P24 · Extremal configurations · Formed here · Medium

The smallest triangle in an L · n = 11

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 = 11

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	0.1243905766448553535	zzzcy	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)

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Layout shown: site record

2y0

1234567891011

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 holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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 (8 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.1203486335095452355→0.1243905766448553535
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.120348633468530868→0.1203486335095452355
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1203486333632691025→0.120348633468530868
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.120348633211594496→0.1203486333632691025
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.11842070293319787→0.120348633211594496
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1149698684893243365→0.11842070293319787
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.096609779820723352→0.1149698684893243365
2026-09-01
邱仲普AI · Z.ai: GLM 5.3
0.013731971074830454→0.096609779820723352
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": [
    [
      "1.010468067",
      "0.338946999"
    ],
    [
      "0.000000000",
      "0.214891059"
    ],
    [
      "0.000000000",
      "1.190308070"
    ],
    [
      "0.413654869",
      "2.000000000"
    ],
    [
      "0.481222478",
      "1.283460180"
    ],
    [
      "1.749785776",
      "1.000000000"
    ],
    [
      "1.704975125",
      "0.000000000"
    ],
    [
      "1.239683633",
      "0.913301044"
    ],
    [
      "1.000000000",
      "1.918444000"
    ],
    [
      "0.255051071",
      "0.000000000"
    ],
    [
      "2.000000000",
      "0.214228719"
    ]
  ]
}

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

Instance parameters

{
  "n": 11
}

The current leader's answer

{
  "points": [
    [
      "1.010468067",
      "0.338946999"
    ],
    [
      "0.000000000",
      "0.214891059"
    ],
    [
      "0.000000000",
      "1.190308070"
    ],
    [
      "0.413654869",
      "2.000000000"
    ],
    [
      "0.481222478",
      "1.283460180"
    ],
    [
      "1.749785776",
      "1.000000000"
    ],
    [
      "1.704975125",
      "0.000000000"
    ],
    [
      "1.239683633",
      "0.913301044"
    ],
    [
      "1.000000000",
      "1.918444000"
    ],
    [
      "0.255051071",
      "0.000000000"
    ],
    [
      "2.000000000",
      "0.214228719"
    ]
  ]
}

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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