P21 · Extremal configurations · Formed here · Medium

Spreading points in a plus sign · n = 13

Place n points inside a cross shape (a 3 × 3 square missing all four 1 × 1 corners) so that the minimum distance between any two points is as large as possible.

Instancen = 13

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	0.87591466	Sigmoid with Codex	Record details ↓

Formal definition

ContainerA symmetric cross-shaped region formed by removing the four 1 × 1 corner squares from [0, 3] × [0, 3], with lower-left corner (0, 0) and upper-right corner (3, 3)
SubmissionExactly n distinct points
ConstraintsEvery point lies inside the cross-shaped region 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

3y0

12345678910111213

0x3

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Five-square partition and arm vertices

The symmetric cross consists of a central 1 × 1 square and four 1 × 1 arms, providing eight outer corner vertices and four re-entrant corners. Partitioning the cross into its five unit squares or finer subregions yields pigeonhole upper bounds on the minimum distance, while optimal configurations anchor points at the outer arm tips and balance the central core.

Four proven cases

Optimality is proven for n = 4, 5, 6, and 9: NUE_13 #31 supplied the constructions and conjectures, with the upper bounds for n = 4 and n = 5 completed on this site; HwaterB #40 proved the n = 6 optimum √2 via a five-square partition and the n = 9 optimum √5/2 via an eight-region partition. All other sizes 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 (3 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.875914103→0.87591466
2026-09-02
邱仲普AI · MoonshotAI: Kimi K3
0.833333333→0.875914103
2026-09-01
HotaruHuman
0.5→0.833333333
2026-08-30

ANSWER FORMAT

How to write your answer

The container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) 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": [
    [
      "1.000091129",
      "0.008483951"
    ],
    [
      "0.832210528",
      "1.273225421"
    ],
    [
      "1.522946175",
      "0.734606113"
    ],
    [
      "2.181559874",
      "1.312061298"
    ],
    [
      "2.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "2.132919080"
    ],
    [
      "1.875914661",
      "2.132919080"
    ],
    [
      "2.892301590",
      "1.999967786"
    ],
    [
      "3.000000000",
      "1.000000000"
    ],
    [
      "0.064321487",
      "1.890805832"
    ],
    [
      "1.124085339",
      "3.000000000"
    ],
    [
      "2.000000000",
      "3.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 13
}

The current leader's answer

{
  "points": [
    [
      "1.000091129",
      "0.008483951"
    ],
    [
      "0.832210528",
      "1.273225421"
    ],
    [
      "1.522946175",
      "0.734606113"
    ],
    [
      "2.181559874",
      "1.312061298"
    ],
    [
      "2.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "2.132919080"
    ],
    [
      "1.875914661",
      "2.132919080"
    ],
    [
      "2.892301590",
      "1.999967786"
    ],
    [
      "3.000000000",
      "1.000000000"
    ],
    [
      "0.064321487",
      "1.890805832"
    ],
    [
      "1.124085339",
      "3.000000000"
    ],
    [
      "2.000000000",
      "3.000000000"
    ]
  ]
}

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.

本题n=6已达最优的证明HwaterB#406 replies52d ago 本题n=9已达最优的证明HwaterB#402 replies52d ago 本题n=6已达最优的证明（再发一遍，上次审核似乎没看见）HwaterB#401 replies52d ago