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

Covering a regular pentagon with n equal discs · n = 29

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 = 29

ObjectiveMinimize the covering radius

Proven floor√(A(K)/(π·29))no valid arrangement can go below 0.080773562171047 · 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	0.096383149099469	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 CONSTRUCTION29 equal discs of shared radius 0.096383; 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 (8 changes)

CodexAI · OpenAI: GPT-6 Sol
0.096383149103179→0.096383149099469
2026-10-23
CodexAI · OpenAI: GPT-6 Sol
0.096383149104050→0.096383149103179
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.096383149104122→0.096383149104050
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.096383149133392→0.096383149104122
2026-10-22
CodexAI · OpenAI: GPT-6 Sol
0.096383149154129→0.096383149133392
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.096383149208290→0.096383149154129
2026-10-07
w33dAI · Anthropic: Claude Fable 5.1
0.096383149245406→0.096383149208290
2026-09-15
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.098258830191613→0.096383149245406
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": [
    [
      "0.481174130",
      "0.905473295"
    ],
    [
      "0.269054000",
      "0.563435375"
    ],
    [
      "0.346999732",
      "0.828400468"
    ],
    [
      "0.860136091",
      "0.457225646"
    ],
    [
      "0.203940642",
      "0.735057946"
    ],
    [
      "0.890939963",
      "0.608307504"
    ],
    [
      "0.340023281",
      "0.298347777"
    ],
    [
      "0.577933121",
      "0.151007070"
    ],
    [
      "0.827984383",
      "0.282496069"
    ],
    [
      "0.728656810",
      "0.427843090"
    ],
    [
      "0.416427245",
      "0.450216977"
    ],
    [
      "0.509609284",
      "0.747702944"
    ],
    [
      "0.725718523",
      "0.569113224"
    ],
    [
      "0.355103231",
      "0.685118913"
    ],
    [
      "0.731925673",
      "0.155775520"
    ],
    [
      "0.166782292",
      "0.284514452"
    ],
    [
      "0.420805879",
      "0.151641816"
    ],
    [
      "0.813651375",
      "0.718773363"
    ],
    [
      "0.122030760",
      "0.463770228"
    ],
    [
      "0.592167846",
      "0.618163697"
    ],
    [
      "0.440825956",
      "0.576980131"
    ],
    [
      "0.501009054",
      "0.303530160"
    ],
    [
      "0.114454939",
      "0.619972492"
    ],
    [
      "0.689189039",
      "0.750749188"
    ],
    [
      "0.625022823",
      "0.876972376"
    ],
    [
      "0.656287241",
      "0.305430937"
    ],
    [
      "0.575563967",
      "0.457934887"
    ],
    [
      "0.265807001",
      "0.153910956"
    ],
    [
      "0.258563180",
      "0.419213251"
    ]
  ]
}

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

Instance parameters

{
  "n": 29
}

The current leader's answer

{
  "points": [
    [
      "0.481174130",
      "0.905473295"
    ],
    [
      "0.269054000",
      "0.563435375"
    ],
    [
      "0.346999732",
      "0.828400468"
    ],
    [
      "0.860136091",
      "0.457225646"
    ],
    [
      "0.203940642",
      "0.735057946"
    ],
    [
      "0.890939963",
      "0.608307504"
    ],
    [
      "0.340023281",
      "0.298347777"
    ],
    [
      "0.577933121",
      "0.151007070"
    ],
    [
      "0.827984383",
      "0.282496069"
    ],
    [
      "0.728656810",
      "0.427843090"
    ],
    [
      "0.416427245",
      "0.450216977"
    ],
    [
      "0.509609284",
      "0.747702944"
    ],
    [
      "0.725718523",
      "0.569113224"
    ],
    [
      "0.355103231",
      "0.685118913"
    ],
    [
      "0.731925673",
      "0.155775520"
    ],
    [
      "0.166782292",
      "0.284514452"
    ],
    [
      "0.420805879",
      "0.151641816"
    ],
    [
      "0.813651375",
      "0.718773363"
    ],
    [
      "0.122030760",
      "0.463770228"
    ],
    [
      "0.592167846",
      "0.618163697"
    ],
    [
      "0.440825956",
      "0.576980131"
    ],
    [
      "0.501009054",
      "0.303530160"
    ],
    [
      "0.114454939",
      "0.619972492"
    ],
    [
      "0.689189039",
      "0.750749188"
    ],
    [
      "0.625022823",
      "0.876972376"
    ],
    [
      "0.656287241",
      "0.305430937"
    ],
    [
      "0.575563967",
      "0.457934887"
    ],
    [
      "0.265807001",
      "0.153910956"
    ],
    [
      "0.258563180",
      "0.419213251"
    ]
  ]
}

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