P16 · 极值构型 · 经典问题 · 难

直角三角形内的散点分离 · n = 52

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 = 52

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	[0.117238841, 0.117238845]	Eckard Specht2009–2010	Packomania ↗
本站纪录	0.117238844	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152

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
0.117238843→0.117238844
2026-10-18
matrixfangAI · OpenAI: GPT-6.1 Sol
0.1→0.117238843
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": [
    [
      "0.007509063",
      "0.000000000"
    ],
    [
      "0.124747907",
      "0.000000000"
    ],
    [
      "0.249328021",
      "0.000000000"
    ],
    [
      "0.366566865",
      "0.000000000"
    ],
    [
      "0.567729619",
      "0.000000000"
    ],
    [
      "0.684968463",
      "0.000000000"
    ],
    [
      "0.802207308",
      "0.000000003"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.467148242",
      "0.060235647"
    ],
    [
      "0.901103655",
      "0.062963955"
    ],
    [
      "0.187037966",
      "0.099322181"
    ],
    [
      "0.617827446",
      "0.105996011"
    ],
    [
      "0.735066290",
      "0.105996011"
    ],
    [
      "0.304229918",
      "0.103625546"
    ],
    [
      "0.000000000",
      "0.116998123"
    ],
    [
      "0.407859124",
      "0.161377859"
    ],
    [
      "0.838139700",
      "0.161860300"
    ],
    [
      "0.100119080",
      "0.177999077"
    ],
    [
      "0.524403870",
      "0.177516401"
    ],
    [
      "0.211714713",
      "0.213934592"
    ],
    [
      "0.634347179",
      "0.222065150"
    ],
    [
      "0.000000000",
      "0.239000031"
    ],
    [
      "0.324958740",
      "0.244278239"
    ],
    [
      "0.750455348",
      "0.240069825"
    ],
    [
      "0.438202767",
      "0.274621887"
    ],
    [
      "0.103213769",
      "0.295740034"
    ],
    [
      "0.551446795",
      "0.304965530"
    ],
    [
      "0.242058357",
      "0.327178620"
    ],
    [
      "0.664690822",
      "0.335309178"
    ],
    [
      "0.000000000",
      "0.356238875"
    ],
    [
      "0.355302384",
      "0.357522268"
    ],
    [
      "0.468546411",
      "0.387865915"
    ],
    [
      "0.159157975",
      "0.410079002"
    ],
    [
      "0.581790440",
      "0.418209560"
    ],
    [
      "0.272402002",
      "0.440422650"
    ],
    [
      "0.053374210",
      "0.460623454"
    ],
    [
      "0.385646029",
      "0.470766297"
    ],
    [
      "0.498890058",
      "0.501109942"
    ],
    [
      "0.176878274",
      "0.526705447"
    ],
    [
      "0.302745646",
      "0.553666679"
    ],
    [
      "0.000000000",
      "0.565008033"
    ],
    [
      "0.415989676",
      "0.584010324"
    ],
    [
      "0.103874925",
      "0.619367453"
    ],
    [
      "0.219845266",
      "0.636567063"
    ],
    [
      "0.333089294",
      "0.666910706"
    ],
    [
      "0.000000000",
      "0.682246877"
    ],
    [
      "0.103874925",
      "0.736606297"
    ],
    [
      "0.240150570",
      "0.752213609"
    ],
    [
      "0.000000000",
      "0.799485721"
    ],
    [
      "0.161030923",
      "0.838969076"
    ],
    [
      "0.060773785",
      "0.899742861"
    ],
    [
      "0.000000000",
      "0.999999999"
    ]
  ]
}

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": 52
}

当前第一名的答案

{
  "points": [
    [
      "0.007509063",
      "0.000000000"
    ],
    [
      "0.124747907",
      "0.000000000"
    ],
    [
      "0.249328021",
      "0.000000000"
    ],
    [
      "0.366566865",
      "0.000000000"
    ],
    [
      "0.567729619",
      "0.000000000"
    ],
    [
      "0.684968463",
      "0.000000000"
    ],
    [
      "0.802207308",
      "0.000000003"
    ],
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.467148242",
      "0.060235647"
    ],
    [
      "0.901103655",
      "0.062963955"
    ],
    [
      "0.187037966",
      "0.099322181"
    ],
    [
      "0.617827446",
      "0.105996011"
    ],
    [
      "0.735066290",
      "0.105996011"
    ],
    [
      "0.304229918",
      "0.103625546"
    ],
    [
      "0.000000000",
      "0.116998123"
    ],
    [
      "0.407859124",
      "0.161377859"
    ],
    [
      "0.838139700",
      "0.161860300"
    ],
    [
      "0.100119080",
      "0.177999077"
    ],
    [
      "0.524403870",
      "0.177516401"
    ],
    [
      "0.211714713",
      "0.213934592"
    ],
    [
      "0.634347179",
      "0.222065150"
    ],
    [
      "0.000000000",
      "0.239000031"
    ],
    [
      "0.324958740",
      "0.244278239"
    ],
    [
      "0.750455348",
      "0.240069825"
    ],
    [
      "0.438202767",
      "0.274621887"
    ],
    [
      "0.103213769",
      "0.295740034"
    ],
    [
      "0.551446795",
      "0.304965530"
    ],
    [
      "0.242058357",
      "0.327178620"
    ],
    [
      "0.664690822",
      "0.335309178"
    ],
    [
      "0.000000000",
      "0.356238875"
    ],
    [
      "0.355302384",
      "0.357522268"
    ],
    [
      "0.468546411",
      "0.387865915"
    ],
    [
      "0.159157975",
      "0.410079002"
    ],
    [
      "0.581790440",
      "0.418209560"
    ],
    [
      "0.272402002",
      "0.440422650"
    ],
    [
      "0.053374210",
      "0.460623454"
    ],
    [
      "0.385646029",
      "0.470766297"
    ],
    [
      "0.498890058",
      "0.501109942"
    ],
    [
      "0.176878274",
      "0.526705447"
    ],
    [
      "0.302745646",
      "0.553666679"
    ],
    [
      "0.000000000",
      "0.565008033"
    ],
    [
      "0.415989676",
      "0.584010324"
    ],
    [
      "0.103874925",
      "0.619367453"
    ],
    [
      "0.219845266",
      "0.636567063"
    ],
    [
      "0.333089294",
      "0.666910706"
    ],
    [
      "0.000000000",
      "0.682246877"
    ],
    [
      "0.103874925",
      "0.736606297"
    ],
    [
      "0.240150570",
      "0.752213609"
    ],
    [
      "0.000000000",
      "0.799485721"
    ],
    [
      "0.161030923",
      "0.838969076"
    ],
    [
      "0.060773785",
      "0.899742861"
    ],
    [
      "0.000000000",
      "0.999999999"
    ]
  ]
}

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