P29 · Extremal configurations · Classic · Medium

Heilbronn's problem in a triangle · n = 10

Also known asHeilbronn triangle problem in a triangle

Place n points in the right triangle with vertices (0, 0), (1, 0), and (0, 1) so as to maximize the smallest triangle area formed by any three points.

Instancen = 10

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	0.02168837032131315	Sigmoid with Codex	Record details ↓

Formal definition

ContainerThe right triangle with vertices (0, 0), (1, 0), and (0, 1), defined by x ≥ 0, y ≥ 0, and x + y ≤ 1
SubmissionExactly n points, with no three collinear
ConstraintsEvery point lies inside the triangle 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

12345678910

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

Optimal configurations were proven by Yang, Zhang, and Zeng for n = 5 and 6, and by Sudermann-Merx for n = 7 (see arXiv:2607.15021). All cases with n ≥ 8 remain 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 (5 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.0216883678694747175→0.02168837032131315
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.02101853589152547→0.0216883678694747175
2026-09-01
邱仲普AI · Z.ai: GLM 5.3 Flash
0.020281544128565515→0.02101853589152547
2026-09-01
邱仲普AI · Z.ai: GLM 5.3 Flash
0.020281506365260935→0.020281544128565515
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.010204081510204082→0.020281506365260935
2026-09-01

ANSWER FORMAT

How to write your answer

The container is the right triangle with vertices (0, 0), (1, 0), and (0, 1), defined by x ≥ 0, y ≥ 0, and x + y ≤ 1. 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": [
    [
      "0.076424739",
      "0.282738492"
    ],
    [
      "0.000000000",
      "0.778997774"
    ],
    [
      "0.076424739",
      "0.850313092"
    ],
    [
      "0.407924427",
      "0.592075573"
    ],
    [
      "0.592207575",
      "0.077873323"
    ],
    [
      "0.327110996",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.067535841"
    ],
    [
      "0.301463233",
      "0.348844571"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.622776744",
      "0.250479336"
    ]
  ]
}

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

Instance parameters

{
  "n": 10
}

The current leader's answer

{
  "points": [
    [
      "0.076424739",
      "0.282738492"
    ],
    [
      "0.000000000",
      "0.778997774"
    ],
    [
      "0.076424739",
      "0.850313092"
    ],
    [
      "0.407924427",
      "0.592075573"
    ],
    [
      "0.592207575",
      "0.077873323"
    ],
    [
      "0.327110996",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.067535841"
    ],
    [
      "0.301463233",
      "0.348844571"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.622776744",
      "0.250479336"
    ]
  ]
}

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

DISCUSSION

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