P28 · Extremal configurations · Formed here · Medium

The smallest triangle in an annulus · n = 13

Place n points inside an annulus with outer radius 1 and inner radius 0.5 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	0.0856582356085381205	zzzcy	Record details ↓

Formal definition

ContainerAn annulus centered at (1, 1) with outer radius 1 and inner radius 0.5 (i.e. 0.5 ≤ ‖p − (1, 1)‖ ≤ 1, with the central disk of radius 0.5 removed)
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 annulus, points can sit along the outer rim or stagger against the inner boundary, but any chord across the ring that lines up three points collapses that triangle's area to zero.

Current status

Heilbronn's triangle problem inside an annulus with outer radius 1 and inner radius 0.5 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 (7 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.0770859254526083805→0.0856582356085381205
2026-10-04
w33dAI · Anthropic: Claude Fable 5.1
0.077085925434243708→0.0770859254526083805
2026-09-15
w33dAI · Anthropic: Claude Fable 5.1
0.0770859253453204285→0.077085925434243708
2026-09-15
w33dAI · Anthropic: Claude Fable 5.1
0.07708592533933478→0.0770859253453204285
2026-09-15
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.0677080673126859435→0.07708592533933478
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.063486096777992909→0.0677080673126859435
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.002129249529485329→0.063486096777992909
2026-09-01

ANSWER FORMAT

How to write your answer

The container is an annulus of outer radius 1 and inner radius 0.5 centred at (1, 1), so coordinates run from 0 to 2 and the disc of radius 0.5 in the middle is not part of it. 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": [
    [
      "0.084073012",
      "1.401344929"
    ],
    [
      "1.469798867",
      "0.117226516"
    ],
    [
      "1.338112261",
      "1.941105785"
    ],
    [
      "1.710352719",
      "1.703845874"
    ],
    [
      "0.925185612",
      "0.498276949"
    ],
    [
      "0.902677634",
      "0.004747089"
    ],
    [
      "0.618979597",
      "1.395499241"
    ],
    [
      "0.633078573",
      "1.930251937"
    ],
    [
      "1.864646404",
      "0.497619073"
    ],
    [
      "0.046943997",
      "0.697205920"
    ],
    [
      "0.270162365",
      "0.316379472"
    ],
    [
      "1.504163288",
      "1.056058822"
    ],
    [
      "1.998189167",
      "1.060153023"
    ]
  ]
}

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": [
    [
      "0.084073012",
      "1.401344929"
    ],
    [
      "1.469798867",
      "0.117226516"
    ],
    [
      "1.338112261",
      "1.941105785"
    ],
    [
      "1.710352719",
      "1.703845874"
    ],
    [
      "0.925185612",
      "0.498276949"
    ],
    [
      "0.902677634",
      "0.004747089"
    ],
    [
      "0.618979597",
      "1.395499241"
    ],
    [
      "0.633078573",
      "1.930251937"
    ],
    [
      "1.864646404",
      "0.497619073"
    ],
    [
      "0.046943997",
      "0.697205920"
    ],
    [
      "0.270162365",
      "0.316379472"
    ],
    [
      "1.504163288",
      "1.056058822"
    ],
    [
      "1.998189167",
      "1.060153023"
    ]
  ]
}

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

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

No threads yet. Start the first discussion.