"
    ],
    [
      "0.509302666",
      "0.116995973"
    ],
    [
      "0.741672475",
      "0.170918196"
    ],
    [
      "0.251722936",
      "0.144568757"
    ],
    [
      "0.262893786",
      "0.426300791"
    ],
    [
      "0.431487599",
      "0.885941056"
    ]
  ]
}

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

Instance parameters

{
  "n": 32
}

The current leader's answer

{
  "points": [
    [
      "0.505300839",
      "0.744714227"
    ],
    [
      "0.521801902",
      "0.417178863"
    ],
    [
      "0.666063246",
      "0.330734234"
    ],
    [
      "0.661065127",
      "0.483423681"
    ],
    [
      "0.378427126",
      "0.327086190"
    ],
    [
      "0.517139621",
      "0.271370171"
    ],
    [
      "0.760386863",
      "0.578063907"
    ],
    [
      "0.265888091",
      "0.582187613"
    ],
    [
      "0.403426194",
      "0.489439029"
    ],
    [
      "0.390254040",
      "0.653643776"
    ],
    [
      "0.528817606",
      "0.577120625"
    ],
    [
      "0.634475990",
      "0.655916448"
    ],
    [
      "0.806027119",
      "0.411960518"
    ],
    [
      "0.136308695",
      "0.375639914"
    ],
    [
      "0.336384502",
      "0.791915378"
    ],
    [
      "0.642318256",
      "0.819313036"
    ],
    [
      "0.533162815",
      "0.914464289"
    ],
    [
      "0.767046704",
      "0.745100254"
    ],
    [
      "0.229831722",
      "0.727124630"
    ],
    [
      "0.812621923",
      "0.291643242"
    ],
    [
      "0.234043261",
      "0.263192887"
    ],
    [
      "0.374675280",
      "0.175184532"
    ],
    [
      "0.643969220",
      "0.175162184"
    ],
    [
      "0.104997125",
      "0.652152438"
    ],
    [
      "0.893064374",
      "0.654428422"
    ],
    [
      "0.905354158",
      "0.506445996"
    ],
    [
      "0.129937517",
      "0.536836165"
    ],
    [
      "0.509302666",
      "0.116995973"
    ],
    [
      "0.741672475",
      "0.170918196"
    ],
    [
      "0.251722936",
      "0.144568757"
    ],
    [
      "0.262893786",
      "0.426300791"
    ],
    [
      "0.431487599",
      "0.885941056"
    ]
  ]
}

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