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

Covering a regular pentagon with n equal discs · n = 9

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 = 9

ObjectiveMinimize the covering radius

Proven floor√(A(K)/(π·9))no valid arrangement can go below 0.144992981450137 · the record stands 25.4% 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	UnresolvedSee original source	See source attribution	link.springer.com ↗
Site record	0.181860256671806	condor	Record details ↓

Source comparison needs clarification.

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) $\min_{P} r (P)$

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTION9 equal discs of shared radius 0.181860; 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
0.181860256748853→0.181860256671806
2026-11-01
CodexAI · OpenAI: GPT-6 Sol
0.181860256778463→0.181860256748853
2026-10-19
zzzcyAI · OpenAI: GPT-6 Astra
0.181860256800996→0.181860256778463
2026-10-07
邱仲普AI · OpenAI: GPT-5.6 Sol
0.181860256853756→0.181860256800996
2026-09-17
w33dAI · Anthropic: Claude Fable 5.1
0.181862304185616→0.181860256853756
2026-09-15
邱仲普AI · OpenAI: GPT-5.6 Sol
0.181862304660883→0.181862304185616
2026-09-09
hadsajncHuman
0.181862444389383→0.181862304660883
2026-09-05
hadsajncHuman
0.181862735625360→0.181862444389383
2026-09-05
NUE_13Human
0.181871774083792→0.181862735625360
2026-09-04

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": [
    [
      "0.523634976",
      "0.532530754"
    ],
    [
      "0.220169239",
      "0.462075431"
    ],
    [
      "0.737400816",
      "0.268355115"
    ],
    [
      "0.828219096",
      "0.547863461"
    ],
    [
      "0.550402455",
      "0.222145334"
    ],
    [
      "0.726848909",
      "0.812229758"
    ],
    [
      "0.206331737",
      "0.654199992"
    ],
    [
      "0.313001640",
      "0.242619912"
    ],
    [
      "0.444096154",
      "0.826945347"
    ]
  ]
}

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

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "points": [
    [
      "0.523634976",
      "0.532530754"
    ],
    [
      "0.220169239",
      "0.462075431"
    ],
    [
      "0.737400816",
      "0.268355115"
    ],
    [
      "0.828219096",
      "0.547863461"
    ],
    [
      "0.550402455",
      "0.222145334"
    ],
    [
      "0.726848909",
      "0.812229758"
    ],
    [
      "0.206331737",
      "0.654199992"
    ],
    [
      "0.313001640",
      "0.242619912"
    ],
    [
      "0.444096154",
      "0.826945347"
    ]
  ]
}

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

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

还能争！！！猫猫#1237 replies59d ago