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

The biggest little polygon · n = 30

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 = 30

ObjectiveMaximize the area

Record comparison

the area · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[0.7816739265, 0.7816739275]	See source attribution	arxiv.org ↗
Site record	0.781673926695356704	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

123456789101112131415161718192021222324252627282930

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 (2 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.781673926650096605→0.781673926695356704
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.781673926428627079→0.781673926650096605
2026-10-02

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": [
    [
      "0.235970667",
      "0.696044399"
    ],
    [
      "0.259429952",
      "0.581599189"
    ],
    [
      "0.311206345",
      "0.476874588"
    ],
    [
      "0.378769340",
      "0.392312226"
    ],
    [
      "0.461676188",
      "0.324856967"
    ],
    [
      "0.556810518",
      "0.276584891"
    ],
    [
      "0.659981238",
      "0.249563179"
    ],
    [
      "0.766530155",
      "0.245003816"
    ],
    [
      "0.871627238",
      "0.263112337"
    ],
    [
      "0.970505487",
      "0.303067212"
    ],
    [
      "1.058679186",
      "0.363055736"
    ],
    [
      "1.132147675",
      "0.440355527"
    ],
    [
      "1.187573777",
      "0.531451055"
    ],
    [
      "1.222427201",
      "0.632144478"
    ],
    [
      "1.235114829",
      "0.737407952"
    ],
    [
      "1.225658381",
      "0.839286253"
    ],
    [
      "1.205254317",
      "0.906277908"
    ],
    [
      "1.180211836",
      "0.971677030"
    ],
    [
      "1.125103161",
      "1.057884031"
    ],
    [
      "1.050440007",
      "1.133162086"
    ],
    [
      "0.961091114",
      "1.191219900"
    ],
    [
      "0.861401878",
      "1.229068016"
    ],
    [
      "0.755947015",
      "1.244947813"
    ],
    [
      "0.649519582",
      "1.238134485"
    ],
    [
      "0.546948688",
      "1.208936760"
    ],
    [
      "0.452887607",
      "1.158679116"
    ],
    [
      "0.371602928",
      "1.089641048"
    ],
    [
      "0.306780010",
      "1.004951150"
    ],
    [
      "0.261350762",
      "0.908427084"
    ],
    [
      "0.237352863",
      "0.804273931"
    ]
  ]
}

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": [
    [
      "0.235970667",
      "0.696044399"
    ],
    [
      "0.259429952",
      "0.581599189"
    ],
    [
      "0.311206345",
      "0.476874588"
    ],
    [
      "0.378769340",
      "0.392312226"
    ],
    [
      "0.461676188",
      "0.324856967"
    ],
    [
      "0.556810518",
      "0.276584891"
    ],
    [
      "0.659981238",
      "0.249563179"
    ],
    [
      "0.766530155",
      "0.245003816"
    ],
    [
      "0.871627238",
      "0.263112337"
    ],
    [
      "0.970505487",
      "0.303067212"
    ],
    [
      "1.058679186",
      "0.363055736"
    ],
    [
      "1.132147675",
      "0.440355527"
    ],
    [
      "1.187573777",
      "0.531451055"
    ],
    [
      "1.222427201",
      "0.632144478"
    ],
    [
      "1.235114829",
      "0.737407952"
    ],
    [
      "1.225658381",
      "0.839286253"
    ],
    [
      "1.205254317",
      "0.906277908"
    ],
    [
      "1.180211836",
      "0.971677030"
    ],
    [
      "1.125103161",
      "1.057884031"
    ],
    [
      "1.050440007",
      "1.133162086"
    ],
    [
      "0.961091114",
      "1.191219900"
    ],
    [
      "0.861401878",
      "1.229068016"
    ],
    [
      "0.755947015",
      "1.244947813"
    ],
    [
      "0.649519582",
      "1.238134485"
    ],
    [
      "0.546948688",
      "1.208936760"
    ],
    [
      "0.452887607",
      "1.158679116"
    ],
    [
      "0.371602928",
      "1.089641048"
    ],
    [
      "0.306780010",
      "1.004951150"
    ],
    [
      "0.261350762",
      "0.908427084"
    ],
    [
      "0.237352863",
      "0.804273931"
    ]
  ]
}

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.