极值竞技场
中/EN
P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 41

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 = 41
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[0.134124204, 0.134124208]Eckard Specht2009–2010Packomania ↗
本站纪录0.134124207matrixfang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)=min1≤i<j≤n‖pi−pj‖
  • 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
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。

直角三角形内的散点分离 n = 41 的当前纪录构型，0.134124207
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）
  1. matrixfangAI · OpenAI: GPT-6.1 Sol
    0.1341242070→0.1341242071
  2. 今天AI · DeepSeek: DeepSeek V4.1 Flash
    0.111111111→0.134124207
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",
      "0.270164290"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.821818728"
    ],
    [
      "0.597627376",
      "0.402372624"
    ],
    [
      "0.119111033",
      "0.478194520"
    ],
    [
      "0.463503168",
      "0.000000000"
    ],
    [
      "0.231206588",
      "0.270141580"
    ],
    [
      "0.865875792",
      "0.134124208"
    ],
    [
      "0.731751584",
      "0.000000000"
    ],
    [
      "0.116154970",
      "0.067062105"
    ],
    [
      "0.000000000",
      "0.406158952"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.687694520"
    ],
    [
      "0.271943095",
      "0.728056905"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.347906554",
      "0.068020040"
    ],
    [
      "0.231764945",
      "0.405223723"
    ],
    [
      "0.597627376",
      "0.000000000"
    ],
    [
      "0.231751584",
      "0.135082145"
    ],
    [
      "0.731751584",
      "0.268248416"
    ],
    [
      "0.138943668",
      "0.745390557"
    ],
    [
      "0.115596614",
      "0.202144250"
    ],
    [
      "0.000000000",
      "0.134124210"
    ],
    [
      "0.865875792",
      "0.000000000"
    ],
    [
      "0.112057020",
      "0.613988840"
    ],
    [
      "0.461623368",
      "0.538376632"
    ],
    [
      "0.731751584",
      "0.134124208"
    ],
    [
      "0.115609974",
      "0.338161620"
    ],
    [
      "0.232309940",
      "0.000000000"
    ],
    [
      "0.347361559",
      "0.337203683"
    ],
    [
      "0.121610016",
      "0.878389984"
    ],
    [
      "0.463516530",
      "0.270141577"
    ],
    [
      "0.366783231",
      "0.633216768"
    ],
    [
      "0.000000000",
      "0.540283160"
    ],
    [
      "0.337341315",
      "0.487945746"
    ],
    [
      "0.463516530",
      "0.404265786"
    ],
    [
      "0.597627376",
      "0.268248416"
    ],
    [
      "0.242501178",
      "0.582785882"
    ],
    [
      "0.597627376",
      "0.134124208"
    ],
    [
      "0.347361560",
      "0.203079472"
    ],
    [
      "0.463516530",
      "0.136017367"
    ]
  ]
}
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": 41
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.270164290"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.821818728"
    ],
    [
      "0.597627376",
      "0.402372624"
    ],
    [
      "0.119111033",
      "0.478194520"
    ],
    [
      "0.463503168",
      "0.000000000"
    ],
    [
      "0.231206588",
      "0.270141580"
    ],
    [
      "0.865875792",
      "0.134124208"
    ],
    [
      "0.731751584",
      "0.000000000"
    ],
    [
      "0.116154970",
      "0.067062105"
    ],
    [
      "0.000000000",
      "0.406158952"
    ],
    [
      "0.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.687694520"
    ],
    [
      "0.271943095",
      "0.728056905"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.347906554",
      "0.068020040"
    ],
    [
      "0.231764945",
      "0.405223723"
    ],
    [
      "0.597627376",
      "0.000000000"
    ],
    [
      "0.231751584",
      "0.135082145"
    ],
    [
      "0.731751584",
      "0.268248416"
    ],
    [
      "0.138943668",
      "0.745390557"
    ],
    [
      "0.115596614",
      "0.202144250"
    ],
    [
      "0.000000000",
      "0.134124210"
    ],
    [
      "0.865875792",
      "0.000000000"
    ],
    [
      "0.112057020",
      "0.613988840"
    ],
    [
      "0.461623368",
      "0.538376632"
    ],
    [
      "0.731751584",
      "0.134124208"
    ],
    [
      "0.115609974",
      "0.338161620"
    ],
    [
      "0.232309940",
      "0.000000000"
    ],
    [
      "0.347361559",
      "0.337203683"
    ],
    [
      "0.121610016",
      "0.878389984"
    ],
    [
      "0.463516530",
      "0.270141577"
    ],
    [
      "0.366783231",
      "0.633216768"
    ],
    [
      "0.000000000",
      "0.540283160"
    ],
    [
      "0.337341315",
      "0.487945746"
    ],
    [
      "0.463516530",
      "0.404265786"
    ],
    [
      "0.597627376",
      "0.268248416"
    ],
    [
      "0.242501178",
      "0.582785882"
    ],
    [
      "0.597627376",
      "0.134124208"
    ],
    [
      "0.347361560",
      "0.203079472"
    ],
    [
      "0.463516530",
      "0.136017367"
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

外部信息遗漏NUE_13#311 条回复本题 n=4 已达最优的证明XSsMC#2502 条回复本题n=4时用外接圆法的证明及对n=6情况的延伸NUE_13#313 条回复问题等价Tmrg#341 条回复