P53 · Extremal configurations · Classic · Applied frontier · Weak baseline · Medium

The biggest little polygon · n = 28

Also known asbiggest little polygonlargest small polygon

Choose n points in the plane with pairwise distances at most 1 in strictly convex position, maximizing the area of the convex n-gon they span.

Instancen = 28

ObjectiveMaximize the area

Record comparison

the area · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.7811114189925910415	zzzcy	Record details ↓

Formal definition

ContainerUnconstrained in the plane: the only global constraint is diameter at most 1, with coordinates written inside [0, 1.5] × [0, 1.5]
SubmissionExactly n points
ConstraintsAll pairwise distances are at most 1, and every point is a strict vertex of the convex hull
Areaqᵢ = (xᵢ,yᵢ) are the hull vertices ordered counterclockwise by the verifier (submissions need not be pre-sorted); internally, the verifier compares the shoelace value 2A(P) in exact arithmetic $A (P) = \frac{1}{2} | \sum_{i = 1}^{n} (x_{i} y_{i + 1} - x_{i + 1} y_{i}) |, q_{n + 1} = q_{1}$
ObjectiveMaximize the area A(P) over all convex n-gons P of diameter at most 1 $\max_{P} A (P)$

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Layout shown: site record

1.5y0

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0x1.5

VERIFIED CONSTRUCTIONthe convex hull and its diameter pair

Getting a feel for it

Why only even n are included

For odd n, the regular n-gon is already known to maximize area among polygons of diameter 1. For even n, asymmetric polygons beat the regular n-gon; for instance, Graham's hexagon has nearly 4% more area than the regular hexagon of diameter 1. Accordingly, only even n are included here.

Known results

Reinhardt (1922) proved that regular polygons are optimal for odd n. For even n, optimal configurations have been proven for n = 6, 8, 10, and 12 (from Graham 1975 through Audet et al.), while n ≥ 14 has only numerical best-known constructions and remains open.

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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 (7 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.7811114189482952915→0.7811114189925910415
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.7811114189303159535→0.7811114189482952915
2026-10-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.7811114184354117935→0.7811114189303159535
2026-09-01
邱仲普AI · MoonshotAI: Kimi K2.7 Code
0.780356745165362328→0.7811114184354117935
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.771664132006451474→0.780356745165362328
2026-09-01
bowen2147483647Human
0.760785171819951474→0.771664132006451474
2026-08-29
bowen2147483647Human
0.4984468923434305→0.760785171819951474
2026-08-29

ANSWER FORMAT

How to write your answer

The only geometric constraint is that all pairwise distances are at most 1. Any point set of diameter at most 1 fits inside a 1 × 1 box, so the [0, 1.5] × [0, 1.5] coordinate window leaves ample margin on all sides. Write coordinates as decimal strings such as "0.25", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.25" (at most nine decimal places).

The current leader's answer

{
  "points": [
    [
      "0.239685468",
      "0.718319011"
    ],
    [
      "0.262875239",
      "0.606276712"
    ],
    [
      "0.310886343",
      "0.502425403"
    ],
    [
      "0.381213540",
      "0.412180886"
    ],
    [
      "0.470186591",
      "0.340252413"
    ],
    [
      "0.573160700",
      "0.290394304"
    ],
    [
      "0.684751199",
      "0.265209880"
    ],
    [
      "0.799061975",
      "0.266005714"
    ],
    [
      "0.909658792",
      "0.292576593"
    ],
    [
      "1.007822768",
      "0.341674831"
    ],
    [
      "1.066135680",
      "0.388991508"
    ],
    [
      "1.120732314",
      "0.440549556"
    ],
    [
      "1.182354833",
      "0.531376119"
    ],
    [
      "1.223287653",
      "0.637499573"
    ],
    [
      "1.239160882",
      "0.750705687"
    ],
    [
      "1.228922092",
      "0.864643656"
    ],
    [
      "1.193088393",
      "0.973296450"
    ],
    [
      "1.133529756",
      "1.070983065"
    ],
    [
      "1.053354717",
      "1.152603899"
    ],
    [
      "0.956747066",
      "1.213899312"
    ],
    [
      "0.848744689",
      "1.251671301"
    ],
    [
      "0.734956116",
      "1.263948818"
    ],
    [
      "0.621135736",
      "1.250049541"
    ],
    [
      "0.512082869",
      "1.210145874"
    ],
    [
      "0.407332882",
      "1.141307182"
    ],
    [
      "0.325273408",
      "1.046557088"
    ],
    [
      "0.271328157",
      "0.943723540"
    ],
    [
      "0.242530766",
      "0.832732584"
    ]
  ]
}

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

Instance parameters

{
  "n": 28
}

The current leader's answer

{
  "points": [
    [
      "0.239685468",
      "0.718319011"
    ],
    [
      "0.262875239",
      "0.606276712"
    ],
    [
      "0.310886343",
      "0.502425403"
    ],
    [
      "0.381213540",
      "0.412180886"
    ],
    [
      "0.470186591",
      "0.340252413"
    ],
    [
      "0.573160700",
      "0.290394304"
    ],
    [
      "0.684751199",
      "0.265209880"
    ],
    [
      "0.799061975",
      "0.266005714"
    ],
    [
      "0.909658792",
      "0.292576593"
    ],
    [
      "1.007822768",
      "0.341674831"
    ],
    [
      "1.066135680",
      "0.388991508"
    ],
    [
      "1.120732314",
      "0.440549556"
    ],
    [
      "1.182354833",
      "0.531376119"
    ],
    [
      "1.223287653",
      "0.637499573"
    ],
    [
      "1.239160882",
      "0.750705687"
    ],
    [
      "1.228922092",
      "0.864643656"
    ],
    [
      "1.193088393",
      "0.973296450"
    ],
    [
      "1.133529756",
      "1.070983065"
    ],
    [
      "1.053354717",
      "1.152603899"
    ],
    [
      "0.956747066",
      "1.213899312"
    ],
    [
      "0.848744689",
      "1.251671301"
    ],
    [
      "0.734956116",
      "1.263948818"
    ],
    [
      "0.621135736",
      "1.250049541"
    ],
    [
      "0.512082869",
      "1.210145874"
    ],
    [
      "0.407332882",
      "1.141307182"
    ],
    [
      "0.325273408",
      "1.046557088"
    ],
    [
      "0.271328157",
      "0.943723540"
    ],
    [
      "0.242530766",
      "0.832732584"
    ]
  ]
}

Submit points, with each coordinate written as a decimal string such as "0.25" (at most nine decimal places). · Verifier v1.0.0

DISCUSSION

Discussion

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