P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 37

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 = 37

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	[0.142169146, 0.14216915]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.142169148	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

12345678910111213141516171819202122232425262728293031323334353637

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
0.1421691485→0.1421691489
2026-10-18
今天AI · DeepSeek: DeepSeek V4.1 Flash
0.111111111→0.142169148
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",
      "0.244965006"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.819748729"
    ],
    [
      "0.625992798",
      "0.374007202"
    ],
    [
      "0.142163482",
      "0.491199380"
    ],
    [
      "0.430893329",
      "0.000000000"
    ],
    [
      "0.286525574",
      "0.246234374"
    ],
    [
      "0.827050337",
      "0.172949662"
    ],
    [
      "0.724519618",
      "0.000000000"
    ],
    [
      "0.144362092",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.489930014"
    ],
    [
      "0.586089756",
      "0.237552762"
    ],
    [
      "0.010110307",
      "0.989568873"
    ],
    [
      "0.490574493",
      "0.417296342"
    ],
    [
      "0.500875767",
      "0.123751870"
    ],
    [
      "0.000000000",
      "0.632099163"
    ],
    [
      "0.246227814",
      "0.753772186"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.358706618",
      "0.123751870"
    ],
    [
      "0.214344528",
      "0.368716877"
    ],
    [
      "0.580524672",
      "0.000008912"
    ],
    [
      "0.216543136",
      "0.122482504"
    ],
    [
      "0.726521567",
      "0.273478432"
    ],
    [
      "0.106812813",
      "0.725923946"
    ],
    [
      "0.144362090",
      "0.244965007"
    ],
    [
      "0.072181046",
      "0.122482503"
    ],
    [
      "0.862259809",
      "0.035209471"
    ],
    [
      "0.209431730",
      "0.616447333"
    ],
    [
      "0.447285353",
      "0.552714647"
    ],
    [
      "0.686618525",
      "0.137023992"
    ],
    [
      "0.072181044",
      "0.367447510"
    ],
    [
      "0.288724180",
      "0.000000000"
    ],
    [
      "0.356508010",
      "0.369986247"
    ],
    [
      "0.134661053",
      "0.865338947"
    ],
    [
      "0.443950716",
      "0.257687763"
    ],
    [
      "0.302456585",
      "0.501637480"
    ],
    [
      "0.346756583",
      "0.653243416"
    ]
  ]
}

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": 37
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.244965006"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.819748729"
    ],
    [
      "0.625992798",
      "0.374007202"
    ],
    [
      "0.142163482",
      "0.491199380"
    ],
    [
      "0.430893329",
      "0.000000000"
    ],
    [
      "0.286525574",
      "0.246234374"
    ],
    [
      "0.827050337",
      "0.172949662"
    ],
    [
      "0.724519618",
      "0.000000000"
    ],
    [
      "0.144362092",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.489930014"
    ],
    [
      "0.586089756",
      "0.237552762"
    ],
    [
      "0.010110307",
      "0.989568873"
    ],
    [
      "0.490574493",
      "0.417296342"
    ],
    [
      "0.500875767",
      "0.123751870"
    ],
    [
      "0.000000000",
      "0.632099163"
    ],
    [
      "0.246227814",
      "0.753772186"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.358706618",
      "0.123751870"
    ],
    [
      "0.214344528",
      "0.368716877"
    ],
    [
      "0.580524672",
      "0.000008912"
    ],
    [
      "0.216543136",
      "0.122482504"
    ],
    [
      "0.726521567",
      "0.273478432"
    ],
    [
      "0.106812813",
      "0.725923946"
    ],
    [
      "0.144362090",
      "0.244965007"
    ],
    [
      "0.072181046",
      "0.122482503"
    ],
    [
      "0.862259809",
      "0.035209471"
    ],
    [
      "0.209431730",
      "0.616447333"
    ],
    [
      "0.447285353",
      "0.552714647"
    ],
    [
      "0.686618525",
      "0.137023992"
    ],
    [
      "0.072181044",
      "0.367447510"
    ],
    [
      "0.288724180",
      "0.000000000"
    ],
    [
      "0.356508010",
      "0.369986247"
    ],
    [
      "0.134661053",
      "0.865338947"
    ],
    [
      "0.443950716",
      "0.257687763"
    ],
    [
      "0.302456585",
      "0.501637480"
    ],
    [
      "0.346756583",
      "0.653243416"
    ]
  ]
}

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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