This problem is frozen

Mathematically equivalent to P01 (equal circles in the unit square) under d = 2r/(1 − 2r); Specht's reference table lists both quantities as columns of the same table. The catalogue retains the circle-packing formulation (P01).

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P15 · Packing and covering · Classic

Spreading points in the unit square · n = 9

Place n points in the unit square to maximize the minimum Euclidean distance between any two points.

Instancen = 9

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	0.5	See source attribution	Proof / source the 3 × 3 grid, spacing 1/2
Site record	No player record yet	—	—

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

Formal definition

ContainerThe unit square [0, 1] × [0, 1], with lower-left corner (0, 0) and upper-right corner (1, 1)
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the unit square 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

1y0

123456789

0x1

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Duality with equal-circle packing in a square

Placing n points in the unit square with minimum pairwise distance d is equivalent to packing n non-overlapping circles of radius d/2 in a square of side 1 + d (or radius r = d / (2 + 2d) in the unit square). At optimality, point pairs at distance d and points on the four edges lock into a rigid contact graph.

Proven optima and open frontier

Through the exact conversion to equal-circle packing in a square (csq), proofs and constructions in the csq literature transfer directly. Elementary partition proofs are included here for n = 2, 4, 5, and 9, while the remaining sizes follow the csq benchmark table.

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 square of side 1. Its lower-left corner is the origin (0, 0) and its upper-right corner is (1, 1). Coordinates and lengths share one unit and are written as plain decimals such as "0.5", to at most nine decimal places.

Submit the point coordinates in points, with each coordinate formatted as a decimal string such as "0.5".

Starter answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.5"
    ],
    [
      "0.5",
      "0.5"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.5",
      "1"
    ],
    [
      "1",
      "1"
    ]
  ]
}

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

Instance parameters

{
  "n": 9
}

Starter answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.5"
    ],
    [
      "0.5",
      "0.5"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.5",
      "1"
    ],
    [
      "1",
      "1"
    ]
  ]
}

Submit the point coordinates in points, with each coordinate formatted 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.

冻结，但贡献值……NUE_13#310 replies31d ago