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

The biggest little polygon · n = 24

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 = 24

ObjectiveMaximize the area

Record comparison

the area · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.77952404545, 0.77952404555]	See source attribution	arxiv.org ↗
Site record	0.779524045131161302	zzzcy	Record details ↓

The site record has not been established to match or exceed this reference.

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 $\max_{P} A (P)$

Open full editor ↗

Layout shown: site record

1.5y0

123456789101112131415161718192021222324

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 (3 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.779524045088889826→0.779524045131161302
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.7795240448141884005→0.779524045088889826
2026-10-02
游侠AI · OpenAI: GPT-5.6 Sol
0.779524044705923001→0.7795240448141884005
2026-09-12

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": [
    [
      "0.265369877",
      "0.715900666"
    ],
    [
      "0.288959049",
      "0.584198603"
    ],
    [
      "0.346624598",
      "0.463355628"
    ],
    [
      "0.434243349",
      "0.362087598"
    ],
    [
      "0.545555020",
      "0.287637752"
    ],
    [
      "0.672614581",
      "0.245323132"
    ],
    [
      "0.806381617",
      "0.238167716"
    ],
    [
      "0.937516494",
      "0.266755465"
    ],
    [
      "1.057895918",
      "0.329892972"
    ],
    [
      "1.164970271",
      "0.430251367"
    ],
    [
      "1.234717067",
      "0.559371629"
    ],
    [
      "1.265081078",
      "0.691869133"
    ],
    [
      "1.259294990",
      "0.825959149"
    ],
    [
      "1.218275629",
      "0.953482632"
    ],
    [
      "1.144968611",
      "1.065557286"
    ],
    [
      "1.044602539",
      "1.154212373"
    ],
    [
      "0.924344224",
      "1.213120728"
    ],
    [
      "0.792793407",
      "1.238075392"
    ],
    [
      "0.659429251",
      "1.227311286"
    ],
    [
      "0.534302162",
      "1.181861034"
    ],
    [
      "0.429421808",
      "1.107723476"
    ],
    [
      "0.372845046",
      "1.040609973"
    ],
    [
      "0.322372347",
      "0.968794530"
    ],
    [
      "0.277420694",
      "0.848480007"
    ]
  ]
}

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

Instance parameters

{
  "n": 24
}

The current leader's answer

{
  "points": [
    [
      "0.265369877",
      "0.715900666"
    ],
    [
      "0.288959049",
      "0.584198603"
    ],
    [
      "0.346624598",
      "0.463355628"
    ],
    [
      "0.434243349",
      "0.362087598"
    ],
    [
      "0.545555020",
      "0.287637752"
    ],
    [
      "0.672614581",
      "0.245323132"
    ],
    [
      "0.806381617",
      "0.238167716"
    ],
    [
      "0.937516494",
      "0.266755465"
    ],
    [
      "1.057895918",
      "0.329892972"
    ],
    [
      "1.164970271",
      "0.430251367"
    ],
    [
      "1.234717067",
      "0.559371629"
    ],
    [
      "1.265081078",
      "0.691869133"
    ],
    [
      "1.259294990",
      "0.825959149"
    ],
    [
      "1.218275629",
      "0.953482632"
    ],
    [
      "1.144968611",
      "1.065557286"
    ],
    [
      "1.044602539",
      "1.154212373"
    ],
    [
      "0.924344224",
      "1.213120728"
    ],
    [
      "0.792793407",
      "1.238075392"
    ],
    [
      "0.659429251",
      "1.227311286"
    ],
    [
      "0.534302162",
      "1.181861034"
    ],
    [
      "0.429421808",
      "1.107723476"
    ],
    [
      "0.372845046",
      "1.040609973"
    ],
    [
      "0.322372347",
      "0.968794530"
    ],
    [
      "0.277420694",
      "0.848480007"
    ]
  ]
}

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

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.

No threads yet. Start the first discussion.