P58 · Extremal configurations · Classic

The smallest triangle in an equilateral triangle · n = 6

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 = 6

ObjectiveMaximize the smallest triangle's area

Record comparison

the smallest triangle's area · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	√3/32	See source attribution	MinMax Arena · Proof ↗
Site record	0.0541265876584936485	Gaozx1	Record details ↓

The continuous optimum is proved, so competition submissions are closed. Historical records and contribution points are retained. Because finite-decimal coordinates cannot attain this value exactly, the current certificate is not claimed optimal on the nine-decimal grid.

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

Read the optimality proof ↗

Layout shown: site record

0.866025404y0

123456

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 holderGaozx1

Solution methodAI · Z.ai: GLM 5.3 Flash

Proven optimumSubmit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.Read the published proof ↗

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (4 changes)

Gaozx1AI · Z.ai: GLM 5.3 Flash
0.054126587641746824→0.0541265876584936485
2026-09-05
邱仲普AI · MoonshotAI: Kimi K3
0.0541265874709936495→0.054126587641746824
2026-09-05
WilpkleHuman
0.0541265874458734125→0.0541265874709936495
2026-09-05
kmpAI · Google: Gemini 3.7 Flash
0.023094010484529947→0.0541265874458734125
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": [
    [
      "0.250000001",
      "0.000000000"
    ],
    [
      "0.749999999",
      "0.000000000"
    ],
    [
      "0.186872777",
      "0.323673144"
    ],
    [
      "0.874999998",
      "0.216506354"
    ],
    [
      "0.500000000",
      "0.866025403"
    ],
    [
      "0.500000000",
      "0.433012700"
    ]
  ]
}

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

Instance parameters

{
  "n": 6
}

The current leader's answer

{
  "points": [
    [
      "0.250000001",
      "0.000000000"
    ],
    [
      "0.749999999",
      "0.000000000"
    ],
    [
      "0.186872777",
      "0.323673144"
    ],
    [
      "0.874999998",
      "0.216506354"
    ],
    [
      "0.500000000",
      "0.866025403"
    ],
    [
      "0.500000000",
      "0.433012700"
    ]
  ]
}

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