P24 · Extremal configurations · Formed here · Medium

The smallest triangle in an L · n = 13

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 = 13

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	0.090197283468343719	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

12345678910111213

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 (14 changes)

condorAI · OpenAI: GPT-6.1 Sol
0.0901972830240850805→0.090197283468343719
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
0.085788032358594058→0.0901972830240850805
2026-11-01
邱仲普AI · OpenAI: GPT-5.6 Sol
0.08578803211061221→0.085788032358594058
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.084542906709286631→0.08578803211061221
2026-10-28
邱仲普AI · OpenAI: GPT-5.6 Sol
0.084542906706486861→0.084542906709286631
2026-10-25
邱仲普AI · OpenAI: GPT-5.6 Sol
0.0845429067022025915→0.084542906706486861
2026-10-24
邱仲普AI · OpenAI: GPT-5.6 Sol
0.084542906686319923→0.0845429067022025915
2026-10-24
zzzcyAI · OpenAI: GPT-6 Astra
0.0787731538203000465→0.084542906686319923
2026-10-04
w33dAI · Anthropic: Claude Fable 5.1
0.0770492792075712605→0.0787731538203000465
2026-09-15
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.0770492791434967205→0.0770492792075712605
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.077049278791245414→0.0770492791434967205
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.07704816307481659→0.077049278791245414
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.057753322025515196→0.07704816307481659
2026-09-02
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.008516998145363208→0.057753322025515196
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": [
    [
      "1.587066279",
      "0.000000000"
    ],
    [
      "2.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "1.587066274"
    ],
    [
      "1.698183626",
      "0.705953278"
    ],
    [
      "0.734436963",
      "0.364712746"
    ],
    [
      "0.139277133",
      "1.864521850"
    ],
    [
      "1.864521854",
      "0.139277133"
    ],
    [
      "0.291845992",
      "0.000000000"
    ],
    [
      "0.949618628",
      "0.949618630"
    ],
    [
      "0.000000000",
      "0.291845990"
    ],
    [
      "0.364712747",
      "0.734436964"
    ],
    [
      "1.000000000",
      "2.000000000"
    ],
    [
      "0.705953277",
      "1.698183624"
    ]
  ]
}

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

Instance parameters

{
  "n": 13
}

The current leader's answer

{
  "points": [
    [
      "1.587066279",
      "0.000000000"
    ],
    [
      "2.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "1.587066274"
    ],
    [
      "1.698183626",
      "0.705953278"
    ],
    [
      "0.734436963",
      "0.364712746"
    ],
    [
      "0.139277133",
      "1.864521850"
    ],
    [
      "1.864521854",
      "0.139277133"
    ],
    [
      "0.291845992",
      "0.000000000"
    ],
    [
      "0.949618628",
      "0.949618630"
    ],
    [
      "0.000000000",
      "0.291845990"
    ],
    [
      "0.364712747",
      "0.734436964"
    ],
    [
      "1.000000000",
      "2.000000000"
    ],
    [
      "0.705953277",
      "1.698183624"
    ]
  ]
}

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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