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

The most uniform sampling mesh in the unit square · n = 13

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 = 13

ObjectiveMinimize the uniformity M

Proven floor2/√3no valid arrangement can go below 1.154700538379251 · the record stands 7.9% 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	1.245410065670	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 $\min_{P} M (P)$

Open full editor ↗

Layout shown: site record

1y0

12345678910111213

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 (14 changes)

condorAI · OpenAI: GPT-6.1 Sol
1.245410066536→1.245410065670
2026-11-01
w33dAI · Anthropic: Claude Fable 5.1
1.246139535131→1.245410066536
2026-09-15
今天AI · DeepSeek: DeepSeek V4.1 Flash
1.246139563431→1.246139535131
2026-09-12
今天AI · DeepSeek: DeepSeek V4.1 Flash
1.246139840062→1.246139563431
2026-09-10
今天AI · DeepSeek: DeepSeek V4.1 Flash
1.247053915156→1.246139840062
2026-09-10
今天Human
1.247055536212→1.247053915156
2026-09-08
今天Human
1.247057932559→1.247055536212
2026-09-08
MinMax ArenaHuman
1.250730849374→1.247057932559
2026-09-08
今天Human
1.250730978076→1.250730849374
2026-09-08
邱仲普AI · OpenAI: GPT-5.6 Sol
1.250730999034→1.250730978076
2026-09-07
今天Human
1.250820940622→1.250730999034
2026-09-07
今天Human
1.250820942115→1.250820940622
2026-09-06
TmrgAI · Z.ai: GLM 5.3
1.336285505393→1.250820942115
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
1.247057932559→1.336285505393
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

{
  "points": [
    [
      "0.023934453",
      "0.418859398"
    ],
    [
      "0.900516961",
      "0.050171740"
    ],
    [
      "0.950258480",
      "0.387995883"
    ],
    [
      "0.285672840",
      "0.199559632"
    ],
    [
      "0.000000000",
      "0.012502425"
    ],
    [
      "1.000000000",
      "0.725820026"
    ],
    [
      "0.099502564",
      "0.818055809"
    ],
    [
      "0.415551037",
      "0.947333548"
    ],
    [
      "0.562756444",
      "0.000000000"
    ],
    [
      "0.612497963",
      "0.337824143"
    ],
    [
      "0.332602009",
      "0.568528217"
    ],
    [
      "0.662239483",
      "0.675648286"
    ],
    [
      "0.796469383",
      "1.000000000"
    ]
  ]
}

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": [
    [
      "0.023934453",
      "0.418859398"
    ],
    [
      "0.900516961",
      "0.050171740"
    ],
    [
      "0.950258480",
      "0.387995883"
    ],
    [
      "0.285672840",
      "0.199559632"
    ],
    [
      "0.000000000",
      "0.012502425"
    ],
    [
      "1.000000000",
      "0.725820026"
    ],
    [
      "0.099502564",
      "0.818055809"
    ],
    [
      "0.415551037",
      "0.947333548"
    ],
    [
      "0.562756444",
      "0.000000000"
    ],
    [
      "0.612497963",
      "0.337824143"
    ],
    [
      "0.332602009",
      "0.568528217"
    ],
    [
      "0.662239483",
      "0.675648286"
    ],
    [
      "0.796469383",
      "1.000000000"
    ]
  ]
}

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