P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 40

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 = 40

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	[0.136295991, 0.136295995]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.136295994	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

12345678910111213141516171819202122232425262728293031323334353637383940

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
0.1362959944→0.1362959945
2026-10-18
今天AI · DeepSeek: DeepSeek V4.1 Flash
0.111111111→0.136295994
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": [
    [
      "0.118035794",
      "0.068147997"
    ],
    [
      "0.000000000",
      "0.817775970"
    ],
    [
      "0.832210136",
      "0.000000000"
    ],
    [
      "0.542341008",
      "0.290419221"
    ],
    [
      "0.251398703",
      "0.580274145"
    ],
    [
      "0.236071588",
      "0.272591990"
    ],
    [
      "0.506785780",
      "0.022546541"
    ],
    [
      "0.382893677",
      "0.617106323"
    ],
    [
      "0.770794783",
      "0.229205217"
    ],
    [
      "0.000000000",
      "0.272591990"
    ],
    [
      "0.506785781",
      "0.158842536"
    ],
    [
      "0.153876121",
      "0.744831290"
    ],
    [
      "0.578044154",
      "0.421955846"
    ],
    [
      "0.236071588",
      "0.136295995"
    ],
    [
      "0.118035794",
      "0.340739987"
    ],
    [
      "0.641203977",
      "0.000000000"
    ],
    [
      "0.674106639",
      "0.325267702"
    ],
    [
      "0.118035794",
      "0.613331977"
    ],
    [
      "0.868671540",
      "0.131328460"
    ],
    [
      "0.008648123",
      "0.982321474"
    ],
    [
      "0.639424392",
      "0.192741091"
    ],
    [
      "0.349089580",
      "0.485068899"
    ],
    [
      "0.446278523",
      "0.387107365"
    ],
    [
      "0.000000000",
      "0.681479975"
    ],
    [
      "0.999999971",
      "0.000000005"
    ],
    [
      "0.481127004",
      "0.518872996"
    ],
    [
      "0.118035794",
      "0.204443992"
    ],
    [
      "0.286517854",
      "0.713482145"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.408887985"
    ],
    [
      "0.236071588",
      "0.000000000"
    ],
    [
      "0.736707056",
      "0.097240733"
    ],
    [
      "0.122526977",
      "0.877473023"
    ],
    [
      "0.372367583",
      "0.136295995"
    ],
    [
      "0.372367583",
      "0.272591990"
    ],
    [
      "0.118035794",
      "0.477035982"
    ],
    [
      "0.236071588",
      "0.408887985"
    ],
    [
      "0.000000000",
      "0.545183980"
    ],
    [
      "0.372367583",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.136295995"
    ]
  ]
}

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": 40
}

当前第一名的答案

{
  "points": [
    [
      "0.118035794",
      "0.068147997"
    ],
    [
      "0.000000000",
      "0.817775970"
    ],
    [
      "0.832210136",
      "0.000000000"
    ],
    [
      "0.542341008",
      "0.290419221"
    ],
    [
      "0.251398703",
      "0.580274145"
    ],
    [
      "0.236071588",
      "0.272591990"
    ],
    [
      "0.506785780",
      "0.022546541"
    ],
    [
      "0.382893677",
      "0.617106323"
    ],
    [
      "0.770794783",
      "0.229205217"
    ],
    [
      "0.000000000",
      "0.272591990"
    ],
    [
      "0.506785781",
      "0.158842536"
    ],
    [
      "0.153876121",
      "0.744831290"
    ],
    [
      "0.578044154",
      "0.421955846"
    ],
    [
      "0.236071588",
      "0.136295995"
    ],
    [
      "0.118035794",
      "0.340739987"
    ],
    [
      "0.641203977",
      "0.000000000"
    ],
    [
      "0.674106639",
      "0.325267702"
    ],
    [
      "0.118035794",
      "0.613331977"
    ],
    [
      "0.868671540",
      "0.131328460"
    ],
    [
      "0.008648123",
      "0.982321474"
    ],
    [
      "0.639424392",
      "0.192741091"
    ],
    [
      "0.349089580",
      "0.485068899"
    ],
    [
      "0.446278523",
      "0.387107365"
    ],
    [
      "0.000000000",
      "0.681479975"
    ],
    [
      "0.999999971",
      "0.000000005"
    ],
    [
      "0.481127004",
      "0.518872996"
    ],
    [
      "0.118035794",
      "0.204443992"
    ],
    [
      "0.286517854",
      "0.713482145"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.408887985"
    ],
    [
      "0.236071588",
      "0.000000000"
    ],
    [
      "0.736707056",
      "0.097240733"
    ],
    [
      "0.122526977",
      "0.877473023"
    ],
    [
      "0.372367583",
      "0.136295995"
    ],
    [
      "0.372367583",
      "0.272591990"
    ],
    [
      "0.118035794",
      "0.477035982"
    ],
    [
      "0.236071588",
      "0.408887985"
    ],
    [
      "0.000000000",
      "0.545183980"
    ],
    [
      "0.372367583",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.136295995"
    ]
  ]
}

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

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

外部信息遗漏NUE_13#311 条回复10 天前 本题 n=4 已达最优的证明XSsMC#2502 条回复45 天前 本题n=4时用外接圆法的证明及对n=6情况的延伸NUE_13#313 条回复52 天前 问题等价Tmrg#341 条回复61 天前