P17 · Extremal configurations · Formed here · Medium

Spreading points in a rectangle · n = 9

Place n points inside a 2 × 1 rectangle 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
External best known	—	—	No reference listed
Site record	0.647111422	Sigmoid with Codex	Record details ↓

Formal definition

ContainerA 2 × 1 rectangle [0, 2] × [0, 1], with lower-left corner (0, 0) and upper-right corner (2, 1)
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the rectangle 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)

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Layout shown: site record

1y0

123456789

0x2

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Strip partitions and staggered chains

Inside a 2 × 1 rectangle, the four corners and the two long horizontal edges provide natural anchor positions. Partitioning the rectangle into vertical strips or unit squares yields tight pigeonhole upper bounds on the minimum distance, while optimal configurations stagger points across the top and bottom edges.

Four to six points solved; remaining cases

HwaterB #40's three-strip argument proves the continuous optimum √13/3 for n = 4, and XSsMC #250's two-square reduction with the classical three-point theorem proves √6 − √2 for n = 5; both cases retain their history and close competition without claiming nine-decimal grid optimality. NUE_13 #31's tightening-strip argument proves the exact optimum 1 for n = 6, attained by the 2 × 3 grid. Optimality proofs have not yet been adopted for n = 7…20, which remain open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

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 (2 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.640312423→0.647111422
2026-09-02
NUE_13Human
0.4→0.640312423
2026-08-28

ANSWER FORMAT

How to write your answer

The container is a 2 × 1 rectangle, with the origin (0, 0) at its lower-left corner and (2, 1) at its upper-right. Coordinates 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".

The current leader's answer

{
  "points": [
    [
      "0.542423124",
      "1.000000000"
    ],
    [
      "2.000000000",
      "0.000000000"
    ],
    [
      "1.457576876",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.647111423",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.542423124"
    ],
    [
      "2.000000000",
      "0.647111423"
    ],
    [
      "1.352888577",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.647111423"
    ]
  ]
}

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

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "points": [
    [
      "0.542423124",
      "1.000000000"
    ],
    [
      "2.000000000",
      "0.000000000"
    ],
    [
      "1.457576876",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.647111423",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.542423124"
    ],
    [
      "2.000000000",
      "0.647111423"
    ],
    [
      "1.352888577",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.647111423"
    ]
  ]
}

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.

谁帮忙求一下n=7的精确解NUE_13#312 replies30d ago 本题n=5已最优的证明XSsMC#2501 replies49d ago 本题n=4已最优的证明HwaterB#402 replies52d ago 捡了个漏小可本州#510 replies63d ago