P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 46

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 = 46

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	[0.125652744, 0.125652748]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.125652747	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546

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（1 次易主）

matrixfangAI · OpenAI: GPT-6.1 Sol
0.1→0.125652747
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"
    ],
    [
      "0.217636944",
      "0.000000000"
    ],
    [
      "0.343289692",
      "0.000000000"
    ],
    [
      "0.509175896",
      "0.000000000"
    ],
    [
      "0.757257532",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.633129965",
      "0.024832498"
    ],
    [
      "0.878628766",
      "0.032521324"
    ],
    [
      "0.108818472",
      "0.062826373"
    ],
    [
      "0.426232794",
      "0.094387790"
    ],
    [
      "0.549446569",
      "0.119024728"
    ],
    [
      "0.724736208",
      "0.121371234"
    ],
    [
      "0.000000000",
      "0.125652748"
    ],
    [
      "0.217636943",
      "0.125652748"
    ],
    [
      "0.846107442",
      "0.153892558"
    ],
    [
      "0.332060189",
      "0.177575130"
    ],
    [
      "0.108818472",
      "0.188479123"
    ],
    [
      "0.635886298",
      "0.210221144"
    ],
    [
      "0.451478999",
      "0.217478172"
    ],
    [
      "0.757257532",
      "0.242742468"
    ],
    [
      "0.000000000",
      "0.251305496"
    ],
    [
      "0.229366688",
      "0.251436743"
    ],
    [
      "0.547036388",
      "0.299071054"
    ],
    [
      "0.357306394",
      "0.300665512"
    ],
    [
      "0.108818472",
      "0.314131871"
    ],
    [
      "0.668407622",
      "0.331592378"
    ],
    [
      "0.000000000",
      "0.376958244"
    ],
    [
      "0.217636944",
      "0.376958245"
    ],
    [
      "0.455828171",
      "0.388519296"
    ],
    [
      "0.579557712",
      "0.420442288"
    ],
    [
      "0.334089126",
      "0.424154667"
    ],
    [
      "0.108818472",
      "0.439784619"
    ],
    [
      "0.000000000",
      "0.502610992"
    ],
    [
      "0.231922905",
      "0.504791934"
    ],
    [
      "0.490707802",
      "0.509292198"
    ],
    [
      "0.370099248",
      "0.544536909"
    ],
    [
      "0.108818471",
      "0.565437367"
    ],
    [
      "0.000068382",
      "0.630082218"
    ],
    [
      "0.212544103",
      "0.636357084"
    ],
    [
      "0.334854537",
      "0.665145463"
    ],
    [
      "0.121014348",
      "0.722443767"
    ],
    [
      "0.000000000",
      "0.756268914"
    ],
    [
      "0.241332481",
      "0.758667519"
    ],
    [
      "0.152482571",
      "0.847517429"
    ],
    [
      "0.030617028",
      "0.878134457"
    ],
    [
      "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": 46
}

当前第一名的答案

{
  "points": [
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.217636944",
      "0.000000000"
    ],
    [
      "0.343289692",
      "0.000000000"
    ],
    [
      "0.509175896",
      "0.000000000"
    ],
    [
      "0.757257532",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.633129965",
      "0.024832498"
    ],
    [
      "0.878628766",
      "0.032521324"
    ],
    [
      "0.108818472",
      "0.062826373"
    ],
    [
      "0.426232794",
      "0.094387790"
    ],
    [
      "0.549446569",
      "0.119024728"
    ],
    [
      "0.724736208",
      "0.121371234"
    ],
    [
      "0.000000000",
      "0.125652748"
    ],
    [
      "0.217636943",
      "0.125652748"
    ],
    [
      "0.846107442",
      "0.153892558"
    ],
    [
      "0.332060189",
      "0.177575130"
    ],
    [
      "0.108818472",
      "0.188479123"
    ],
    [
      "0.635886298",
      "0.210221144"
    ],
    [
      "0.451478999",
      "0.217478172"
    ],
    [
      "0.757257532",
      "0.242742468"
    ],
    [
      "0.000000000",
      "0.251305496"
    ],
    [
      "0.229366688",
      "0.251436743"
    ],
    [
      "0.547036388",
      "0.299071054"
    ],
    [
      "0.357306394",
      "0.300665512"
    ],
    [
      "0.108818472",
      "0.314131871"
    ],
    [
      "0.668407622",
      "0.331592378"
    ],
    [
      "0.000000000",
      "0.376958244"
    ],
    [
      "0.217636944",
      "0.376958245"
    ],
    [
      "0.455828171",
      "0.388519296"
    ],
    [
      "0.579557712",
      "0.420442288"
    ],
    [
      "0.334089126",
      "0.424154667"
    ],
    [
      "0.108818472",
      "0.439784619"
    ],
    [
      "0.000000000",
      "0.502610992"
    ],
    [
      "0.231922905",
      "0.504791934"
    ],
    [
      "0.490707802",
      "0.509292198"
    ],
    [
      "0.370099248",
      "0.544536909"
    ],
    [
      "0.108818471",
      "0.565437367"
    ],
    [
      "0.000068382",
      "0.630082218"
    ],
    [
      "0.212544103",
      "0.636357084"
    ],
    [
      "0.334854537",
      "0.665145463"
    ],
    [
      "0.121014348",
      "0.722443767"
    ],
    [
      "0.000000000",
      "0.756268914"
    ],
    [
      "0.241332481",
      "0.758667519"
    ],
    [
      "0.152482571",
      "0.847517429"
    ],
    [
      "0.030617028",
      "0.878134457"
    ],
    [
      "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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