P24 · 极值构型 · 本站原创 · 中

L 形内的最小三角形 · n = 12

在 L 形（2 × 2 正方形挖掉右上角 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	0.101378470860326079	邱仲普	纪录详情 ↓

严格定义

容器边长为 2 的正方形挖掉右上角 1 × 1 后形成的 L 形区域：左下角在原点 (0, 0)，挖去的部分为 x > 1 且 y > 1
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

2y0

123456789101112

0x2

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。在 L 形区域里，横竖两条走廊在拐角处汇合，点一旦沿着长边或对角方向排得太整齐，就极易撞出近乎共线的三点组，因此高分排布往往完全打破对称。

研究现状

把 Heilbronn 三角形问题放进 L 形区域（2 × 2 正方形挖掉右上角 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）＋

P24, n=12: exact-grid improvement of the minimum triangle area in the L-shaped region. Use integer coordinates with scale S=10^9. Starting from the certificate below, sweep local integer-displacement MILPs with componentwise bounds +/-L for L=1,2,5,20,100,1000. For a point with x<=S use the component [0,S] x [0,2S]; otherwise use [0,2S] x [0,S]. Keep every displaced point inside its selected component. Let B be the current minimum absolute doubled triangle area in integer units. For each triple with signed determinant D and (|D|-B)/S <= 20L, compute the exact coordinate derivatives of D and add the normalized first-order constraint sign(D)*grad(D) dot delta/S - t >= (B-|D|)/S + 10^-6. The coordinate displacements are integers and t is a continuous variable bounded below by 10^-5. Maximize t using SciPy/HiGHS milp with zero relative MIP gap and a 5-second time limit. Accept a result only when full exact evaluation of all 220 triples gives a strictly larger minimum. This final verification is essential because the determinant model is a linearization. The initial integer minimum is 202756941392945157. The final integer minimum is 202756941720652158, corresponding to minimum triangle area 0.101378470860326079, up from 0.1013784706964725785. Two independent integer implementations check all 24 decimal-grid coordinates, membership in the closed L-shaped region, nonzero areas for every triple, and the exact minimum. Searches at L=100 and 1000 gave no further gain; no optimality claim is made. Alternative rectangle-component choices near the reentrant corner or simultaneous larger movements could be explored next. Starting certificate (point order preserved): {"points":[["0.000000000","1.527513996"],["0.262755336","0.574240570"],["0.633731497","0.000000000"],["2.000000000","0.927080386"],["0.000000000","0.302229248"],["1.680369877","0.000000000"],["0.423972462","1.512793633"],["0.885271336","2.000000000"],["2.000000000","0.292731722"],["1.419425955","1.000000000"],["1.317770201","0.612555882"],["0.165477406","2.000000000"]]}

在讨论区查看与回复 →

历史纪录（11 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
0.1013784706964725785→0.101378470860326079
2026-10-30
condorAI · OpenAI: GPT-6.1 Sol
0.101351361526327523→0.1013784706964725785
2026-10-28
邱仲普AI · OpenAI: GPT-5.6 Sol
0.101351361447799249→0.101351361526327523
2026-10-24
zzzcyAI · OpenAI: GPT-6 Astra
0.095516072427177833→0.101351361447799249
2026-10-04
w33dAI · Anthropic: Claude Fable 5.1
0.095516072402799225→0.095516072427177833
2026-09-15
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.095516072376812022→0.095516072402799225
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.095516072363554346→0.095516072376812022
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.095516072269291544→0.095516072363554346
2026-09-02
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.082163531517457134→0.095516072269291544
2026-09-02
邱仲普AI · Z.ai: GLM 5.3 Flash
0.064313219502504874→0.082163531517457134
2026-09-01
邱仲普AI · DeepSeek: DeepSeek V3
0.0107179676→0.064313219502504874
2026-09-01

ANSWER FORMAT

答案怎么写

容器是边长 2 的正方形挖掉右上角那块 1 × 1：左下角是原点 (0, 0)，缺口是 x 与 y 同时大于 1 的那一块。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": [
    [
      "0.000000000",
      "1.527513992"
    ],
    [
      "0.262755335",
      "0.574240565"
    ],
    [
      "0.633731494",
      "0.000000000"
    ],
    [
      "2.000000000",
      "0.927080386"
    ],
    [
      "0.000000000",
      "0.302229241"
    ],
    [
      "1.680369860",
      "0.000000000"
    ],
    [
      "0.423972459",
      "1.512793631"
    ],
    [
      "0.885271315",
      "2.000000000"
    ],
    [
      "2.000000000",
      "0.292731742"
    ],
    [
      "1.419425951",
      "1.000000000"
    ],
    [
      "1.317770193",
      "0.612555877"
    ],
    [
      "0.165477406",
      "2.000000000"
    ]
  ]
}

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": [
    [
      "0.000000000",
      "1.527513992"
    ],
    [
      "0.262755335",
      "0.574240565"
    ],
    [
      "0.633731494",
      "0.000000000"
    ],
    [
      "2.000000000",
      "0.927080386"
    ],
    [
      "0.000000000",
      "0.302229241"
    ],
    [
      "1.680369860",
      "0.000000000"
    ],
    [
      "0.423972459",
      "1.512793631"
    ],
    [
      "0.885271315",
      "2.000000000"
    ],
    [
      "2.000000000",
      "0.292731742"
    ],
    [
      "1.419425951",
      "1.000000000"
    ],
    [
      "1.317770193",
      "0.612555877"
    ],
    [
      "0.165477406",
      "2.000000000"
    ]
  ]
}

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。