P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 42

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 = 42

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	[0.132046096, 0.1320461]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.132046099	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

123456789101112131415161718192021222324252627282930313233343536373839404142

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
0.132046098→0.132046099
2026-10-18
今天AI · DeepSeek: DeepSeek V4.1 Flash
0.111111111→0.132046098
2026-10-18

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": [
    [
      "0.975727964",
      "0.006633390"
    ],
    [
      "0.744462552",
      "0.093370693"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.399194112",
      "0.254071692"
    ],
    [
      "0.238415301",
      "0.466755520"
    ],
    [
      "0.558521576",
      "0.279239493"
    ],
    [
      "0.132046102",
      "0.000000000"
    ],
    [
      "0.066023051",
      "0.114355276"
    ],
    [
      "0.685876510",
      "0.314123489"
    ],
    [
      "0.495171183",
      "0.163382253"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.634256651",
      "0.169714968"
    ],
    [
      "0.179153963",
      "0.820846037"
    ],
    [
      "0.651091860",
      "0.000000000"
    ],
    [
      "0.359481765",
      "0.126895691"
    ],
    [
      "0.110913985",
      "0.432410405"
    ],
    [
      "0.114355278",
      "0.570087206"
    ],
    [
      "0.000000000",
      "0.636110258"
    ],
    [
      "0.523615081",
      "0.034436074"
    ],
    [
      "0.372023948",
      "0.627976052"
    ],
    [
      "0.264092202",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.360756652"
    ],
    [
      "0.592505818",
      "0.407494182"
    ],
    [
      "0.000000000",
      "0.228710552"
    ],
    [
      "0.333171061",
      "0.368426968"
    ],
    [
      "0.272524656",
      "0.727475344"
    ],
    [
      "0.000000000",
      "0.768156358"
    ],
    [
      "0.198069151",
      "0.114355277"
    ],
    [
      "0.063232142",
      "0.884078179"
    ],
    [
      "0.000000000",
      "0.504064158"
    ],
    [
      "0.872617896",
      "0.127382104"
    ],
    [
      "0.396138302",
      "0.000000000"
    ],
    [
      "0.243261272",
      "0.598712668"
    ],
    [
      "0.837833245",
      "0.000000000"
    ],
    [
      "0.465150884",
      "0.372610186"
    ],
    [
      "0.270431436",
      "0.224808308"
    ],
    [
      "0.204408385",
      "0.339163584"
    ],
    [
      "0.367177977",
      "0.496018904"
    ],
    [
      "0.499135125",
      "0.500864875"
    ],
    [
      "0.116060239",
      "0.704453496"
    ],
    [
      "0.779247203",
      "0.220752797"
    ],
    [
      "0.132046100",
      "0.228710553"
    ]
  ]
}

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": 42
}

当前第一名的答案

{
  "points": [
    [
      "0.975727964",
      "0.006633390"
    ],
    [
      "0.744462552",
      "0.093370693"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.399194112",
      "0.254071692"
    ],
    [
      "0.238415301",
      "0.466755520"
    ],
    [
      "0.558521576",
      "0.279239493"
    ],
    [
      "0.132046102",
      "0.000000000"
    ],
    [
      "0.066023051",
      "0.114355276"
    ],
    [
      "0.685876510",
      "0.314123489"
    ],
    [
      "0.495171183",
      "0.163382253"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.634256651",
      "0.169714968"
    ],
    [
      "0.179153963",
      "0.820846037"
    ],
    [
      "0.651091860",
      "0.000000000"
    ],
    [
      "0.359481765",
      "0.126895691"
    ],
    [
      "0.110913985",
      "0.432410405"
    ],
    [
      "0.114355278",
      "0.570087206"
    ],
    [
      "0.000000000",
      "0.636110258"
    ],
    [
      "0.523615081",
      "0.034436074"
    ],
    [
      "0.372023948",
      "0.627976052"
    ],
    [
      "0.264092202",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.360756652"
    ],
    [
      "0.592505818",
      "0.407494182"
    ],
    [
      "0.000000000",
      "0.228710552"
    ],
    [
      "0.333171061",
      "0.368426968"
    ],
    [
      "0.272524656",
      "0.727475344"
    ],
    [
      "0.000000000",
      "0.768156358"
    ],
    [
      "0.198069151",
      "0.114355277"
    ],
    [
      "0.063232142",
      "0.884078179"
    ],
    [
      "0.000000000",
      "0.504064158"
    ],
    [
      "0.872617896",
      "0.127382104"
    ],
    [
      "0.396138302",
      "0.000000000"
    ],
    [
      "0.243261272",
      "0.598712668"
    ],
    [
      "0.837833245",
      "0.000000000"
    ],
    [
      "0.465150884",
      "0.372610186"
    ],
    [
      "0.270431436",
      "0.224808308"
    ],
    [
      "0.204408385",
      "0.339163584"
    ],
    [
      "0.367177977",
      "0.496018904"
    ],
    [
      "0.499135125",
      "0.500864875"
    ],
    [
      "0.116060239",
      "0.704453496"
    ],
    [
      "0.779247203",
      "0.220752797"
    ],
    [
      "0.132046100",
      "0.228710553"
    ]
  ]
}

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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