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

The most uniform sampling mesh in the unit square · n = 30

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 = 30

ObjectiveMinimize the uniformity M

Record comparison

the uniformity M · Lower is better

Record	Value / interval	Author / holder	Source
Proven optimum	2/√3	MinMax Arena	Site paper ↗
Site record	1.154700544784	今天	Record details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

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 $\min_{P} M (P)$

Read the optimality proof ↗

Layout shown: site record

1y0

123456789101112131415161718192021222324252627282930

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 holder今天

Solution methodHuman

Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (4 changes)

今天Human
1.174828798279→1.154700544784
2026-09-06
今天Human
1.177777780901→1.174828798279
2026-09-06
TmrgHuman
1.376548418491→1.177777780901
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
4.517539502523→1.376548418491
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": [
    [
      "0.000000000",
      "0.018874776"
    ],
    [
      "0.000000000",
      "0.403774955"
    ],
    [
      "0.000000000",
      "0.788675135"
    ],
    [
      "0.111111111",
      "0.981125224"
    ],
    [
      "0.111111111",
      "0.596225045"
    ],
    [
      "0.111111111",
      "0.211324865"
    ],
    [
      "0.222222222",
      "0.788675135"
    ],
    [
      "0.222222222",
      "0.403774955"
    ],
    [
      "0.222222222",
      "0.018874776"
    ],
    [
      "0.333333333",
      "0.981125224"
    ],
    [
      "0.333333333",
      "0.596225045"
    ],
    [
      "0.333333333",
      "0.211324865"
    ],
    [
      "0.444444444",
      "0.788675135"
    ],
    [
      "0.444444444",
      "0.403774955"
    ],
    [
      "0.444444444",
      "0.018874776"
    ],
    [
      "0.555555556",
      "0.981125224"
    ],
    [
      "0.555555556",
      "0.596225045"
    ],
    [
      "0.555555556",
      "0.211324865"
    ],
    [
      "0.666666667",
      "0.788675135"
    ],
    [
      "0.666666667",
      "0.403774955"
    ],
    [
      "0.666666667",
      "0.018874776"
    ],
    [
      "0.777777778",
      "0.981125224"
    ],
    [
      "0.777777778",
      "0.596225045"
    ],
    [
      "0.777777778",
      "0.211324865"
    ],
    [
      "0.888888889",
      "0.788675135"
    ],
    [
      "0.888888889",
      "0.403774955"
    ],
    [
      "0.888888889",
      "0.018874776"
    ],
    [
      "1.000000000",
      "0.211324865"
    ],
    [
      "1.000000000",
      "0.596225045"
    ],
    [
      "1.000000000",
      "0.981125224"
    ]
  ]
}

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

Instance parameters

{
  "n": 30
}

The current leader's answer

{
  "points": [
    [
      "0.000000000",
      "0.018874776"
    ],
    [
      "0.000000000",
      "0.403774955"
    ],
    [
      "0.000000000",
      "0.788675135"
    ],
    [
      "0.111111111",
      "0.981125224"
    ],
    [
      "0.111111111",
      "0.596225045"
    ],
    [
      "0.111111111",
      "0.211324865"
    ],
    [
      "0.222222222",
      "0.788675135"
    ],
    [
      "0.222222222",
      "0.403774955"
    ],
    [
      "0.222222222",
      "0.018874776"
    ],
    [
      "0.333333333",
      "0.981125224"
    ],
    [
      "0.333333333",
      "0.596225045"
    ],
    [
      "0.333333333",
      "0.211324865"
    ],
    [
      "0.444444444",
      "0.788675135"
    ],
    [
      "0.444444444",
      "0.403774955"
    ],
    [
      "0.444444444",
      "0.018874776"
    ],
    [
      "0.555555556",
      "0.981125224"
    ],
    [
      "0.555555556",
      "0.596225045"
    ],
    [
      "0.555555556",
      "0.211324865"
    ],
    [
      "0.666666667",
      "0.788675135"
    ],
    [
      "0.666666667",
      "0.403774955"
    ],
    [
      "0.666666667",
      "0.018874776"
    ],
    [
      "0.777777778",
      "0.981125224"
    ],
    [
      "0.777777778",
      "0.596225045"
    ],
    [
      "0.777777778",
      "0.211324865"
    ],
    [
      "0.888888889",
      "0.788675135"
    ],
    [
      "0.888888889",
      "0.403774955"
    ],
    [
      "0.888888889",
      "0.018874776"
    ],
    [
      "1.000000000",
      "0.211324865"
    ],
    [
      "1.000000000",
      "0.596225045"
    ],
    [
      "1.000000000",
      "0.981125224"
    ]
  ]
}

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

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.

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