P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 44

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 = 44

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	[0.12864309, 0.128643094]	Claudia Orquídea López Soto2011-02-26	Packomania ↗
本站纪录	0.128643093	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344

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（4 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.12864309300000001→0.12864309300000006
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.12864309300000000→0.12864309300000001
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.128643093000000000→0.128643093000000003
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.111111111→0.128643093
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": [
    [
      "0.263393609",
      "0.000000000"
    ],
    [
      "0.391885593",
      "0.006233412"
    ],
    [
      "0.520377577",
      "0.000000000"
    ],
    [
      "0.649020671",
      "0.000000000"
    ],
    [
      "0.777663767",
      "0.000000001"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.128770047",
      "0.000000000"
    ],
    [
      "0.888831884",
      "0.064735581"
    ],
    [
      "0.455380968",
      "0.118114522"
    ],
    [
      "0.583872952",
      "0.111881110"
    ],
    [
      "0.712516046",
      "0.111881110"
    ],
    [
      "0.205558019",
      "0.114909051"
    ],
    [
      "0.000000000",
      "0.129234522"
    ],
    [
      "0.330075384",
      "0.147227338"
    ],
    [
      "0.824096303",
      "0.175903697"
    ],
    [
      "0.115310880",
      "0.206585108"
    ],
    [
      "0.419059197",
      "0.241523494"
    ],
    [
      "0.603413574",
      "0.239031452"
    ],
    [
      "0.239828244",
      "0.238903394"
    ],
    [
      "0.000000000",
      "0.263615331"
    ],
    [
      "0.729789537",
      "0.263583323"
    ],
    [
      "0.328094800",
      "0.332487904"
    ],
    [
      "0.147552306",
      "0.331122395"
    ],
    [
      "0.512449170",
      "0.329995856"
    ],
    [
      "0.636708864",
      "0.363291136"
    ],
    [
      "0.000000004",
      "0.392258424"
    ],
    [
      "0.237130392",
      "0.423452303"
    ],
    [
      "0.421484772",
      "0.420960265"
    ],
    [
      "0.545744460",
      "0.454255540"
    ],
    [
      "0.112381303",
      "0.454864247"
    ],
    [
      "0.330520362",
      "0.511924666"
    ],
    [
      "0.000000000",
      "0.520901517"
    ],
    [
      "0.454780047",
      "0.545219953"
    ],
    [
      "0.112381299",
      "0.583507340"
    ],
    [
      "0.239555958",
      "0.602889071"
    ],
    [
      "0.363815643",
      "0.636184357"
    ],
    [
      "0.000000000",
      "0.649544611"
    ],
    [
      "0.112381299",
      "0.712150434"
    ],
    [
      "0.263642765",
      "0.729428531"
    ],
    [
      "0.000000002",
      "0.778187705"
    ],
    [
      "0.176089519",
      "0.823910481"
    ],
    [
      "0.065183372",
      "0.889093853"
    ],
    [
      "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": 44
}

当前第一名的答案

{
  "points": [
    [
      "0.263393609",
      "0.000000000"
    ],
    [
      "0.391885593",
      "0.006233412"
    ],
    [
      "0.520377577",
      "0.000000000"
    ],
    [
      "0.649020671",
      "0.000000000"
    ],
    [
      "0.777663767",
      "0.000000001"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.128770047",
      "0.000000000"
    ],
    [
      "0.888831884",
      "0.064735581"
    ],
    [
      "0.455380968",
      "0.118114522"
    ],
    [
      "0.583872952",
      "0.111881110"
    ],
    [
      "0.712516046",
      "0.111881110"
    ],
    [
      "0.205558019",
      "0.114909051"
    ],
    [
      "0.000000000",
      "0.129234522"
    ],
    [
      "0.330075384",
      "0.147227338"
    ],
    [
      "0.824096303",
      "0.175903697"
    ],
    [
      "0.115310880",
      "0.206585108"
    ],
    [
      "0.419059197",
      "0.241523494"
    ],
    [
      "0.603413574",
      "0.239031452"
    ],
    [
      "0.239828244",
      "0.238903394"
    ],
    [
      "0.000000000",
      "0.263615331"
    ],
    [
      "0.729789537",
      "0.263583323"
    ],
    [
      "0.328094800",
      "0.332487904"
    ],
    [
      "0.147552306",
      "0.331122395"
    ],
    [
      "0.512449170",
      "0.329995856"
    ],
    [
      "0.636708864",
      "0.363291136"
    ],
    [
      "0.000000004",
      "0.392258424"
    ],
    [
      "0.237130392",
      "0.423452303"
    ],
    [
      "0.421484772",
      "0.420960265"
    ],
    [
      "0.545744460",
      "0.454255540"
    ],
    [
      "0.112381303",
      "0.454864247"
    ],
    [
      "0.330520362",
      "0.511924666"
    ],
    [
      "0.000000000",
      "0.520901517"
    ],
    [
      "0.454780047",
      "0.545219953"
    ],
    [
      "0.112381299",
      "0.583507340"
    ],
    [
      "0.239555958",
      "0.602889071"
    ],
    [
      "0.363815643",
      "0.636184357"
    ],
    [
      "0.000000000",
      "0.649544611"
    ],
    [
      "0.112381299",
      "0.712150434"
    ],
    [
      "0.263642765",
      "0.729428531"
    ],
    [
      "0.000000002",
      "0.778187705"
    ],
    [
      "0.176089519",
      "0.823910481"
    ],
    [
      "0.065183372",
      "0.889093853"
    ],
    [
      "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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