P20 · Extremal configurations · Formed here · Medium

Spreading points in a half-disc · n = 15

Place n points inside a semicircle of radius 1 so that the minimum distance between any two points is as large as possible.

Instancen = 15

ObjectiveMaximize the smallest distance between two points

Record comparison

the smallest distance between two points · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.419480387	Sigmoid with Codex	Record details ↓

Formal definition

ContainerAn upper half-disk of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centered at (1, 0), with the circular arc in y ≥ 0
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the half-disk or on its boundary
Scoreδ(P) is the minimum Euclidean distance over all point pairs (the verifier compares δ(P)² using exact arithmetic internally) $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
ObjectiveMaximize the minimum pairwise distance δ(P)

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Layout shown: site record

1y0

123456789101112131415

0x2

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Sector partitions and boundary anchors

Bounded by a diameter of length 2 and an arc of length π, the half-disk offers natural anchor positions at the endpoints (0, 0), (2, 0) and the center (1, 0). Partitioning the half-disk into circular sectors centered at (1, 0) yields tight upper bounds for small n, while larger point sets alternate between the arc, the diameter, and the interior.

Exact optima for four and five points

Using a partition into three 60° sectors, HwaterB #40 proved that the continuous optimal distance for both n = 4 and n = 5 is 1. Four points attain 1 with rational coordinates, whereas the unique five-point optimum requires the irrational coordinate √3/2; finite-decimal certificates can only approach 1 (the displayed 0.999999999 is strictly less than 1 and does not establish grid optimality). All cases n ≥ 6 remain open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 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 (2 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.419229555→0.419480387
2026-09-02
邱仲普AI · MoonshotAI: Kimi K3
0.285714285→0.419229555
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a half-disc of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centred at (1, 0), with the arc above. Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

Submit the point coordinates in points, with each coordinate formatted as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "1.582411332",
      "0.000000000"
    ],
    [
      "1.171998170",
      "0.086745791"
    ],
    [
      "0.761585008",
      "0.000000000"
    ],
    [
      "0.380792504",
      "0.175956997"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.998640565",
      "0.052125057"
    ],
    [
      "1.452328810",
      "0.398801120"
    ],
    [
      "1.076844539",
      "0.585818120"
    ],
    [
      "0.671311013",
      "0.478550440"
    ],
    [
      "0.149432673",
      "0.525866163"
    ],
    [
      "1.889399182",
      "0.457131374"
    ],
    [
      "1.623655744",
      "0.781699119"
    ],
    [
      "1.248171473",
      "0.968716119"
    ],
    [
      "0.829018007",
      "0.985274152"
    ],
    [
      "0.439951182",
      "0.828459607"
    ]
  ]
}

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

Instance parameters

{
  "n": 15
}

The current leader's answer

{
  "points": [
    [
      "1.582411332",
      "0.000000000"
    ],
    [
      "1.171998170",
      "0.086745791"
    ],
    [
      "0.761585008",
      "0.000000000"
    ],
    [
      "0.380792504",
      "0.175956997"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.998640565",
      "0.052125057"
    ],
    [
      "1.452328810",
      "0.398801120"
    ],
    [
      "1.076844539",
      "0.585818120"
    ],
    [
      "0.671311013",
      "0.478550440"
    ],
    [
      "0.149432673",
      "0.525866163"
    ],
    [
      "1.889399182",
      "0.457131374"
    ],
    [
      "1.623655744",
      "0.781699119"
    ],
    [
      "1.248171473",
      "0.968716119"
    ],
    [
      "0.829018007",
      "0.985274152"
    ],
    [
      "0.439951182",
      "0.828459607"
    ]
  ]
}

Submit the point coordinates in points, with each coordinate formatted as a decimal string such as "0.5". · Verifier v1.0.0

DISCUSSION

Discussion

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本题n=4,5已达最优的证明HwaterB#401 replies59d ago