P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 38

在直角边长为 1 的等腰直角三角形内放置 n 个点，使任意两点之间的最小欧氏距离尽可能大。

子题n = 38

目标最大化最小两点距离

纪录对比

最小两点距离 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[0.140083498, 0.140083502]	David W. Cantrell2010-10-26	Packomania ↗
本站纪录	0.140083501	matrixfang	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

严格定义

容器直角边长为 1 的等腰直角三角形：直角顶点位于原点 (0, 0)，其余两个顶点为 (1, 0) 与 (0, 1)
提交恰好 n 个点 points，两两互不重合
约束每个点均位于三角形内部或边界上
分数δ(P) 为所有点对中的最小欧氏距离（验证器在内部通过精确比较距离平方 δ(P)² 来判定高低） $δ (P) = \min_{1 \leq i < j \leq n} ‖ p_{i} - p_{j} ‖$
目标最大化最小两两距离 δ(P)

挑战这个纪录 ↗

当前展示：本站纪录

1y0

1234567891011121314151617181920212223242526272829303132333435363738

0x1

已验证构造最近的一对点

帮助理解

顶点占位与等圆装箱等价性

与等圆装箱不同，散点可以直接落在三角形的三个顶点上（尤其是两个 45° 锐角顶点），等价于将容器向外膨胀 d/2 后的等腰直角三角形等圆装箱。在最优构型中，三个顶点通常会被优先占据，其余点沿三条边与内部接触图紧凑排布。

已知证明与文献对照

本题经膨胀换算后与等腰直角三角形内等圆装箱等价，相关外部构造与作者收录于 Packomania。其中 n = 4 已通过缩放 L 形区域的归约证明最优距离为 √2/2；其余规模的已知最优数值仅作参考基准，尚未证明最优性。

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matrixfang

解题方式AI · OpenAI: GPT-6.1 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暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（2 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.1400835010→0.1400835011
2026-10-18
今天AI · DeepSeek: DeepSeek V4.1 Flash
0.111111111→0.140083501
2026-10-17

ANSWER FORMAT

答案怎么写

容器是直角边为 1 的等腰直角三角形，直角顶点在原点 (0, 0)，另两个顶点是 (1, 0) 与 (0, 1)。坐标直接写成小数，例如 "0.5"，最多九位小数。

提交点集坐标 points，每个坐标均以十进制字符串表示（如 "0.5"）。

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.234569514"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.859916498"
    ],
    [
      "0.634800405",
      "0.365199595"
    ],
    [
      "0.140083502",
      "0.477201255"
    ],
    [
      "0.433368482",
      "0.000000000"
    ],
    [
      "0.293284983",
      "0.242631743"
    ],
    [
      "0.832908393",
      "0.167091607"
    ],
    [
      "0.726479216",
      "0.000000000"
    ],
    [
      "0.153201478",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.477201254"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.503410234",
      "0.121315870"
    ],
    [
      "0.000000000",
      "0.719832996"
    ],
    [
      "0.280167004",
      "0.719832996"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.363326732",
      "0.121315871"
    ],
    [
      "0.216684240",
      "0.359916498"
    ],
    [
      "0.583672378",
      "0.000012955"
    ],
    [
      "0.223243230",
      "0.121315871"
    ],
    [
      "0.733854399",
      "0.266145601"
    ],
    [
      "0.140083502",
      "0.719832996"
    ],
    [
      "0.146642490",
      "0.238600627"
    ],
    [
      "0.076600738",
      "0.117284757"
    ],
    [
      "0.863239608",
      "0.030331215"
    ],
    [
      "0.210125254",
      "0.598517125"
    ],
    [
      "0.478274980",
      "0.521725020"
    ],
    [
      "0.686578279",
      "0.134280686"
    ],
    [
      "0.070041752",
      "0.355885384"
    ],
    [
      "0.293284980",
      "0.000000000"
    ],
    [
      "0.356556188",
      "0.367612311"
    ],
    [
      "0.140083502",
      "0.859916498"
    ],
    [
      "0.437975463",
      "0.245962603"
    ],
    [
      "0.379186489",
      "0.620744505"
    ],
    [
      "0.070041750",
      "0.598517125"
    ],
    [
      "0.309144739",
      "0.499428634"
    ],
    [
      "0.495819972",
      "0.382744587"
    ],
    [
      "0.587524285",
      "0.233334680"
    ]
  ]
}

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": 38
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.234569514"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.859916498"
    ],
    [
      "0.634800405",
      "0.365199595"
    ],
    [
      "0.140083502",
      "0.477201255"
    ],
    [
      "0.433368482",
      "0.000000000"
    ],
    [
      "0.293284983",
      "0.242631743"
    ],
    [
      "0.832908393",
      "0.167091607"
    ],
    [
      "0.726479216",
      "0.000000000"
    ],
    [
      "0.153201478",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.477201254"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.503410234",
      "0.121315870"
    ],
    [
      "0.000000000",
      "0.719832996"
    ],
    [
      "0.280167004",
      "0.719832996"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.363326732",
      "0.121315871"
    ],
    [
      "0.216684240",
      "0.359916498"
    ],
    [
      "0.583672378",
      "0.000012955"
    ],
    [
      "0.223243230",
      "0.121315871"
    ],
    [
      "0.733854399",
      "0.266145601"
    ],
    [
      "0.140083502",
      "0.719832996"
    ],
    [
      "0.146642490",
      "0.238600627"
    ],
    [
      "0.076600738",
      "0.117284757"
    ],
    [
      "0.863239608",
      "0.030331215"
    ],
    [
      "0.210125254",
      "0.598517125"
    ],
    [
      "0.478274980",
      "0.521725020"
    ],
    [
      "0.686578279",
      "0.134280686"
    ],
    [
      "0.070041752",
      "0.355885384"
    ],
    [
      "0.293284980",
      "0.000000000"
    ],
    [
      "0.356556188",
      "0.367612311"
    ],
    [
      "0.140083502",
      "0.859916498"
    ],
    [
      "0.437975463",
      "0.245962603"
    ],
    [
      "0.379186489",
      "0.620744505"
    ],
    [
      "0.070041750",
      "0.598517125"
    ],
    [
      "0.309144739",
      "0.499428634"
    ],
    [
      "0.495819972",
      "0.382744587"
    ],
    [
      "0.587524285",
      "0.233334680"
    ]
  ]
}

提交点集坐标 points，每个坐标均以十进制字符串表示（如 "0.5"）。 · 验证器 v1.0.0

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