This problem is frozen

This narrow annulus may degenerate over a wide range of n, making evenly spaced single-ring configurations provably optimal. Frozen pending a degeneracy audit.

Closed to new records since 2026-08-27. 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.

P27 · Extremal configurations · Formed here

Spreading points in an annulus · n = 4

Place n points inside an annulus with outer radius 1 and inner radius 0.5 so that the minimum distance between any two points is as large as possible.

Instancen = 4

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	1.414213562	See source attribution	Proof / source the inscribed square, on the cardinal points of the outer rim
Site record	No player record yet	—	—

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

Formal definition

ContainerAn annulus centered at (1, 1) with outer radius 1 and inner radius 0.5 (i.e. 0.5 ≤ ‖p − (1, 1)‖ ≤ 1, with the central disk of radius 0.5 removed)
SubmissionExactly n distinct points
ConstraintsEvery point must lie inside the container or on its boundary
Scoreδ(P) is the minimum Euclidean distance over all pairs of points; internally, the verifier compares δ(P)² using exact arithmetic $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
ObjectiveMaximize δ(P)

Inspect in editor ↗

Layout shown: starter layout

2y0

1234

0x2

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Geometric intuition

The annulus has a band width of 0.5 (outer radius 1, inner radius 0.5): for small n, points space out roughly evenly around the outer rim, but as n grows a single ring is no longer enough and points must stagger between the inner and outer boundaries while the central hole blocks shortcuts across the middle.

Current status

Spreading points inside an annulus with outer radius 1 and inner radius 0.5 is an original variant introduced on this site, with no prior published literature. Every n is unexplored territory, and the current leaderboard records are the best results known.

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 an annulus of outer radius 1 and inner radius 0.5 centred at (1, 1), so coordinates run from 0 to 2 and the disc of radius 0.5 in the middle is not part of it. Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

Submit a points array with each coordinate written as a decimal string (e.g. "0.5").

Starter answer

{
  "points": [
    [
      "1",
      "0"
    ],
    [
      "2",
      "1"
    ],
    [
      "1",
      "2"
    ],
    [
      "0",
      "1"
    ]
  ]
}

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

Instance parameters

{
  "n": 4
}

Starter answer

{
  "points": [
    [
      "1",
      "0"
    ],
    [
      "2",
      "1"
    ],
    [
      "1",
      "2"
    ],
    [
      "0",
      "1"
    ]
  ]
}

Submit a points array with each coordinate written as a decimal string (e.g. "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.

草，冻结了，我就说NUE_13#311 replies52d ago