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.

Closed to new records since 2026-08-28. Existing records remain in the history and on their holders' profiles, and this URL stays active; since records can no longer change hands, they no longer count toward contribution points as of 2026-10-04.

P07 · Extremal configurations · Classic

Spreading points in a circle · n = 11

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

Instancen = 11

ObjectiveMaximize the smallest distance between two points

Record comparison

the smallest distance between two points · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	2 sin(π/9)	See source attribution	web.archive.org ↗
Site record	No player record yet	—	—

FinishedA construction reaching the theoretical ceiling is already published, leaving nothing further to improve

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)

Inspect in editor ↗

Layout shown: starter layout

2y0

1234567891011

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.

The optimum is proven; no further improvement is possibleSubmit 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": [
    [
      "1.968460740",
      "1.249166199"
    ],
    [
      "0.922790638",
      "1.997014901"
    ],
    [
      "0.175164448",
      "0.434627281"
    ],
    [
      "1.413552247",
      "0.089519611"
    ],
    [
      "1.093230356",
      "1.334548739"
    ],
    [
      "1.581723023",
      "1.813386946"
    ],
    [
      "0.299985372",
      "1.714128504"
    ],
    [
      "0.470425090",
      "1.051662342"
    ],
    [
      "0.731553888",
      "0.036705297"
    ],
    [
      "1.902044913",
      "0.568357818"
    ],
    [
      "1.026814530",
      "0.653740358"
    ]
  ]
}

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

Instance parameters

{
  "n": 11
}

Starter answer

{
  "points": [
    [
      "1.968460740",
      "1.249166199"
    ],
    [
      "0.922790638",
      "1.997014901"
    ],
    [
      "0.175164448",
      "0.434627281"
    ],
    [
      "1.413552247",
      "0.089519611"
    ],
    [
      "1.093230356",
      "1.334548739"
    ],
    [
      "1.581723023",
      "1.813386946"
    ],
    [
      "0.299985372",
      "1.714128504"
    ],
    [
      "0.470425090",
      "1.051662342"
    ],
    [
      "0.731553888",
      "0.036705297"
    ],
    [
      "1.902044913",
      "0.568357818"
    ],
    [
      "1.026814530",
      "0.653740358"
    ]
  ]
}

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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