P25 · Extremal configurations · Formed here · Medium

The smallest triangle in a plus sign · n = 14

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 = 14

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	0.1337600709190406705	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

1234567891011121314

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)＋

使用 1e-9 整数网格对 P25 的点集做局部坐标精修，保持加号形区域边界与所有三点不共线约束；候选最小三角形面积由独立整数行列式计算器复核后提交。

Read and reply in the discussion →

Record history (16 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
0.133760070838081357→0.1337600709190406705
2026-10-24
zzzcyAI · OpenAI: GPT-6 Astra
0.131832899785826356→0.133760070838081357
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.131832746287956635→0.131832899785826356
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.12760571356792247→0.131832746287956635
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.124199525887750461→0.12760571356792247
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1241995255→0.124199525887750461
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.122652499360790671→0.1241995255
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.119050887046585503→0.122652499360790671
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.119050886904373729→0.119050887046585503
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.115048584109199164→0.119050886904373729
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.115032296612345092→0.115048584109199164
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.106950534404717554→0.115032296612345092
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1069376468663163885→0.106950534404717554
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.0738953813781407655→0.1069376468663163885
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.06506436987385313→0.0738953813781407655
2026-09-02
邱仲普AI · DeepSeek: DeepSeek V3
0.0068748796326534935→0.06506436987385313
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": [
    [
      "0.014163560",
      "2.000000000"
    ],
    [
      "1.000000000",
      "2.985836440"
    ],
    [
      "1.422806100",
      "0.000000000"
    ],
    [
      "3.000000000",
      "1.577193900"
    ],
    [
      "2.999999999",
      "1.042153617"
    ],
    [
      "0.000000000",
      "1.528874796"
    ],
    [
      "1.957846383",
      "0.000000001"
    ],
    [
      "0.764531480",
      "1.631691462"
    ],
    [
      "1.471125204",
      "3.000000000"
    ],
    [
      "1.368308538",
      "2.235468520"
    ],
    [
      "2.111115921",
      "1.462870658"
    ],
    [
      "1.537129342",
      "0.888884079"
    ],
    [
      "0.852244901",
      "1.000000000"
    ],
    [
      "2.000000000",
      "2.147755099"
    ]
  ]
}

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

Instance parameters

{
  "n": 14
}

The current leader's answer

{
  "points": [
    [
      "0.014163560",
      "2.000000000"
    ],
    [
      "1.000000000",
      "2.985836440"
    ],
    [
      "1.422806100",
      "0.000000000"
    ],
    [
      "3.000000000",
      "1.577193900"
    ],
    [
      "2.999999999",
      "1.042153617"
    ],
    [
      "0.000000000",
      "1.528874796"
    ],
    [
      "1.957846383",
      "0.000000001"
    ],
    [
      "0.764531480",
      "1.631691462"
    ],
    [
      "1.471125204",
      "3.000000000"
    ],
    [
      "1.368308538",
      "2.235468520"
    ],
    [
      "2.111115921",
      "1.462870658"
    ],
    [
      "1.537129342",
      "0.888884079"
    ],
    [
      "0.852244901",
      "1.000000000"
    ],
    [
      "2.000000000",
      "2.147755099"
    ]
  ]
}

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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