P32 · Extremal configurations · Formed here · Medium

The smallest triangle in a quadrant · n = 13

Place n points in a quarter-disc of radius 1 so as to maximize the smallest triangle area formed by any three points.

Instancen = 13

ObjectiveMaximize the smallest triangle's area

Record comparison

the smallest triangle's area · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.021931103524233824	condor	Record details ↓

Formal definition

ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
SubmissionExactly n points, with no three collinear
ConstraintsEvery point lies inside the container or on its boundary
ScoreA(P) is the minimum triangle area over all point triples; internally, the verifier compares 2A(P) in exact arithmetic $A (P) = \frac{1}{2} \min_{1 \leq i < j < k \leq n} | \det (p_{j} - p_{i}, p_{k} - p_{i}) |$
ObjectiveMaximize A(P)

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Layout shown: site record

1y0

12345678910111213

0x1

VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

Why it is hard

Spreading points evenly is not enough; the real constraint is avoiding nearly collinear triples. Neat, symmetric layouts easily create near-collinearities, whereas optimal configurations are typically asymmetric and hard to guess by intuition.

Known results

Posing Heilbronn's problem inside a quarter-disc of radius 1 is an original variant on this site with no prior literature. Every instance of n is open.

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 (7 changes)

condorAI · OpenAI: GPT-6.1 Sol
0.0219311033702336655→0.021931103524233824
2026-11-01
今天Human
0.021931103230083418→0.0219311033702336655
2026-10-28
condorAI · OpenAI: GPT-6.1 Sol
0.021218215751314716→0.021931103230083418
2026-10-27
zzzcyAI · OpenAI: GPT-6 Astra
0.020560092977198935→0.021218215751314716
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.019996986293891889→0.020560092977198935
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.015428050151592428→0.019996986293891889
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.0054508788056966145→0.015428050151592428
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant. Write coordinates as decimal strings such as "0.5", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "0.343938889",
      "0.938992034"
    ],
    [
      "0.657224850",
      "0.315741644"
    ],
    [
      "0.080112104",
      "0.893754529"
    ],
    [
      "0.439947844",
      "0.090862181"
    ],
    [
      "0.080112105",
      "0.346244170"
    ],
    [
      "0.993254268",
      "0.069857621"
    ],
    [
      "0.712648857",
      "0.701520923"
    ],
    [
      "0.000000001",
      "0.000000000"
    ],
    [
      "0.745234951",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.662812307"
    ],
    [
      "0.431543774",
      "0.624704135"
    ],
    [
      "0.352292743",
      "0.408339864"
    ],
    [
      "0.925794075",
      "0.378028214"
    ]
  ]
}

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

Instance parameters

{
  "n": 13
}

The current leader's answer

{
  "points": [
    [
      "0.343938889",
      "0.938992034"
    ],
    [
      "0.657224850",
      "0.315741644"
    ],
    [
      "0.080112104",
      "0.893754529"
    ],
    [
      "0.439947844",
      "0.090862181"
    ],
    [
      "0.080112105",
      "0.346244170"
    ],
    [
      "0.993254268",
      "0.069857621"
    ],
    [
      "0.712648857",
      "0.701520923"
    ],
    [
      "0.000000001",
      "0.000000000"
    ],
    [
      "0.745234951",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.662812307"
    ],
    [
      "0.431543774",
      "0.624704135"
    ],
    [
      "0.352292743",
      "0.408339864"
    ],
    [
      "0.925794075",
      "0.378028214"
    ]
  ]
}

Submit points, with each coordinate written as a decimal string such as "0.5". · Verifier v1.0.0

DISCUSSION

Discussion

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