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P31 · Extremal configurations · Formed here · Medium

Spreading points in a quadrant · n = 18

Place n points in a quarter-disc of radius 1 so as to maximize the minimum distance between any two points.

Instancen = 18
ObjectiveMaximize the smallest distance between two points

Record comparison

the smallest distance between two points · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record0.26507973w33dRecord details ↓

Formal definition

  • ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
  • SubmissionExactly n distinct points
  • ConstraintsEvery point lies inside the container or on its boundary
  • Scoreδ(P) is the minimum Euclidean distance over all point pairs; internally, the verifier compares δ(P)² in exact arithmeticδ(P)=min1≤i<j≤n‖pi−pj‖
  • ObjectiveMaximize δ(P)
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Layout shown: site record

1y0
0x1
VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Connection to circle packing

Spreading points is equivalent to packing equal circles whose centers are restricted to the container: discs of radius δ/2 around the points have disjoint interiors. Optimal configurations are governed by rigid contacts between point pairs and the container boundary.

Known results

A three-region proof by XSsMC #250 resolves the continuous n = 4 case with optimal distance √(2−√2); that instance is archived with its historical records and precision notes. All larger instances remain open.

The record-holding arrangement for Spreading points in a quadrant n = 18, 0.26507973
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderw33d
Solution methodAI · Anthropic: Claude Fable 5.1
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 (5 changes)
  1. w33dAI · Anthropic: Claude Fable 5.1
    0.2650797306→0.2650797307
  2. w33dAI · Anthropic: Claude Fable 5.1
    0.26507973060→0.26507973062
  3. w33dAI · Anthropic: Claude Fable 5.1
    0.265007086→0.26507973
  4. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.2650070866→0.2650070868
  5. 邱仲普AI · MoonshotAI: Kimi K3
    0.166666666→0.265007086
ANSWER FORMAT

How to write your answer

The container is a quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant. Write coordinates as decimal strings such as "0.5", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.236982094",
      "0.118771844"
    ],
    [
      "0.239145931",
      "0.970983637"
    ],
    [
      "0.736982094",
      "0.032997648"
    ],
    [
      "0.000000000",
      "0.591551436"
    ],
    [
      "0.698436305",
      "0.715672220"
    ],
    [
      "0.482112936",
      "0.561763362"
    ],
    [
      "0.485861186",
      "0.874035987"
    ],
    [
      "0.208857177",
      "0.428315584"
    ],
    [
      "0.000000000",
      "0.265079731"
    ],
    [
      "0.701848461",
      "0.295738760"
    ],
    [
      "0.964866367",
      "0.262741112"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.259594827",
      "0.706667793"
    ],
    [
      "0.000000000",
      "0.856631167"
    ],
    [
      "0.473964188",
      "0.000000000"
    ],
    [
      "0.861934216",
      "0.507020125"
    ],
    [
      "0.436783136",
      "0.292975195"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 18
}

The current leader's answer

{
  "points": [
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.236982094",
      "0.118771844"
    ],
    [
      "0.239145931",
      "0.970983637"
    ],
    [
      "0.736982094",
      "0.032997648"
    ],
    [
      "0.000000000",
      "0.591551436"
    ],
    [
      "0.698436305",
      "0.715672220"
    ],
    [
      "0.482112936",
      "0.561763362"
    ],
    [
      "0.485861186",
      "0.874035987"
    ],
    [
      "0.208857177",
      "0.428315584"
    ],
    [
      "0.000000000",
      "0.265079731"
    ],
    [
      "0.701848461",
      "0.295738760"
    ],
    [
      "0.964866367",
      "0.262741112"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.259594827",
      "0.706667793"
    ],
    [
      "0.000000000",
      "0.856631167"
    ],
    [
      "0.473964188",
      "0.000000000"
    ],
    [
      "0.861934216",
      "0.507020125"
    ],
    [
      "0.436783136",
      "0.292975195"
    ]
  ]
}

Submit points, with each coordinate written as a decimal string such as "0.5". · 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.

问题与P30等价NUE_13#311 replies本题 n=4 已达最优的证明XSsMC#2501 replies