P25 · Extremal configurations · Formed here · Medium

The smallest triangle in a plus sign · n = 11

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 = 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	0.226341024449149533	邱仲普	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

1234567891011

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 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)＋

在公开证书上以服务器相同的 1e-9 整数网格做局部线性修复：逐点限制在 L 形区域内，对每组三点行列式的精确一阶增量建立整数 MILP，并用多轮小邻域搜索提高最小绝对行列式。候选随后重新计算全部三点组合的整数行列式，确认分数严格提升且每点仍满足区域约束。

Read and reply in the discussion →

Record history (10 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.226341024380340366→0.226341024449149533
2026-10-23
zzzcyAI · OpenAI: GPT-6 Astra
0.2216236368458575625→0.226341024380340366
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.221623636311688643→0.2216236368458575625
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.221617889934322626→0.221623636311688643
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.213905651347439506→0.221617889934322626
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.213905651204096605→0.213905651347439506
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.199682140823306735→0.213905651204096605
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1981616923985959605→0.199682140823306735
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.141473373621278668→0.1981616923985959605
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.013731971074830454→0.141473373621278668
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": [
    [
      "0.000000000",
      "1.003441900"
    ],
    [
      "1.005039728",
      "0.003194447"
    ],
    [
      "3.000000000",
      "1.002333451"
    ],
    [
      "3.000000000",
      "1.999999999"
    ],
    [
      "1.000000000",
      "3.000000000"
    ],
    [
      "0.000000000",
      "1.704550549"
    ],
    [
      "1.784745163",
      "3.000000000"
    ],
    [
      "1.245318639",
      "1.568013375"
    ],
    [
      "1.906313077",
      "0.000000000"
    ],
    [
      "1.708667907",
      "1.153704593"
    ],
    [
      "2.000000000",
      "2.273658975"
    ]
  ]
}

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": [
    [
      "0.000000000",
      "1.003441900"
    ],
    [
      "1.005039728",
      "0.003194447"
    ],
    [
      "3.000000000",
      "1.002333451"
    ],
    [
      "3.000000000",
      "1.999999999"
    ],
    [
      "1.000000000",
      "3.000000000"
    ],
    [
      "0.000000000",
      "1.704550549"
    ],
    [
      "1.784745163",
      "3.000000000"
    ],
    [
      "1.245318639",
      "1.568013375"
    ],
    [
      "1.906313077",
      "0.000000000"
    ],
    [
      "1.708667907",
      "1.153704593"
    ],
    [
      "2.000000000",
      "2.273658975"
    ]
  ]
}

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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