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

Covering a regular pentagon with n equal discs · n = 28

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 = 28

ObjectiveMinimize the covering radius

Proven floor√(A(K)/(π·28))no valid arrangement can go below 0.082203293735757 · 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	0.098192465900833	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 CONSTRUCTION28 equal discs of shared radius 0.098192; 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
0.098192465901411→0.098192465900833
2026-10-23
CodexAI · OpenAI: GPT-6 Sol
0.098192465901860→0.098192465901411
2026-10-23
CodexAI · OpenAI: GPT-6 Sol
0.098192465932034→0.098192465901860
2026-10-21
CodexAI · OpenAI: GPT-6 Sol
0.098192465936759→0.098192465932034
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.098192466015496→0.098192465936759
2026-10-07
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.100998684838539→0.098192466015496
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": [
    [
      "0.514167559",
      "0.915537375"
    ],
    [
      "0.276905111",
      "0.572487911"
    ],
    [
      "0.355272399",
      "0.867186550"
    ],
    [
      "0.887644598",
      "0.473148620"
    ],
    [
      "0.213289795",
      "0.745479121"
    ],
    [
      "0.887162601",
      "0.635074975"
    ],
    [
      "0.582850482",
      "0.152684820"
    ],
    [
      "0.829393915",
      "0.293871450"
    ],
    [
      "0.421170751",
      "0.458162141"
    ],
    [
      "0.489520738",
      "0.751358475"
    ],
    [
      "0.733655723",
      "0.554591061"
    ],
    [
      "0.342338543",
      "0.699853205"
    ],
    [
      "0.733826047",
      "0.163153588"
    ],
    [
      "0.167086071",
      "0.288008570"
    ],
    [
      "0.267215060",
      "0.155495695"
    ],
    [
      "0.423987249",
      "0.153744274"
    ],
    [
      "0.771174183",
      "0.732285589"
    ],
    [
      "0.123757559",
      "0.469927234"
    ],
    [
      "0.618331626",
      "0.650043132"
    ],
    [
      "0.460817468",
      "0.580182185"
    ],
    [
      "0.575812162",
      "0.475367145"
    ],
    [
      "0.507539173",
      "0.309058983"
    ],
    [
      "0.117858325",
      "0.624165354"
    ],
    [
      "0.642028402",
      "0.824805961"
    ],
    [
      "0.656943631",
      "0.318495520"
    ],
    [
      "0.343777287",
      "0.302806260"
    ],
    [
      "0.265653309",
      "0.421959490"
    ],
    [
      "0.751212814",
      "0.418441008"
    ]
  ]
}

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

Instance parameters

{
  "n": 28
}

The current leader's answer

{
  "points": [
    [
      "0.514167559",
      "0.915537375"
    ],
    [
      "0.276905111",
      "0.572487911"
    ],
    [
      "0.355272399",
      "0.867186550"
    ],
    [
      "0.887644598",
      "0.473148620"
    ],
    [
      "0.213289795",
      "0.745479121"
    ],
    [
      "0.887162601",
      "0.635074975"
    ],
    [
      "0.582850482",
      "0.152684820"
    ],
    [
      "0.829393915",
      "0.293871450"
    ],
    [
      "0.421170751",
      "0.458162141"
    ],
    [
      "0.489520738",
      "0.751358475"
    ],
    [
      "0.733655723",
      "0.554591061"
    ],
    [
      "0.342338543",
      "0.699853205"
    ],
    [
      "0.733826047",
      "0.163153588"
    ],
    [
      "0.167086071",
      "0.288008570"
    ],
    [
      "0.267215060",
      "0.155495695"
    ],
    [
      "0.423987249",
      "0.153744274"
    ],
    [
      "0.771174183",
      "0.732285589"
    ],
    [
      "0.123757559",
      "0.469927234"
    ],
    [
      "0.618331626",
      "0.650043132"
    ],
    [
      "0.460817468",
      "0.580182185"
    ],
    [
      "0.575812162",
      "0.475367145"
    ],
    [
      "0.507539173",
      "0.309058983"
    ],
    [
      "0.117858325",
      "0.624165354"
    ],
    [
      "0.642028402",
      "0.824805961"
    ],
    [
      "0.656943631",
      "0.318495520"
    ],
    [
      "0.343777287",
      "0.302806260"
    ],
    [
      "0.265653309",
      "0.421959490"
    ],
    [
      "0.751212814",
      "0.418441008"
    ]
  ]
}

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