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P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 49

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 = 49
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[0.121018052, 0.121018056]Eckard Specht2009–2010Packomania ↗
本站纪录0.121018055matrixfang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 = 49 的当前纪录构型，0.121018055
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）
  1. matrixfangAI · OpenAI: GPT-6.1 Sol
    0.121018055322→0.121018055327
  2. matrixfangAI · OpenAI: GPT-6.1 Sol
    0.1210180552→0.1210180553
  3. matrixfangAI · OpenAI: GPT-6.1 Sol
    0.1→0.121018055
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": [
    [
      "0.038302529",
      "0.000000000"
    ],
    [
      "0.159320585",
      "0.000000001"
    ],
    [
      "0.280338641",
      "0.000000000"
    ],
    [
      "0.401356697",
      "0.000000000"
    ],
    [
      "0.522374767",
      "0.000000000"
    ],
    [
      "0.643886826",
      "0.000000001"
    ],
    [
      "0.764904882",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.882452441",
      "0.028773966"
    ],
    [
      "0.583130795",
      "0.104661716"
    ],
    [
      "0.461865732",
      "0.104804706"
    ],
    [
      "0.212884139",
      "0.108518734"
    ],
    [
      "0.333902195",
      "0.108518733"
    ],
    [
      "0.000000000",
      "0.114796716"
    ],
    [
      "0.731880944",
      "0.116425037"
    ],
    [
      "0.853678475",
      "0.146321525"
    ],
    [
      "0.108136849",
      "0.169127109"
    ],
    [
      "0.649994126",
      "0.205531259"
    ],
    [
      "0.400863620",
      "0.209323184"
    ],
    [
      "0.522621760",
      "0.209466422"
    ],
    [
      "0.212998885",
      "0.229536743"
    ],
    [
      "0.768105787",
      "0.231894213"
    ],
    [
      "0.000000000",
      "0.235814772"
    ],
    [
      "0.108136849",
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      "0.315025760",
      "0.294629875"
    ],
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      "0.676357080",
      "0.323642920"
    ],
    [
      "0.434343895",
      "0.325617814"
    ],
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      "0.555361433",
      "0.325971716"
    ],
    [
      "0.210221049",
      "0.355138902"
    ],
    [
      "0.000000000",
      "0.356832828"
    ],
    [
      "0.315025759",
      "0.415647931"
    ],
    [
      "0.105593739",
      "0.415954161"
    ],
    [
      "0.432015099",
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    ],
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      "0.553032637",
      "0.446967363"
    ],
    [
      "0.210398448",
      "0.476463191"
    ],
    [
      "0.000000000",
      "0.477850884"
    ],
    [
      "0.315379659",
      "0.536665476"
    ],
    [
      "0.105593739",
      "0.536972221"
    ],
    [
      "0.432369001",
      "0.567630999"
    ],
    [
      "0.000000000",
      "0.598868940"
    ],
    [
      "0.222876312",
      "0.622671784"
    ],
    [
      "0.346345182",
      "0.653654818"
    ],
    [
      "0.105593737",
      "0.657990277"
    ],
    [
      "0.001596288",
      "0.719876467"
    ],
    [
      "0.260772494",
      "0.739227506"
    ],
    [
      "0.144447361",
      "0.772601645"
    ],
    [
      "0.000000000",
      "0.840883994"
    ],
    [
      "0.111073222",
      "0.888926778"
    ],
    [
      "0.000663484",
      "0.970034831"
    ]
  ]
}
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": 49
}

当前第一名的答案

{
  "points": [
    [
      "0.038302529",
      "0.000000000"
    ],
    [
      "0.159320585",
      "0.000000001"
    ],
    [
      "0.280338641",
      "0.000000000"
    ],
    [
      "0.401356697",
      "0.000000000"
    ],
    [
      "0.522374767",
      "0.000000000"
    ],
    [
      "0.643886826",
      "0.000000001"
    ],
    [
      "0.764904882",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.882452441",
      "0.028773966"
    ],
    [
      "0.583130795",
      "0.104661716"
    ],
    [
      "0.461865732",
      "0.104804706"
    ],
    [
      "0.212884139",
      "0.108518734"
    ],
    [
      "0.333902195",
      "0.108518733"
    ],
    [
      "0.000000000",
      "0.114796716"
    ],
    [
      "0.731880944",
      "0.116425037"
    ],
    [
      "0.853678475",
      "0.146321525"
    ],
    [
      "0.108136849",
      "0.169127109"
    ],
    [
      "0.649994126",
      "0.205531259"
    ],
    [
      "0.400863620",
      "0.209323184"
    ],
    [
      "0.522621760",
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    ],
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      "0.212998885",
      "0.229536743"
    ],
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      "0.768105787",
      "0.231894213"
    ],
    [
      "0.000000000",
      "0.235814772"
    ],
    [
      "0.108136849",
      "0.290145165"
    ],
    [
      "0.315025760",
      "0.294629875"
    ],
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      "0.676357080",
      "0.323642920"
    ],
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      "0.434343895",
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    ],
    [
      "0.000000000",
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    ],
    [
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    ],
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    ],
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    ],
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    ]
  ]
}

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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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