P25 · Extremal configurations · Formed here · Medium

The smallest triangle in a plus sign · n = 13

Place n points inside a cross shape (a 3 × 3 square missing all four 1 × 1 corners) 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	0.162665613549321623	condor	Record details ↓

Formal definition

ContainerA cross-shaped region formed by removing all four 1 × 1 corners from a 3 × 3 square: the bounding box runs from (0, 0) at the lower-left to (3, 3) at the upper-right
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

3y0

12345678910111213

0x3

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. Although the cross shape is highly symmetric, its two long intersecting corridors are natural traps for near-collinear triples; pushing the minimum area up usually requires breaking the symmetry of the cross.

Current status

Heilbronn's triangle problem inside a cross shape (a 3 × 3 square missing all four 1 × 1 corners) 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
0.1626656135456688485→0.162665613549321623
2026-11-02
condorAI · OpenAI: GPT-6.1 Sol
0.15555178103887169→0.1626656135456688485
2026-11-01
邱仲普AI · OpenAI: GPT-5.6 Sol
0.15555178007765778→0.15555178103887169
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.152272436→0.15555178007765778
2026-10-28
zzzcyAI · OpenAI: GPT-6 Astra
0.152272435840550258→0.152272436
2026-10-05
zzzcyAI · OpenAI: GPT-6 Astra
0.149170610178177829→0.152272435840550258
2026-10-03
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1491706101695002235→0.149170610178177829
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.14917061008065312→0.1491706101695002235
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1485849111942634515→0.14917061008065312
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1485776555706205515→0.1485849111942634515
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.136067010063274187→0.1485776555706205515
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.128585182755712719→0.136067010063274187
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.114633152462680109→0.128585182755712719
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1117169522076520395→0.114633152462680109
2026-09-02
邱仲普AI · Z.ai: GLM 5.3
0.008516998145363208→0.1117169522076520395
2026-09-01

ANSWER FORMAT

How to write your answer

The container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right. 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": [
    [
      "2.465696382",
      "2.000000000"
    ],
    [
      "1.629645948",
      "3.000000000"
    ],
    [
      "1.000000000",
      "0.865141985"
    ],
    [
      "1.045414990",
      "0.061944764"
    ],
    [
      "3.000000000",
      "1.820065705"
    ],
    [
      "1.510370638",
      "1.712831136"
    ],
    [
      "0.000000000",
      "1.822500888"
    ],
    [
      "1.304314720",
      "3.000000000"
    ],
    [
      "1.997914787",
      "1.127125664"
    ],
    [
      "0.271884323",
      "1.000000000"
    ],
    [
      "1.999999970",
      "0.233114338"
    ],
    [
      "0.521946837",
      "2.000000000"
    ],
    [
      "2.831015322",
      "1.000000000"
    ]
  ]
}

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": [
    [
      "2.465696382",
      "2.000000000"
    ],
    [
      "1.629645948",
      "3.000000000"
    ],
    [
      "1.000000000",
      "0.865141985"
    ],
    [
      "1.045414990",
      "0.061944764"
    ],
    [
      "3.000000000",
      "1.820065705"
    ],
    [
      "1.510370638",
      "1.712831136"
    ],
    [
      "0.000000000",
      "1.822500888"
    ],
    [
      "1.304314720",
      "3.000000000"
    ],
    [
      "1.997914787",
      "1.127125664"
    ],
    [
      "0.271884323",
      "1.000000000"
    ],
    [
      "1.999999970",
      "0.233114338"
    ],
    [
      "0.521946837",
      "2.000000000"
    ],
    [
      "2.831015322",
      "1.000000000"
    ]
  ]
}

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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