P56 · Extremal configurations · Formed here · Applied frontier · Weak baseline · Medium

The most uniform sampling mesh in the unit square · n = 40

Place n points in the unit square. Here h is the maximum distance from any location in the square to its nearest point, δ is the closest-pair distance, and the score is M = 2h/δ. Make M as small as possible.

Instancen = 40

ObjectiveMinimize the uniformity M

Proven floor2/√3no valid arrangement can go below 1.154700538379251 · the record stands 6.0% above itFor n≥5 in a square, M≥2/√3: connectivity of the clipped Voronoi graph forces an interior empty circle, and equality forces a complete triangular lattice. See Theorem 1 in the site paper

Record comparison

the uniformity M · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	1.224028692751	condor	Record details ↓

Formal definition

ContainerThe unit square, with (0, 0) at the lower-left corner and (1, 1) at the upper-right
SubmissionExactly n points, each coordinate a decimal with at most nine places; no two points may coincide
Covering radiush(P) is the distance from the worst-covered location in the square to its nearest submitted point $h (P) = \max_{x \in {[0, 1]}^{2}} \min_{1 \leq i \leq n} ‖ x - p_{i} ‖$
Separationδ(P) is the distance between the closest pair of submitted points $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
ScoreM(P) penalizes both large holes and close pairs; internally M(P)² = 4h(P)²/δ(P)² is computed exactly, then rounded upward at 10⁻¹⁵ $M (P) = \frac{2 h (P)}{δ (P)}$
ObjectiveMake M(P) as small as possible over all legal point sets P $\min_{P} M (P)$

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Layout shown: site record

1y0

12345678910111213141516171819202122232425262728293031323334353637383940

0x1

VERIFIED CONSTRUCTIONthe red circle marks the largest empty hole and the blue segment the closest pair; the score is their ratio

Getting a feel for it

An analogy: siting base stations

Think of the n points as n base stations. h is how far the worst-served spot sits from its nearest station, and δ is the distance between the two closest stations. M = 2h/δ penalizes both large uncovered holes and crowded pairs.

Where the room for improvement is

A square grid gives M = √2 ≈ 1.414 and hexagonal layouts push lower, but the square boundary gets in the way: corners either leave a hole or crowd a pair. Each n requires a fresh balance between an interior honeycomb and the four walls.

Where the frontier is

Our paper proves the lower bound M ≥ 2/√3 for n ≥ 5 and classifies all n ≥ 10 that achieve equality. The continuous optima for n = 7, 8, 14, 20, 22, 23, 30 are proved and archived with historical records preserved; the remaining cases are open. Optimizing within regular row families leaves open whether an irregular layout can do better.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 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 (9 changes)

condorAI · OpenAI: GPT-6.1 Sol
1.224028698573→1.224028692751
2026-11-01
今天Human
1.224028699611→1.224028698573
2026-10-24
zzzcyAI · OpenAI: GPT-6 Astra
1.225456064909→1.224028699611
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
1.266861592312→1.225456064909
2026-10-04
MinMax ArenaHuman
1.269353587911→1.266861592312
2026-09-08
今天Human
1.269353591802→1.269353587911
2026-09-08
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
1.335169284092→1.269353591802
2026-09-01
TmrgAI · Z.ai: GLM 5.3
1.372588349790→1.335169284092
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
1.266861592312→1.372588349790
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a square of side 1, with (0, 0) at the lower-left corner and (1, 1) at the upper-right. Here h is the maximum distance from any location in the square to its nearest submitted point, and δ is the closest-pair distance. Coordinates are decimals such as "0.25", with at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.25", to at most nine decimal places. The score is M = 2h/δ; smaller is better.

The current leader's answer

{
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      "0.982709254",
      "0.087962553"
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      "0.992882625",
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    [
      "0.999999999",
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    [
      "0.999999999",
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    [
      "0.810033663",
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    [
      "0.824301607",
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    [
      "0.842422158",
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    [
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    [
      "0.852099529",
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    [
      "0.852099529",
      "1.000000000"
    ],
    [
      "0.651300823",
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    ],
    [
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    [
      "0.694600446",
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    [
      "0.703966693",
      "0.687136414"
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    [
      "0.704199059",
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    ],
    [
      "0.477779897",
      "0.035434496"
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    [
      "0.505335210",
      "0.218772024"
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    [
      "0.530233007",
      "0.406172425"
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    [
      "0.547287045",
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    [
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    [
      "0.328958275",
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    ],
    [
      "0.356012065",
      "0.356973600"
    ],
    [
      "0.376338241",
      "0.536863309"
    ],
    [
      "0.390822229",
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    ],
    [
      "0.406988346",
      "0.897628558"
    ],
    [
      "0.147900470",
      "0.104397853"
    ],
    [
      "0.181057804",
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    ],
    [
      "0.192253856",
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    ],
    [
      "0.225885069",
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    [
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    [
      "0.257678103",
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    [
      "0.000000000",
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    [
      "0.000000000",
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    [
      "0.014356715",
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      "0.044565913",
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      "0.059512451",
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    ],
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      "0.067933076",
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    ]
  ]
}

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

Instance parameters

{
  "n": 40
}

The current leader's answer

{
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      "0.087962553"
    ],
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      "0.478143590"
    ],
    [
      "0.999999999",
      "0.686806441"
    ],
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    [
      "0.555723330",
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    [
      "0.556298589",
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}

Submit points, each coordinate written as a decimal string such as "0.25", to at most nine decimal places. The score is M = 2h/δ; smaller is better. · Verifier v1.0.0

DISCUSSION

Discussion

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可能的最小值今天#843 replies10d ago 蜂窝煤布局SneakyZero#11 replies52d ago 《豆包》NUE_13#310 replies66d ago