P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 39

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 = 39

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	[0.137790333, 0.137790337]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.137790335	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

123456789101112131415161718192021222324252627282930313233343536373839

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（3 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.13779033593→0.13779033596
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.1377903358→0.1377903359
2026-10-18
今天AI · DeepSeek: DeepSeek V4.1 Flash
0.111111111→0.137790335
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": [
    [
      "0.603945718",
      "0.396010938"
    ],
    [
      "0.173984295",
      "0.826015704"
    ],
    [
      "0.000000000",
      "0.315111124"
    ],
    [
      "0.407578854",
      "0.000000000"
    ],
    [
      "0.962509789",
      "0.006189577"
    ],
    [
      "0.120185202",
      "0.382503215"
    ],
    [
      "0.368849258",
      "0.631150742"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.600465364"
    ],
    [
      "0.401978660",
      "0.240936168"
    ],
    [
      "0.000000000",
      "0.452901460"
    ],
    [
      "0.189107076",
      "0.125398369"
    ],
    [
      "0.470872246",
      "0.360267019"
    ],
    [
      "0.609377517",
      "0.122020944"
    ],
    [
      "0.701399869",
      "0.298600131"
    ],
    [
      "0.000000000",
      "0.039530452"
    ],
    [
      "0.872176703",
      "0.127823297"
    ],
    [
      "0.043112145",
      "0.869127850"
    ],
    [
      "0.506534911",
      "0.493465089"
    ],
    [
      "0.130916724",
      "0.695072626"
    ],
    [
      "0.257975535",
      "0.244743717"
    ],
    [
      "0.545369190",
      "0.000000000"
    ],
    [
      "0.116371819",
      "0.526683412"
    ],
    [
      "0.235754014",
      "0.595487978"
    ],
    [
      "0.000000000",
      "0.738255700"
    ],
    [
      "0.471587181",
      "0.122020944"
    ],
    [
      "0.739863808",
      "0.166287236"
    ],
    [
      "0.368744614",
      "0.493360445"
    ],
    [
      "0.333081908",
      "0.360265185"
    ],
    [
      "0.271416777",
      "0.728583223"
    ],
    [
      "0.120185202",
      "0.244712879"
    ],
    [
      "0.269788518",
      "0.000000000"
    ],
    [
      "0.131998182",
      "0.000000000"
    ],
    [
      "0.820723992",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.177320788"
    ],
    [
      "0.683046591",
      "0.005577631"
    ],
    [
      "0.542507227",
      "0.242505195"
    ],
    [
      "0.235649385",
      "0.457697624"
    ],
    [
      "0.326897414",
      "0.125398368"
    ]
  ]
}

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": 39
}

当前第一名的答案

{
  "points": [
    [
      "0.603945718",
      "0.396010938"
    ],
    [
      "0.173984295",
      "0.826015704"
    ],
    [
      "0.000000000",
      "0.315111124"
    ],
    [
      "0.407578854",
      "0.000000000"
    ],
    [
      "0.962509789",
      "0.006189577"
    ],
    [
      "0.120185202",
      "0.382503215"
    ],
    [
      "0.368849258",
      "0.631150742"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.600465364"
    ],
    [
      "0.401978660",
      "0.240936168"
    ],
    [
      "0.000000000",
      "0.452901460"
    ],
    [
      "0.189107076",
      "0.125398369"
    ],
    [
      "0.470872246",
      "0.360267019"
    ],
    [
      "0.609377517",
      "0.122020944"
    ],
    [
      "0.701399869",
      "0.298600131"
    ],
    [
      "0.000000000",
      "0.039530452"
    ],
    [
      "0.872176703",
      "0.127823297"
    ],
    [
      "0.043112145",
      "0.869127850"
    ],
    [
      "0.506534911",
      "0.493465089"
    ],
    [
      "0.130916724",
      "0.695072626"
    ],
    [
      "0.257975535",
      "0.244743717"
    ],
    [
      "0.545369190",
      "0.000000000"
    ],
    [
      "0.116371819",
      "0.526683412"
    ],
    [
      "0.235754014",
      "0.595487978"
    ],
    [
      "0.000000000",
      "0.738255700"
    ],
    [
      "0.471587181",
      "0.122020944"
    ],
    [
      "0.739863808",
      "0.166287236"
    ],
    [
      "0.368744614",
      "0.493360445"
    ],
    [
      "0.333081908",
      "0.360265185"
    ],
    [
      "0.271416777",
      "0.728583223"
    ],
    [
      "0.120185202",
      "0.244712879"
    ],
    [
      "0.269788518",
      "0.000000000"
    ],
    [
      "0.131998182",
      "0.000000000"
    ],
    [
      "0.820723992",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.177320788"
    ],
    [
      "0.683046591",
      "0.005577631"
    ],
    [
      "0.542507227",
      "0.242505195"
    ],
    [
      "0.235649385",
      "0.457697624"
    ],
    [
      "0.326897414",
      "0.125398368"
    ]
  ]
}

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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