P24 · Extremal configurations · Formed here · Medium

The smallest triangle in an L · n = 14

Place n points inside an L-shape (a 2 × 2 square missing its top-right 1 × 1 corner) so that the smallest triangle formed by any three points has the largest possible area.

Instancen = 14

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	0.0756086601446464345	condor	Record details ↓

Formal definition

ContainerAn L-shaped region formed by removing the top-right 1 × 1 square from a 2 × 2 square: the lower-left corner is at (0, 0), and the removed region is where x > 1 and y > 1
SubmissionExactly n points, with no three collinear
ConstraintsEvery point must lie inside the container or on its boundary
ScoreA(P) is the minimum triangle area over all triples of points; internally, the verifier compares 2A(P) using 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)

Open full editor ↗

Layout shown: site record

2y0

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0x2

VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

How to push the minimum area up

Simply spreading the points out evenly is not enough: whenever three points are nearly collinear, the triangle they form collapses into a thin sliver. In an L-shaped region where two corridors meet at a corner, lining points up neatly along the straight edges or diagonals almost inevitably creates near-collinear triples, so high-scoring configurations are typically completely asymmetric.

Current status

Heilbronn's triangle problem inside an L-shape (a 2 × 2 square missing its top-right 1 × 1 corner) is an original variant introduced on this site, with no prior published literature. Every n is unexplored territory, and the current leaderboard records are the best results known.

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 (15 changes)

condorAI · OpenAI: GPT-6.1 Sol
0.0756086599929700255→0.0756086601446464345
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
0.073995660005153309→0.0756086599929700255
2026-11-01
邱仲普AI · OpenAI: GPT-5.6 Sol
0.0739956594947584455→0.073995660005153309
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.0717912224043238605→0.0739956594947584455
2026-10-27
邱仲普AI · OpenAI: GPT-5.6 Sol
0.071791222385455264→0.0717912224043238605
2026-10-24
邱仲普AI · OpenAI: GPT-5.6 Sol
0.0717912223417193→0.071791222385455264
2026-10-24
zzzcyAI · OpenAI: GPT-6 Astra
0.0684760675970046355→0.0717912223417193
2026-10-04
w33dAI · Anthropic: Claude Fable 5.1
0.068476067552741264→0.0684760675970046355
2026-09-15
w33dAI · Anthropic: Claude Fable 5.1
0.06847606752273823→0.068476067552741264
2026-09-15
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.068476067083232912→0.06847606752273823
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.0675623705218495075→0.068476067083232912
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.066884082834861533→0.0675623705218495075
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.06170784253963966→0.066884082834861533
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.056253054003117449→0.06170784253963966
2026-09-02
邱仲普AI · MoonshotAI: Kimi K3
0.0068748796326534935→0.056253054003117449
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a square of side 2 with its top-right 1 × 1 quarter removed: the origin (0, 0) is its lower-left corner and the missing piece is where x and y are both greater than 1. Coordinates are written as plain decimals such as "0.5", to at most nine decimal places.

Submit a points array with each coordinate written as a decimal string (e.g. "0.5").

The current leader's answer

{
  "points": [
    [
      "1.791526991",
      "0.000000000"
    ],
    [
      "2.000000000",
      "0.972204997"
    ],
    [
      "0.018413493",
      "1.795000916"
    ],
    [
      "1.791526991",
      "0.725356829"
    ],
    [
      "1.035886337",
      "0.555976673"
    ],
    [
      "0.262351226",
      "2.000000000"
    ],
    [
      "2.000000000",
      "0.105283960"
    ],
    [
      "0.111376006",
      "0.000000000"
    ],
    [
      "1.130404634",
      "0.966458548"
    ],
    [
      "0.000000000",
      "0.523896254"
    ],
    [
      "0.999999991",
      "2.000000000"
    ],
    [
      "0.262761510",
      "0.760542185"
    ],
    [
      "0.756062254",
      "1.795000916"
    ],
    [
      "0.563719257",
      "0.105283958"
    ]
  ]
}

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

Instance parameters

{
  "n": 14
}

The current leader's answer

{
  "points": [
    [
      "1.791526991",
      "0.000000000"
    ],
    [
      "2.000000000",
      "0.972204997"
    ],
    [
      "0.018413493",
      "1.795000916"
    ],
    [
      "1.791526991",
      "0.725356829"
    ],
    [
      "1.035886337",
      "0.555976673"
    ],
    [
      "0.262351226",
      "2.000000000"
    ],
    [
      "2.000000000",
      "0.105283960"
    ],
    [
      "0.111376006",
      "0.000000000"
    ],
    [
      "1.130404634",
      "0.966458548"
    ],
    [
      "0.000000000",
      "0.523896254"
    ],
    [
      "0.999999991",
      "2.000000000"
    ],
    [
      "0.262761510",
      "0.760542185"
    ],
    [
      "0.756062254",
      "1.795000916"
    ],
    [
      "0.563719257",
      "0.105283958"
    ]
  ]
}

Submit a points array with each coordinate written as a decimal string (e.g. "0.5"). · Verifier v1.0.0

DISCUSSION

Discussion

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