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 · Hard

Spreading points in a circle · n = 20

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

Instancen = 20

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	0.485163607	See source attribution	web.archive.org ↗
Site record	0.421738225	NUE_13	Record details ↓

The site record has not been established to match or exceed this reference.

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: site record

2y0

1234567891011121314151617181920

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

Record details

Inspect the layout, solver notes and record history.

Record holderNUE_13

Solution methodHuman

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 (1 changes)

NUE_13Human
0.156434464→0.421738225
2026-08-25

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".

The current leader's answer

{
  "points": [
    [
      "1.833912990",
      "0.495652174"
    ],
    [
      "1.684223909",
      "0.924943332"
    ],
    [
      "1.926247627",
      "1.290414285"
    ],
    [
      "1.624327143",
      "1.613204150"
    ],
    [
      "1.258856323",
      "1.885963014"
    ],
    [
      "0.770434571",
      "1.896521792"
    ],
    [
      "0.372448078",
      "1.677267362"
    ],
    [
      "0.104368402",
      "1.307117219"
    ],
    [
      "0.080708788",
      "0.869544773"
    ],
    [
      "0.235776249",
      "0.469290947"
    ],
    [
      "0.555652386",
      "0.144347879"
    ],
    [
      "1.000993375",
      "0.048969579"
    ],
    [
      "1.506796010",
      "0.173933620"
    ],
    [
      "1.363498732",
      "0.637230660"
    ],
    [
      "1.388100411",
      "1.234037146"
    ],
    [
      "1.027825928",
      "1.501739236"
    ],
    [
      "0.650160671",
      "1.314037119"
    ],
    [
      "0.532153177",
      "0.866795824"
    ],
    [
      "0.889725781",
      "0.518281498"
    ],
    [
      "1.026832553",
      "0.995056959"
    ]
  ]
}

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

Instance parameters

{
  "n": 20
}

The current leader's answer

{
  "points": [
    [
      "1.833912990",
      "0.495652174"
    ],
    [
      "1.684223909",
      "0.924943332"
    ],
    [
      "1.926247627",
      "1.290414285"
    ],
    [
      "1.624327143",
      "1.613204150"
    ],
    [
      "1.258856323",
      "1.885963014"
    ],
    [
      "0.770434571",
      "1.896521792"
    ],
    [
      "0.372448078",
      "1.677267362"
    ],
    [
      "0.104368402",
      "1.307117219"
    ],
    [
      "0.080708788",
      "0.869544773"
    ],
    [
      "0.235776249",
      "0.469290947"
    ],
    [
      "0.555652386",
      "0.144347879"
    ],
    [
      "1.000993375",
      "0.048969579"
    ],
    [
      "1.506796010",
      "0.173933620"
    ],
    [
      "1.363498732",
      "0.637230660"
    ],
    [
      "1.388100411",
      "1.234037146"
    ],
    [
      "1.027825928",
      "1.501739236"
    ],
    [
      "0.650160671",
      "1.314037119"
    ],
    [
      "0.532153177",
      "0.866795824"
    ],
    [
      "0.889725781",
      "0.518281498"
    ],
    [
      "1.026832553",
      "0.995056959"
    ]
  ]
}

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

No threads yet. Start the first discussion.