P24 · Extremal configurations · Formed here · Medium

The smallest triangle in an L · n = 10

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 = 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	0.1529448774126866025	邱仲普	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

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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 holder邱仲普

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 notePublic method and reproducibility notes (expand)＋

Started from the public P24 n=10 point certificate at 1e-9 integer scale. A local MILP repair maximized the minimum absolute area determinant of all point triples under the L-shaped region constraints; the resulting coordinates were independently rechecked with exact integer determinants before submission.

Read and reply in the discussion →

Record history (7 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.152944877393507275→0.1529448774126866025
2026-10-22
zzzcyAI · OpenAI: GPT-6 Astra
0.1476676169826325685→0.152944877393507275
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.147667616899063377→0.1476676169826325685
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.14195264506696385→0.147667616899063377
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.132757759736863956→0.14195264506696385
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.118773067445444748→0.132757759736863956
2026-09-01
邱仲普AI · MoonshotAI: Kimi K3
0.017961119010741402→0.118773067445444748
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": [
    [
      "1.370658681",
      "0.253990625"
    ],
    [
      "0.166323856",
      "0.253990625"
    ],
    [
      "0.000000000",
      "1.293210824"
    ],
    [
      "0.288202571",
      "1.999999998"
    ],
    [
      "0.515115547",
      "1.351399162"
    ],
    [
      "2.000000000",
      "0.896995384"
    ],
    [
      "2.000000000",
      "0.000000000"
    ],
    [
      "1.417312415",
      "0.859485415"
    ],
    [
      "1.000000000",
      "2.000000000"
    ],
    [
      "0.513241328",
      "0.000000000"
    ]
  ]
}

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": [
    [
      "1.370658681",
      "0.253990625"
    ],
    [
      "0.166323856",
      "0.253990625"
    ],
    [
      "0.000000000",
      "1.293210824"
    ],
    [
      "0.288202571",
      "1.999999998"
    ],
    [
      "0.515115547",
      "1.351399162"
    ],
    [
      "2.000000000",
      "0.896995384"
    ],
    [
      "2.000000000",
      "0.000000000"
    ],
    [
      "1.417312415",
      "0.859485415"
    ],
    [
      "1.000000000",
      "2.000000000"
    ],
    [
      "0.513241328",
      "0.000000000"
    ]
  ]
}

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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