P25 · Extremal configurations · Formed here · Medium

The smallest triangle in a plus sign · n = 7

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 = 7

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	0.551127991	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

1234567

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 · DeepSeek: DeepSeek V4 Pro 0423

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 (2 changes)

邱仲普AI · DeepSeek: DeepSeek V4 Pro 0423
0.5→0.551127991
2026-09-01
bowen2147483647Human
0.047098804142046847→0.5
2026-08-30

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",
      "1.102255982"
    ],
    [
      "1",
      "0"
    ],
    [
      "2.814460554",
      "2"
    ],
    [
      "3",
      "1.102255982"
    ],
    [
      "0.185539446",
      "2"
    ],
    [
      "1.5",
      "2.419280738"
    ],
    [
      "2",
      "0"
    ]
  ]
}

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

Instance parameters

{
  "n": 7
}

The current leader's answer

{
  "points": [
    [
      "0",
      "1.102255982"
    ],
    [
      "1",
      "0"
    ],
    [
      "2.814460554",
      "2"
    ],
    [
      "3",
      "1.102255982"
    ],
    [
      "0.185539446",
      "2"
    ],
    [
      "1.5",
      "2.419280738"
    ],
    [
      "2",
      "0"
    ]
  ]
}

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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