P58 · Extremal configurations · Classic · Medium

The smallest triangle in an equilateral triangle · n = 13

Also known asHeilbronn triangle problem in an equilateral triangle

Place n points inside an equilateral triangle of side 1 so that the smallest triangle formed by any three of them is as large as possible.

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	0.011499314290281675	Sigmoid with Codex	Record details ↓

Formal definition

ContainerThe container is an equilateral triangle of side 1: its base runs from (0, 0) to (1, 0) and its apex is at (1/2, √3/2).
SubmissionExactly n points, no three collinear
ConstraintsEvery point lies inside the container or on its boundary
ScoreA(P) is the smallest triangle area over all triples; the verifier compares 2A(P) exactly $A (P) = \frac{1}{2} \min_{1 \leq i < j < k \leq n} | \det (p_{j} - p_{i}, p_{k} - p_{i}) |$
ObjectiveMake A(P) as large as possible

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Layout shown: site record

0.866025404y0

12345678910111213

0x1

VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

Where the room for improvement is

Spreading points evenly is only part of the job, because any three nearly collinear points immediately collapse the area of the triangle they form. Regular grids naturally create collinear or near-collinear triples, so strong configurations usually break symmetry.

Where the frontier is

Affine maps between triangles preserve area ratios, so the classical Heilbronn optima for n = 5 and 6 carry over directly; those continuous cases are solved and archived, with nine-decimal grid notes retained. Case n = 11 includes a site-verifiable version of AlphaEvolve's published construction, while the remaining sizes are still open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 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 (3 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.008362990930081632→0.011499314290281675
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.0028436872080270245→0.008362990930081632
2026-09-01
lirdHuman
0.0028389993623336325→0.0028436872080270245
2026-08-27

ANSWER FORMAT

How to write your answer

The container is an equilateral triangle of side 1: its base runs from (0, 0) to (1, 0) and its apex is at (1/2, √3/2). Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

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

The current leader's answer

{
  "points": [
    [
      "0.297623885",
      "0"
    ],
    [
      "0.958294776",
      "0.072235566"
    ],
    [
      "0.128530523",
      "0.222621396"
    ],
    [
      "0.61600761",
      "0"
    ],
    [
      "0.391426253",
      "0.677970157"
    ],
    [
      "0.653757434",
      "0.599709716"
    ],
    [
      "0.423258682",
      "0.141488029"
    ],
    [
      "0.755357031",
      "0.141488029"
    ],
    [
      "0.510045868",
      "0.422285844"
    ],
    [
      "0.096883301",
      "0.072235566"
    ],
    [
      "0.319534102",
      "0.403541005"
    ],
    [
      "0.766455277",
      "0.317799473"
    ],
    [
      "0.50122477",
      "0.777192188"
    ]
  ]
}

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": [
    [
      "0.297623885",
      "0"
    ],
    [
      "0.958294776",
      "0.072235566"
    ],
    [
      "0.128530523",
      "0.222621396"
    ],
    [
      "0.61600761",
      "0"
    ],
    [
      "0.391426253",
      "0.677970157"
    ],
    [
      "0.653757434",
      "0.599709716"
    ],
    [
      "0.423258682",
      "0.141488029"
    ],
    [
      "0.755357031",
      "0.141488029"
    ],
    [
      "0.510045868",
      "0.422285844"
    ],
    [
      "0.096883301",
      "0.072235566"
    ],
    [
      "0.319534102",
      "0.403541005"
    ],
    [
      "0.766455277",
      "0.317799473"
    ],
    [
      "0.50122477",
      "0.777192188"
    ]
  ]
}

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

DISCUSSION

Discussion

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比较有可能的值大小糖#1813 replies52d ago