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

The biggest little polygon · n = 26

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 = 26

ObjectiveMaximize the area

Record comparison

the area · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.780411119912473194	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 (10 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.7804111199124100395→0.780411119912473194
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.780411119853786162→0.7804111199124100395
2026-10-03
zzzcyAI · OpenAI: GPT-6 Astra
0.780411119839984386→0.780411119853786162
2026-10-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.780411119396640365→0.780411119839984386
2026-09-01
邱仲普AI · MoonshotAI: Kimi K2.7 Code
0.7795958531186793935→0.780411119396640365
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.7673393644686637795→0.7795958531186793935
2026-09-01
NUE_13Human
0.7672661598047937795→0.7673393644686637795
2026-08-29
张虹膜Human
0.7507254934622937795→0.7672661598047937795
2026-08-29
bowen2147483647Human
0.5237351124278018115→0.7507254934622937795
2026-08-29
hpt114Human
0.497776581631905516→0.5237351124278018115
2026-08-28

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": [
    [
      "0.272164690",
      "0.811004913"
    ],
    [
      "0.272264254",
      "0.730060723"
    ],
    [
      "0.278910175",
      "0.649389072"
    ],
    [
      "0.315830241",
      "0.536928371"
    ],
    [
      "0.381318395",
      "0.433196567"
    ],
    [
      "0.470611634",
      "0.348182761"
    ],
    [
      "0.578054822",
      "0.287529065"
    ],
    [
      "0.697078051",
      "0.254975829"
    ],
    [
      "0.820449939",
      "0.252505127"
    ],
    [
      "0.940684075",
      "0.280267669"
    ],
    [
      "1.050488609",
      "0.336581467"
    ],
    [
      "1.143221416",
      "0.418051822"
    ],
    [
      "1.213353661",
      "0.519916231"
    ],
    [
      "1.256967283",
      "0.637327474"
    ],
    [
      "1.271394384",
      "0.771761842"
    ],
    [
      "1.245814965",
      "0.904526539"
    ],
    [
      "1.192569208",
      "1.017894881"
    ],
    [
      "1.114192327",
      "1.113561028"
    ],
    [
      "1.014993006",
      "1.187020601"
    ],
    [
      "0.900877822",
      "1.233988420"
    ],
    [
      "0.778748293",
      "1.251635235"
    ],
    [
      "0.656011472",
      "1.238892470"
    ],
    [
      "0.540114927",
      "1.196534201"
    ],
    [
      "0.438099747",
      "1.127138158"
    ],
    [
      "0.356201270",
      "1.034979119"
    ],
    [
      "0.299584950",
      "0.926151068"
    ]
  ]
}

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

Instance parameters

{
  "n": 26
}

The current leader's answer

{
  "points": [
    [
      "0.272164690",
      "0.811004913"
    ],
    [
      "0.272264254",
      "0.730060723"
    ],
    [
      "0.278910175",
      "0.649389072"
    ],
    [
      "0.315830241",
      "0.536928371"
    ],
    [
      "0.381318395",
      "0.433196567"
    ],
    [
      "0.470611634",
      "0.348182761"
    ],
    [
      "0.578054822",
      "0.287529065"
    ],
    [
      "0.697078051",
      "0.254975829"
    ],
    [
      "0.820449939",
      "0.252505127"
    ],
    [
      "0.940684075",
      "0.280267669"
    ],
    [
      "1.050488609",
      "0.336581467"
    ],
    [
      "1.143221416",
      "0.418051822"
    ],
    [
      "1.213353661",
      "0.519916231"
    ],
    [
      "1.256967283",
      "0.637327474"
    ],
    [
      "1.271394384",
      "0.771761842"
    ],
    [
      "1.245814965",
      "0.904526539"
    ],
    [
      "1.192569208",
      "1.017894881"
    ],
    [
      "1.114192327",
      "1.113561028"
    ],
    [
      "1.014993006",
      "1.187020601"
    ],
    [
      "0.900877822",
      "1.233988420"
    ],
    [
      "0.778748293",
      "1.251635235"
    ],
    [
      "0.656011472",
      "1.238892470"
    ],
    [
      "0.540114927",
      "1.196534201"
    ],
    [
      "0.438099747",
      "1.127138158"
    ],
    [
      "0.356201270",
      "1.034979119"
    ],
    [
      "0.299584950",
      "0.926151068"
    ]
  ]
}

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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