P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 45

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 = 45

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	[0.126898598, 0.126898602]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.1268986	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445

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
0.12689860084→0.12689860088
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.111111111→0.1268986
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": [
    [
      "0.003228810",
      "0.000000000"
    ],
    [
      "0.233418690",
      "0.000000000"
    ],
    [
      "0.360317291",
      "0.000000000"
    ],
    [
      "0.487599107",
      "0.000000000"
    ],
    [
      "0.614497708",
      "0.000000000"
    ],
    [
      "0.836242498",
      "0.000000000"
    ],
    [
      "0.999487351",
      "0.000212343"
    ],
    [
      "0.118323750",
      "0.053445390"
    ],
    [
      "0.725370103",
      "0.061729790"
    ],
    [
      "0.423958199",
      "0.109786565"
    ],
    [
      "0.554015481",
      "0.117546012"
    ],
    [
      "0.881412972",
      "0.118587028"
    ],
    [
      "0.000000000",
      "0.126857518"
    ],
    [
      "0.230189880",
      "0.126857518"
    ],
    [
      "0.669107473",
      "0.175474115"
    ],
    [
      "0.115094940",
      "0.180302908"
    ],
    [
      "0.334227338",
      "0.199517427"
    ],
    [
      "0.791682110",
      "0.208317889"
    ],
    [
      "0.456801975",
      "0.232361201"
    ],
    [
      "0.000000000",
      "0.253756119"
    ],
    [
      "0.579376611",
      "0.265204976"
    ],
    [
      "0.244496471",
      "0.289248283"
    ],
    [
      "0.701951249",
      "0.298048751"
    ],
    [
      "0.118866275",
      "0.307145456"
    ],
    [
      "0.367071113",
      "0.322092063"
    ],
    [
      "0.489645750",
      "0.354935838"
    ],
    [
      "0.014815887",
      "0.379786847"
    ],
    [
      "0.612220387",
      "0.387779612"
    ],
    [
      "0.277340251",
      "0.411822926"
    ],
    [
      "0.151710054",
      "0.429720091"
    ],
    [
      "0.399914888",
      "0.444666699"
    ],
    [
      "0.522489526",
      "0.477510474"
    ],
    [
      "0.000000000",
      "0.505817575"
    ],
    [
      "0.305886206",
      "0.535630709"
    ],
    [
      "0.118156566",
      "0.552102355"
    ],
    [
      "0.432758664",
      "0.567241335"
    ],
    [
      "0.219313860",
      "0.628721262"
    ],
    [
      "0.018994486",
      "0.631286553"
    ],
    [
      "0.343027803",
      "0.656972197"
    ],
    [
      "0.121606913",
      "0.718996546"
    ],
    [
      "0.247564794",
      "0.752435205"
    ],
    [
      "0.000000000",
      "0.756755531"
    ],
    [
      "0.157833933",
      "0.842166067"
    ],
    [
      "0.036211699",
      "0.878377766"
    ],
    [
      "0.000000000",
      "1.000000000"
    ]
  ]
}

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": 45
}

当前第一名的答案

{
  "points": [
    [
      "0.003228810",
      "0.000000000"
    ],
    [
      "0.233418690",
      "0.000000000"
    ],
    [
      "0.360317291",
      "0.000000000"
    ],
    [
      "0.487599107",
      "0.000000000"
    ],
    [
      "0.614497708",
      "0.000000000"
    ],
    [
      "0.836242498",
      "0.000000000"
    ],
    [
      "0.999487351",
      "0.000212343"
    ],
    [
      "0.118323750",
      "0.053445390"
    ],
    [
      "0.725370103",
      "0.061729790"
    ],
    [
      "0.423958199",
      "0.109786565"
    ],
    [
      "0.554015481",
      "0.117546012"
    ],
    [
      "0.881412972",
      "0.118587028"
    ],
    [
      "0.000000000",
      "0.126857518"
    ],
    [
      "0.230189880",
      "0.126857518"
    ],
    [
      "0.669107473",
      "0.175474115"
    ],
    [
      "0.115094940",
      "0.180302908"
    ],
    [
      "0.334227338",
      "0.199517427"
    ],
    [
      "0.791682110",
      "0.208317889"
    ],
    [
      "0.456801975",
      "0.232361201"
    ],
    [
      "0.000000000",
      "0.253756119"
    ],
    [
      "0.579376611",
      "0.265204976"
    ],
    [
      "0.244496471",
      "0.289248283"
    ],
    [
      "0.701951249",
      "0.298048751"
    ],
    [
      "0.118866275",
      "0.307145456"
    ],
    [
      "0.367071113",
      "0.322092063"
    ],
    [
      "0.489645750",
      "0.354935838"
    ],
    [
      "0.014815887",
      "0.379786847"
    ],
    [
      "0.612220387",
      "0.387779612"
    ],
    [
      "0.277340251",
      "0.411822926"
    ],
    [
      "0.151710054",
      "0.429720091"
    ],
    [
      "0.399914888",
      "0.444666699"
    ],
    [
      "0.522489526",
      "0.477510474"
    ],
    [
      "0.000000000",
      "0.505817575"
    ],
    [
      "0.305886206",
      "0.535630709"
    ],
    [
      "0.118156566",
      "0.552102355"
    ],
    [
      "0.432758664",
      "0.567241335"
    ],
    [
      "0.219313860",
      "0.628721262"
    ],
    [
      "0.018994486",
      "0.631286553"
    ],
    [
      "0.343027803",
      "0.656972197"
    ],
    [
      "0.121606913",
      "0.718996546"
    ],
    [
      "0.247564794",
      "0.752435205"
    ],
    [
      "0.000000000",
      "0.756755531"
    ],
    [
      "0.157833933",
      "0.842166067"
    ],
    [
      "0.036211699",
      "0.878377766"
    ],
    [
      "0.000000000",
      "1.000000000"
    ]
  ]
}

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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