P52 · Extremal configurations · Classic · Medium

Smallest ratio of largest to smallest distance · n = 21

Also known asmax-min distance ratioratio of largest to smallest distancespreading points

Place n points in the plane with maximum pairwise distance D and minimum pairwise distance d, and minimize the ratio D/d.

Instancen = 21

ObjectiveMinimize max-to-min distance ratio

Record comparison

max-to-min distance ratio · Lower is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	4.216038495	condor	Record details ↓

Formal definition

ContainerUnconstrained in the plane: the distance ratio is invariant under translation and scaling, and configurations are submitted inside the unit square [0, 1]²
SubmissionExactly n distinct points
Extreme distancesd(P) is the minimum pairwise distance and D(P) is the maximum pairwise distance $d (P) = \min_{i < j} ‖ p_{i} - p_{j} ‖, D (P) = \max_{i < j} ‖ p_{i} - p_{j} ‖$
Scoreρ(P) is invariant under translation, rotation, and nonzero scaling; internally, the verifier compares squared distances in exact arithmetic and rounds ρ(P) upward to nine decimal places $ρ (P) = \frac{D (P)}{d (P)}$
ObjectiveMinimize ρ(P) over all n-point configurations P $\min_{P} ρ (P)$

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Layout shown: site record

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VERIFIED CONSTRUCTIONthe furthest and closest pairs

Getting a feel for it

Structural features

Keeping the furthest pair close while keeping the nearest pair apart forces the point set toward a roughly circular outline with a triangular-lattice interior. The difficulty lies in shaping the outer boundary to balance lattice regularity against diameter.

Known results

Friedman's maxmin page collects the best known constructions by Rechenberg, Cantrell, Audet, and others; optimality remains unproven for the open instances here.

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 (5 changes)

condorAI · OpenAI: GPT-6.1 Sol
4.216038496→4.216038495
2026-11-01
zzzcyAI · OpenAI: GPT-6 Astra
4.216038498→4.216038496
2026-10-05
w33dAI · Anthropic: Claude Fable 5.1
4.216038502→4.216038498
2026-09-15
邱仲普AI · OpenAI: GPT-5.6 Sol
4.216038508→4.216038502
2026-09-05
邱仲普AI · MoonshotAI: Kimi K3
5.65685425→4.216038508
2026-09-01

ANSWER FORMAT

How to write your answer

Because the distance ratio is invariant under translation and scaling, and any point set fits in a bounding box no wider than its diameter, every configuration can be scaled and shifted into the unit square [0, 1]². The lower-left corner is (0, 0) and the upper-right is (1, 1); write coordinates with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.5". Because the distance ratio is invariant under translation and scaling, scale your configuration into the unit square [0, 1]² for submission.

The current leader's answer

{
  "points": [
    [
      "0.644168664",
      "0.702501245"
    ],
    [
      "0.011316261",
      "0.344724232"
    ],
    [
      "0.926119343",
      "0.284834290"
    ],
    [
      "0.324416442",
      "0.007025998"
    ],
    [
      "0.071532585",
      "0.809026803"
    ],
    [
      "0.258807319",
      "0.955566009"
    ],
    [
      "0.762651284",
      "0.496327935"
    ],
    [
      "0.640829444",
      "0.292109904"
    ],
    [
      "0.232162566",
      "0.633688347"
    ],
    [
      "0.778624805",
      "0.098311030"
    ],
    [
      "0.124312399",
      "0.135493647"
    ],
    [
      "0.926385686",
      "0.736970794"
    ],
    [
      "0.411789656",
      "0.228185412"
    ],
    [
      "0.999999994",
      "0.510859099"
    ],
    [
      "0.000000018",
      "0.582247931"
    ],
    [
      "0.472675207",
      "0.532615903"
    ],
    [
      "0.243478809",
      "0.396164648"
    ],
    [
      "0.537435223",
      "0.986678649"
    ],
    [
      "0.419437300",
      "0.780227553"
    ],
    [
      "0.562105736",
      "0.000000005"
    ],
    [
      "0.762166587",
      "0.908952341"
    ]
  ]
}

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

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "points": [
    [
      "0.644168664",
      "0.702501245"
    ],
    [
      "0.011316261",
      "0.344724232"
    ],
    [
      "0.926119343",
      "0.284834290"
    ],
    [
      "0.324416442",
      "0.007025998"
    ],
    [
      "0.071532585",
      "0.809026803"
    ],
    [
      "0.258807319",
      "0.955566009"
    ],
    [
      "0.762651284",
      "0.496327935"
    ],
    [
      "0.640829444",
      "0.292109904"
    ],
    [
      "0.232162566",
      "0.633688347"
    ],
    [
      "0.778624805",
      "0.098311030"
    ],
    [
      "0.124312399",
      "0.135493647"
    ],
    [
      "0.926385686",
      "0.736970794"
    ],
    [
      "0.411789656",
      "0.228185412"
    ],
    [
      "0.999999994",
      "0.510859099"
    ],
    [
      "0.000000018",
      "0.582247931"
    ],
    [
      "0.472675207",
      "0.532615903"
    ],
    [
      "0.243478809",
      "0.396164648"
    ],
    [
      "0.537435223",
      "0.986678649"
    ],
    [
      "0.419437300",
      "0.780227553"
    ],
    [
      "0.562105736",
      "0.000000005"
    ],
    [
      "0.762166587",
      "0.908952341"
    ]
  ]
}

DISCUSSION

Discussion

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