P31 · Extremal configurations · Formed here

Spreading points in a quadrant · n = 4

Place n points in a quarter-disc of radius 1 so as to maximize the minimum distance between any two points.

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	√(2−√2)	See source attribution	MinMax Arena · Proof ↗
Site record	0.765366864	qwert	Record details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

Formal definition

ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the container or on its boundary
Scoreδ(P) is the minimum Euclidean distance over all point pairs; internally, the verifier compares δ(P)² in exact arithmetic $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
ObjectiveMaximize δ(P)

Read the optimality proof ↗

Layout shown: site record

1y0

1234

0x1

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Connection to circle packing

Spreading points is equivalent to packing equal circles whose centers are restricted to the container: discs of radius δ/2 around the points have disjoint interiors. Optimal configurations are governed by rigid contacts between point pairs and the container boundary.

Known results

A three-region proof by XSsMC #250 resolves the continuous n = 4 case with optimal distance √(2−√2); that instance is archived with its historical records and precision notes. All larger instances remain open.

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderqwert

Solution methodHuman

Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)

qwertHuman
0.764852284→0.765366864
2026-08-26
NUE_13Human
0.333333333→0.764852284
2026-08-26

ANSWER FORMAT

How to write your answer

The container is a quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant. Write coordinates as decimal strings such as "0.5", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.707106781",
      "0.707106781"
    ],
    [
      "1.000000000",
      "0.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 4
}

The current leader's answer

{
  "points": [
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.707106781",
      "0.707106781"
    ],
    [
      "1.000000000",
      "0.000000000"
    ]
  ]
}

Submit points, with each coordinate written as a decimal string such as "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.

问题与P30等价NUE_13#311 replies23d ago 本题 n=4 已达最优的证明XSsMC#2501 replies45d ago