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P17 · Extremal configurations · Formed here · Medium

Spreading points in a rectangle · n = 7

Place n points inside a 2 × 1 rectangle to maximize the minimum Euclidean distance between any two points.

Instancen = 7
ObjectiveMaximize the smallest distance between two points

Record comparison

the smallest distance between two points · Higher is better
RecordValue / intervalAuthor / holderSource
External best known——No reference listed
Site record0.788288881deeppigRecord 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)=min1≤i<j≤n‖pi−pj‖
  • ObjectiveMaximize the minimum pairwise distance δ(P)
Open full editor ↗

Layout shown: site record

1y0
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
The record-holding arrangement for Spreading points in a rectangle n = 7, 0.788288881
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderdeeppig
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 (9 changes)
  1. deeppigHuman
    0.787673244→0.788288881
  2. deeppigHuman
    0.786299647→0.787673244
  3. 小可本州Human
    0.7862996470→0.7862996477
  4. HwaterBHuman
    0.78629964→0.786299647
  5. HwaterBHuman
    0.783465869→0.78629964
  6. NUE_13Human
    0.748497828→0.783465869
  7. qwertHuman
    0.625→0.748497828
  8. NUE_13Human
    0.559016994→0.625
  9. NUE_13Human
    0.5→0.559016994
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": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.609425435",
      "0.500000000"
    ],
    [
      "1.218850871",
      "0.000000000"
    ],
    [
      "1.218850871",
      "1.000000000"
    ],
    [
      "2.000000000",
      "0.105855559"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "2.000000000",
      "0.894144441"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 7
}

The current leader's answer

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.609425435",
      "0.500000000"
    ],
    [
      "1.218850871",
      "0.000000000"
    ],
    [
      "1.218850871",
      "1.000000000"
    ],
    [
      "2.000000000",
      "0.105855559"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "2.000000000",
      "0.894144441"
    ]
  ]
}

Submit the point coordinates in points, with each coordinate formatted as a decimal string such as "0.5". · Verifier v1.0.0

DISCUSSION

Discussion

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