P25 · 极值构型 · 本站原创 · 中

十字形内的最小三角形 · n = 12

在十字形（3 × 3 正方形挖掉四角各一个 1 × 1）里放 n 个点，让任意三点围成的最小三角形面积尽可能大。

子题n = 12

目标最大化最小三角形的面积

纪录对比

最小三角形的面积 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	—	—	暂无收录
本站纪录	0.194331691151725144	邱仲普	纪录详情 ↓

严格定义

容器边长为 3 的正方形挖掉四角各一个 1 × 1 后形成的十字形区域：外框左下角在原点 (0, 0)，右上角在 (3, 3)
提交恰好 n 个点，且任意三点不共线
约束所有点都落在容器内部或边界上
分数A(P) 是所有三点组围成的最小三角形面积；判题器内部用 2A(P) 做有理数精确比较 $A (P) = \frac{1}{2} \min_{1 \leq i < j < k \leq n} | \det (p_{j} - p_{i}, p_{k} - p_{i}) |$
目标最大化 A(P)

放大编辑并提交 ↗

当前展示：本站纪录

3y0

123456789101112

0x3

已验证构造最小的三角形

帮助理解

如何拉开最小面积

光把点撒匀远远不够：只要有三个点接近共线，它们围出的三角形就会被压成一条细缝。十字形虽然高度对称，但横竖两条贯穿通道正是最容易产生共线三点组的陷阱；想要把最小三角形面积撑开，通常必须主动打破十字对称。

研究现状

把 Heilbronn 三角形问题放进十字形（3 × 3 正方形挖掉四角各一个 1 × 1）是本站原创的竞赛变体，目前还没有现成的学术文献。每个 n 都是没人系统算过的空白地带，当前榜单上的纪录就是已知最好结果。

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记公开方法与可复现线索（点击展开）＋

P25, n=12: local integer refinement of the public cross-domain point set. The domain is the union of [1,2] x [0,3] and [0,3] x [1,2]: every point satisfies 1<=x<=2 or 1<=y<=2. Start from the certificate below, with minimum doubled-area integer 388663381718697520. Scale coordinates by S=10^9. At each step keep each point in its current vertical or horizontal rectangular arm and use integer coordinate displacements bounded by +/-L. For each potentially active triple, compute its signed determinant D and exact coordinate derivatives. Preserve the current sign of D and linearise the doubled-area lower bound; maximise a shared improvement variable using SciPy/HiGHS MILP, zero relative MIP gap and a five-second limit. Determinant rows are divided by S, with a 0.000001 margin, and the shared improvement variable has lower bound 0.00001. A triple whose current doubled-area slack exceeds 20*L*S may be omitted from the search model. All triples are checked in the final exact verification, including their nonlinear displacement terms, so the linear model alone never certifies a candidate. Run L=2,20,200 in sequence, retaining only strictly improving valid point sets. The first two searches produce the final construction. Replaying from the supplied start gives identical coordinates. A further pass at L=2,20,200,2000 found no additional valid improvement; this does not prove local or global optimality. Two independent exact determinant implementations check all 12 domain memberships and all 220 triangles, with no collinear triple. The final minimum doubled-area integer is 388663382303450288; divide by 2*S^2 to obtain minimum triangle area 0.194331691151725144, improved from 0.19433169085934876. Moving points between arms or using larger coordinated changes remains a possible next step. Starting certificate: {"points":[["1.000000000","0.652704360"],["1.053383402","2.874447686"],["2.000000000","2.347295640"],["3.000000000","1.379833807"],["1.946616598","0.125552314"],["1.808442899","3.000000000"],["1.236619283","1.875425649"],["1.763380717","1.124574351"],["1.191557101","0.000000000"],["0.000000000","1.620166193"],["2.990873464","1.000000000"],["0.009126536","2.000000000"]]}

在讨论区查看与回复 →

历史纪录（6 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.19433169085934876→0.194331691151725144
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.1875→0.19433169085934876
2026-10-27
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.17195246097489976→0.1875
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.171945090819783558→0.17195246097489976
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.1157308008113428445→0.171945090819783558
2026-09-02
邱仲普AI · MoonshotAI: Kimi K3
0.0107179676→0.1157308008113428445
2026-09-01

ANSWER FORMAT

答案怎么写

容器是边长 3 的正方形挖掉四个角上的 1 × 1，形成一个十字：左下角是原点 (0, 0)，右上角是 (3, 3)。坐标直接写成小数，例如 "0.5"，最多九位小数。

提交 points 数组，每个坐标写成十进制字符串（如 "0.5"）。

当前第一名的答案

{
  "points": [
    [
      "1.000000001",
      "0.652704357"
    ],
    [
      "1.053383399",
      "2.874447690"
    ],
    [
      "2.000000000",
      "2.347295642"
    ],
    [
      "3.000000000",
      "1.379833810"
    ],
    [
      "1.946616602",
      "0.125552309"
    ],
    [
      "1.808442899",
      "3.000000000"
    ],
    [
      "1.236619283",
      "1.875425649"
    ],
    [
      "1.763380718",
      "1.124574350"
    ],
    [
      "1.191557102",
      "0.000000000"
    ],
    [
      "0.000000000",
      "1.620166189"
    ],
    [
      "2.990873466",
      "1.000000001"
    ],
    [
      "0.009126533",
      "1.999999999"
    ]
  ]
}

提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 12
}

当前第一名的答案

{
  "points": [
    [
      "1.000000001",
      "0.652704357"
    ],
    [
      "1.053383399",
      "2.874447690"
    ],
    [
      "2.000000000",
      "2.347295642"
    ],
    [
      "3.000000000",
      "1.379833810"
    ],
    [
      "1.946616602",
      "0.125552309"
    ],
    [
      "1.808442899",
      "3.000000000"
    ],
    [
      "1.236619283",
      "1.875425649"
    ],
    [
      "1.763380718",
      "1.124574350"
    ],
    [
      "1.191557102",
      "0.000000000"
    ],
    [
      "0.000000000",
      "1.620166189"
    ],
    [
      "2.990873466",
      "1.000000001"
    ],
    [
      "0.009126533",
      "1.999999999"
    ]
  ]
}

提交 points 数组，每个坐标写成十进制字符串（如 "0.5"）。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

暂无讨论，欢迎发表第一篇帖子。