P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 48

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 = 48

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	[0.122507179, 0.122507183]	Claudia Orquídea López Soto2011-02-26	Packomania ↗
本站纪录	0.122507182	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

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748

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
0.12250718200000000→0.12250718200000006
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.1→0.122507182
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"
    ],
    [
      "0.208101122",
      "0.000000000"
    ],
    [
      "0.420289786",
      "0.000000000"
    ],
    [
      "0.632478450",
      "0.000000000"
    ],
    [
      "0.754985632",
      "0.000000004"
    ],
    [
      "0.877492814",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.314195454",
      "0.061253591"
    ],
    [
      "0.526384118",
      "0.061253592"
    ],
    [
      "0.104050561",
      "0.064664445"
    ],
    [
      "0.420289785",
      "0.122507184"
    ],
    [
      "0.632478454",
      "0.122507182"
    ],
    [
      "0.754985636",
      "0.122507186"
    ],
    [
      "0.877492818",
      "0.122507182"
    ],
    [
      "0.210144893",
      "0.125918036"
    ],
    [
      "0.000000000",
      "0.129328890"
    ],
    [
      "0.526384117",
      "0.183760775"
    ],
    [
      "0.316239224",
      "0.187171629"
    ],
    [
      "0.106094332",
      "0.190582481"
    ],
    [
      "0.632478450",
      "0.245014364"
    ],
    [
      "0.754985632",
      "0.245014368"
    ],
    [
      "0.422333556",
      "0.248425220"
    ],
    [
      "0.000000000",
      "0.251836073"
    ],
    [
      "0.212188663",
      "0.251836074"
    ],
    [
      "0.106094331",
      "0.313089666"
    ],
    [
      "0.321044700",
      "0.317901706"
    ],
    [
      "0.517279634",
      "0.325841316"
    ],
    [
      "0.632478454",
      "0.367521546"
    ],
    [
      "0.000000000",
      "0.374343259"
    ],
    [
      "0.212188663",
      "0.374343257"
    ],
    [
      "0.430653974",
      "0.412466975"
    ],
    [
      "0.106170546",
      "0.435728996"
    ],
    [
      "0.312467378",
      "0.444715048"
    ],
    [
      "0.545852794",
      "0.454147205"
    ],
    [
      "0.000000000",
      "0.496850442"
    ],
    [
      "0.206450139",
      "0.506101974"
    ],
    [
      "0.459227135",
      "0.540772865"
    ],
    [
      "0.339003041",
      "0.564313829"
    ],
    [
      "0.100278840",
      "0.567222060"
    ],
    [
      "0.206466744",
      "0.628609155"
    ],
    [
      "0.000000000",
      "0.637593673"
    ],
    [
      "0.315462077",
      "0.684537923"
    ],
    [
      "0.106094333",
      "0.698847262"
    ],
    [
      "0.215089672",
      "0.754776018"
    ],
    [
      "0.000000000",
      "0.760100856"
    ],
    [
      "0.144851577",
      "0.855148423"
    ],
    [
      "0.024902005",
      "0.880050428"
    ],
    [
      "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": 48
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.208101122",
      "0.000000000"
    ],
    [
      "0.420289786",
      "0.000000000"
    ],
    [
      "0.632478450",
      "0.000000000"
    ],
    [
      "0.754985632",
      "0.000000004"
    ],
    [
      "0.877492814",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.314195454",
      "0.061253591"
    ],
    [
      "0.526384118",
      "0.061253592"
    ],
    [
      "0.104050561",
      "0.064664445"
    ],
    [
      "0.420289785",
      "0.122507184"
    ],
    [
      "0.632478454",
      "0.122507182"
    ],
    [
      "0.754985636",
      "0.122507186"
    ],
    [
      "0.877492818",
      "0.122507182"
    ],
    [
      "0.210144893",
      "0.125918036"
    ],
    [
      "0.000000000",
      "0.129328890"
    ],
    [
      "0.526384117",
      "0.183760775"
    ],
    [
      "0.316239224",
      "0.187171629"
    ],
    [
      "0.106094332",
      "0.190582481"
    ],
    [
      "0.632478450",
      "0.245014364"
    ],
    [
      "0.754985632",
      "0.245014368"
    ],
    [
      "0.422333556",
      "0.248425220"
    ],
    [
      "0.000000000",
      "0.251836073"
    ],
    [
      "0.212188663",
      "0.251836074"
    ],
    [
      "0.106094331",
      "0.313089666"
    ],
    [
      "0.321044700",
      "0.317901706"
    ],
    [
      "0.517279634",
      "0.325841316"
    ],
    [
      "0.632478454",
      "0.367521546"
    ],
    [
      "0.000000000",
      "0.374343259"
    ],
    [
      "0.212188663",
      "0.374343257"
    ],
    [
      "0.430653974",
      "0.412466975"
    ],
    [
      "0.106170546",
      "0.435728996"
    ],
    [
      "0.312467378",
      "0.444715048"
    ],
    [
      "0.545852794",
      "0.454147205"
    ],
    [
      "0.000000000",
      "0.496850442"
    ],
    [
      "0.206450139",
      "0.506101974"
    ],
    [
      "0.459227135",
      "0.540772865"
    ],
    [
      "0.339003041",
      "0.564313829"
    ],
    [
      "0.100278840",
      "0.567222060"
    ],
    [
      "0.206466744",
      "0.628609155"
    ],
    [
      "0.000000000",
      "0.637593673"
    ],
    [
      "0.315462077",
      "0.684537923"
    ],
    [
      "0.106094333",
      "0.698847262"
    ],
    [
      "0.215089672",
      "0.754776018"
    ],
    [
      "0.000000000",
      "0.760100856"
    ],
    [
      "0.144851577",
      "0.855148423"
    ],
    [
      "0.024902005",
      "0.880050428"
    ],
    [
      "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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