P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 53

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 = 53

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	[0.115964649, 0.115964653]	David W. Cantrell2010-10-26	Packomania ↗
本站纪录	0.115964652	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253

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
0.11596465200000→0.11596465200002
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.1→0.115964652
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": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.200856672",
      "0.000000000"
    ],
    [
      "0.836001216",
      "0.000000000"
    ],
    [
      "0.604071886",
      "0.000000000"
    ],
    [
      "0.720036539",
      "0.000000000"
    ],
    [
      "0.401713422",
      "0.000000000"
    ],
    [
      "0.502892654",
      "0.056661835"
    ],
    [
      "0.301285047",
      "0.057982256"
    ],
    [
      "0.100428336",
      "0.057982324"
    ],
    [
      "0.918000608",
      "0.081999392"
    ],
    [
      "0.730676520",
      "0.115475501"
    ],
    [
      "0.614711867",
      "0.115475501"
    ],
    [
      "0.403232449",
      "0.115954703"
    ],
    [
      "0.000000000",
      "0.115964653"
    ],
    [
      "0.200856672",
      "0.115964653"
    ],
    [
      "0.836001216",
      "0.163998784"
    ],
    [
      "0.507120799",
      "0.172549381"
    ],
    [
      "0.301285047",
      "0.173946909"
    ],
    [
      "0.100428336",
      "0.173946977"
    ],
    [
      "0.741306471",
      "0.230966804"
    ],
    [
      "0.607163697",
      "0.231194237"
    ],
    [
      "0.401724742",
      "0.231909554"
    ],
    [
      "0.200856752",
      "0.231929305"
    ],
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      "0.000000000",
      "0.231929306"
    ],
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      "0.506354313",
      "0.288511500"
    ],
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      "0.100428336",
      "0.289911630"
    ],
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      "0.301285127",
      "0.289911561"
    ],
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      "0.674133344",
      "0.325866656"
    ],
    [
      "0.406672833",
      "0.347768593"
    ],
    [
      "0.200856670",
      "0.347893957"
    ],
    [
      "0.000000001",
      "0.347893959"
    ],
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      "0.573323960",
      "0.383183919"
    ],
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      "0.301285045",
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      "0.100428336",
      "0.405876285"
    ],
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      "0.401843463",
      "0.463632642"
    ],
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      "0.200856751",
      "0.463858610"
    ],
    [
      "0.000000000",
      "0.463858612"
    ],
    [
      "0.516006697",
      "0.483993303"
    ],
    [
      "0.100428335",
      "0.521840938"
    ],
    [
      "0.311213373",
      "0.539558863"
    ],
    [
      "0.422204124",
      "0.577795876"
    ],
    [
      "0.200856671",
      "0.579823262"
    ],
    [
      "0.000000001",
      "0.579823265"
    ],
    [
      "0.340204732",
      "0.659795268"
    ],
    [
      "0.112485711",
      "0.660998518"
    ],
    [
      "0.228902071",
      "0.692345509"
    ],
    [
      "0.000000000",
      "0.695787918"
    ],
    [
      "0.083725381",
      "0.776024201"
    ],
    [
      "0.196351830",
      "0.803648170"
    ],
    [
      "0.000000000",
      "0.856260484"
    ],
    [
      "0.111349350",
      "0.888650650"
    ],
    [
      "0.005798231",
      "0.985504416"
    ]
  ]
}

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": 53
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.200856672",
      "0.000000000"
    ],
    [
      "0.836001216",
      "0.000000000"
    ],
    [
      "0.604071886",
      "0.000000000"
    ],
    [
      "0.720036539",
      "0.000000000"
    ],
    [
      "0.401713422",
      "0.000000000"
    ],
    [
      "0.502892654",
      "0.056661835"
    ],
    [
      "0.301285047",
      "0.057982256"
    ],
    [
      "0.100428336",
      "0.057982324"
    ],
    [
      "0.918000608",
      "0.081999392"
    ],
    [
      "0.730676520",
      "0.115475501"
    ],
    [
      "0.614711867",
      "0.115475501"
    ],
    [
      "0.403232449",
      "0.115954703"
    ],
    [
      "0.000000000",
      "0.115964653"
    ],
    [
      "0.200856672",
      "0.115964653"
    ],
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      "0.163998784"
    ],
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      "0.507120799",
      "0.172549381"
    ],
    [
      "0.301285047",
      "0.173946909"
    ],
    [
      "0.100428336",
      "0.173946977"
    ],
    [
      "0.741306471",
      "0.230966804"
    ],
    [
      "0.607163697",
      "0.231194237"
    ],
    [
      "0.401724742",
      "0.231909554"
    ],
    [
      "0.200856752",
      "0.231929305"
    ],
    [
      "0.000000000",
      "0.231929306"
    ],
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      "0.506354313",
      "0.288511500"
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      "0.100428336",
      "0.289911630"
    ],
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      "0.301285127",
      "0.289911561"
    ],
    [
      "0.674133344",
      "0.325866656"
    ],
    [
      "0.406672833",
      "0.347768593"
    ],
    [
      "0.200856670",
      "0.347893957"
    ],
    [
      "0.000000001",
      "0.347893959"
    ],
    [
      "0.573323960",
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    ],
    [
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    ],
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      "0.405876285"
    ],
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      "0.401843463",
      "0.463632642"
    ],
    [
      "0.200856751",
      "0.463858610"
    ],
    [
      "0.000000000",
      "0.463858612"
    ],
    [
      "0.516006697",
      "0.483993303"
    ],
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      "0.521840938"
    ],
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      "0.311213373",
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      "0.200856671",
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    ],
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    ],
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      "0.228902071",
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    [
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      "0.776024201"
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      "0.000000000",
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      "0.111349350",
      "0.888650650"
    ],
    [
      "0.005798231",
      "0.985504416"
    ]
  ]
}

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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