This problem is frozen

Mathematically equivalent to P02 (equal circles in a disc): the packing radius r and minimum point separation d are related by the strictly monotone bijection d = 2r/(1 − r), so any solution to one converts directly into a solution to the other. The catalogue retains P02, the standard formulation in the literature.

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P07 · Extremal configurations · Classic · Hard

Spreading points in a circle · n = 17

Place n points inside a circle of radius 1 to maximize the minimum Euclidean distance between any two points.

Instancen = 17

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	0.527421466	See source attribution	web.archive.org ↗
Site record	No player record yet	—	—

Formal definition

ContainerA circle of radius 1 centered at (1, 1), with both coordinates in [0, 2]
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the circle 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)

Open full editor ↗

Layout shown: starter layout

2y0

1234567891011121314151617

0x2

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Duality with equal-circle packing

Placing points in a unit disk with minimum pairwise distance d is equivalent to packing non-overlapping circles of radius d/2 inside a concentric disk of radius 1 + d/2 (or radius r = d / (2 + d) after rescaling to the unit disk). At optimality, pairs at distance d form a rigid contact graph, with many points lying on the boundary.

Proven optima and open frontier

Through the exact scaling between point spreading in a disk and equal-circle packing in a disk (cci), proofs and best-known constructions from the cci literature transfer directly. Optimality is proven here for n = 4, while the remaining sizes follow the open cci frontier.

Source

LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so coordinates run from 0 to 2. Coordinates are written as plain decimals such as "1.5", to at most nine decimal places.

Submit the point coordinates in points, with each coordinate formatted as a decimal string such as "1.5".

Starter answer

{
  "points": [
    [
      "1.5",
      "1"
    ],
    [
      "1.466236115",
      "1.180620833"
    ],
    [
      "1.369504459",
      "1.336847822"
    ],
    [
      "1.222869178",
      "1.447581646"
    ],
    [
      "1.04613418",
      "1.497867088"
    ],
    [
      "0.863168505",
      "1.480912822"
    ],
    [
      "0.698682682",
      "1.399008614"
    ],
    [
      "0.574891432",
      "1.263216081"
    ],
    [
      "0.50851345",
      "1.091874759"
    ],
    [
      "0.50851345",
      "0.908125241"
    ],
    [
      "0.574891432",
      "0.736783919"
    ],
    [
      "0.698682682",
      "0.600991386"
    ],
    [
      "0.863168505",
      "0.519087178"
    ],
    [
      "1.04613418",
      "0.502132912"
    ],
    [
      "1.222869178",
      "0.552418354"
    ],
    [
      "1.369504459",
      "0.663152178"
    ],
    [
      "1.466236115",
      "0.819379167"
    ]
  ]
}

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

Instance parameters

{
  "n": 17
}

Starter answer

{
  "points": [
    [
      "1.5",
      "1"
    ],
    [
      "1.466236115",
      "1.180620833"
    ],
    [
      "1.369504459",
      "1.336847822"
    ],
    [
      "1.222869178",
      "1.447581646"
    ],
    [
      "1.04613418",
      "1.497867088"
    ],
    [
      "0.863168505",
      "1.480912822"
    ],
    [
      "0.698682682",
      "1.399008614"
    ],
    [
      "0.574891432",
      "1.263216081"
    ],
    [
      "0.50851345",
      "1.091874759"
    ],
    [
      "0.50851345",
      "0.908125241"
    ],
    [
      "0.574891432",
      "0.736783919"
    ],
    [
      "0.698682682",
      "0.600991386"
    ],
    [
      "0.863168505",
      "0.519087178"
    ],
    [
      "1.04613418",
      "0.502132912"
    ],
    [
      "1.222869178",
      "0.552418354"
    ],
    [
      "1.369504459",
      "0.663152178"
    ],
    [
      "1.466236115",
      "0.819379167"
    ]
  ]
}

Submit the point coordinates in points, with each coordinate formatted as a decimal string such as "1.5". · Verifier v1.0.0

DISCUSSION

Discussion

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