P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 50

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 = 50

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	[0.119954346, 0.11995435]	David W. Cantrell2010-10-26	Packomania ↗
本站纪录	0.119954348	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950

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（1 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.1→0.119954348
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": [
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.424520100",
      "0.000000000"
    ],
    [
      "0.793781396",
      "0.000000000"
    ],
    [
      "0.210501616",
      "0.000000000"
    ],
    [
      "0.673827047",
      "0.000000000"
    ],
    [
      "0.000071044",
      "0.000000000"
    ],
    [
      "0.544474449",
      "0.000000000"
    ],
    [
      "0.317510858",
      "0.054203948"
    ],
    [
      "0.105286330",
      "0.057608936"
    ],
    [
      "0.896890698",
      "0.061298594"
    ],
    [
      "0.609150748",
      "0.101024859"
    ],
    [
      "0.729382653",
      "0.107811573"
    ],
    [
      "0.417957475",
      "0.119774696"
    ],
    [
      "0.207784775",
      "0.119923578"
    ],
    [
      "0.000000000",
      "0.119954328"
    ],
    [
      "0.835592104",
      "0.164407896"
    ],
    [
      "0.312912100",
      "0.177692872"
    ],
    [
      "0.103901261",
      "0.179900752"
    ],
    [
      "0.520566630",
      "0.181906871"
    ],
    [
      "0.634904566",
      "0.218181962"
    ],
    [
      "0.415521255",
      "0.239825047"
    ],
    [
      "0.207784775",
      "0.239877927"
    ],
    [
      "0.000000000",
      "0.239908677"
    ],
    [
      "0.750771570",
      "0.249228430"
    ],
    [
      "0.103901261",
      "0.299855101"
    ],
    [
      "0.311668289",
      "0.299855101"
    ],
    [
      "0.525799758",
      "0.306440768"
    ],
    [
      "0.665951034",
      "0.334048966"
    ],
    [
      "0.415582323",
      "0.359779381"
    ],
    [
      "0.207784776",
      "0.359832276"
    ],
    [
      "0.000000000",
      "0.359863026"
    ],
    [
      "0.581130500",
      "0.418869500"
    ],
    [
      "0.103901261",
      "0.419809450"
    ],
    [
      "0.315489329",
      "0.426917816"
    ],
    [
      "0.470913046",
      "0.466210377"
    ],
    [
      "0.207784774",
      "0.479786625"
    ],
    [
      "0.000000000",
      "0.479817375"
    ],
    [
      "0.103901261",
      "0.539763800"
    ],
    [
      "0.307370834",
      "0.546657115"
    ],
    [
      "0.423572169",
      "0.576427831"
    ],
    [
      "0.000000000",
      "0.599771724"
    ],
    [
      "0.221562889",
      "0.630539959"
    ],
    [
      "0.103901261",
      "0.659718149"
    ],
    [
      "0.337141550",
      "0.662858450"
    ],
    [
      "0.000035504",
      "0.719726068"
    ],
    [
      "0.252321016",
      "0.747678984"
    ],
    [
      "0.135614764",
      "0.775404361"
    ],
    [
      "0.000000000",
      "0.839680412"
    ],
    [
      "0.107889387",
      "0.892110613"
    ],
    [
      "0.005997699",
      "0.985005688"
    ]
  ]
}

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": 50
}

当前第一名的答案

{
  "points": [
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.424520100",
      "0.000000000"
    ],
    [
      "0.793781396",
      "0.000000000"
    ],
    [
      "0.210501616",
      "0.000000000"
    ],
    [
      "0.673827047",
      "0.000000000"
    ],
    [
      "0.000071044",
      "0.000000000"
    ],
    [
      "0.544474449",
      "0.000000000"
    ],
    [
      "0.317510858",
      "0.054203948"
    ],
    [
      "0.105286330",
      "0.057608936"
    ],
    [
      "0.896890698",
      "0.061298594"
    ],
    [
      "0.609150748",
      "0.101024859"
    ],
    [
      "0.729382653",
      "0.107811573"
    ],
    [
      "0.417957475",
      "0.119774696"
    ],
    [
      "0.207784775",
      "0.119923578"
    ],
    [
      "0.000000000",
      "0.119954328"
    ],
    [
      "0.835592104",
      "0.164407896"
    ],
    [
      "0.312912100",
      "0.177692872"
    ],
    [
      "0.103901261",
      "0.179900752"
    ],
    [
      "0.520566630",
      "0.181906871"
    ],
    [
      "0.634904566",
      "0.218181962"
    ],
    [
      "0.415521255",
      "0.239825047"
    ],
    [
      "0.207784775",
      "0.239877927"
    ],
    [
      "0.000000000",
      "0.239908677"
    ],
    [
      "0.750771570",
      "0.249228430"
    ],
    [
      "0.103901261",
      "0.299855101"
    ],
    [
      "0.311668289",
      "0.299855101"
    ],
    [
      "0.525799758",
      "0.306440768"
    ],
    [
      "0.665951034",
      "0.334048966"
    ],
    [
      "0.415582323",
      "0.359779381"
    ],
    [
      "0.207784776",
      "0.359832276"
    ],
    [
      "0.000000000",
      "0.359863026"
    ],
    [
      "0.581130500",
      "0.418869500"
    ],
    [
      "0.103901261",
      "0.419809450"
    ],
    [
      "0.315489329",
      "0.426917816"
    ],
    [
      "0.470913046",
      "0.466210377"
    ],
    [
      "0.207784774",
      "0.479786625"
    ],
    [
      "0.000000000",
      "0.479817375"
    ],
    [
      "0.103901261",
      "0.539763800"
    ],
    [
      "0.307370834",
      "0.546657115"
    ],
    [
      "0.423572169",
      "0.576427831"
    ],
    [
      "0.000000000",
      "0.599771724"
    ],
    [
      "0.221562889",
      "0.630539959"
    ],
    [
      "0.103901261",
      "0.659718149"
    ],
    [
      "0.337141550",
      "0.662858450"
    ],
    [
      "0.000035504",
      "0.719726068"
    ],
    [
      "0.252321016",
      "0.747678984"
    ],
    [
      "0.135614764",
      "0.775404361"
    ],
    [
      "0.000000000",
      "0.839680412"
    ],
    [
      "0.107889387",
      "0.892110613"
    ],
    [
      "0.005997699",
      "0.985005688"
    ]
  ]
}

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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