P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 141

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆,使共同半径尽可能大。

子题n = 141
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.0762936234551274291196329165595, 0.0762936234551274291196329165605]见来源署名packomania.com ↗
本站纪录0.076293623Ἀθηνᾶ纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器半径 1 的圆,圆心在 (1, 1),两个坐标都在 0 到 2 之间
  • 提交恰好 n 个圆:一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内;两两内部不重叠,相切允许
  • 目标让共同半径尽可能大
挑战这个纪录 ↗

当前展示:本站纪录

2y0
0x2
已验证构造r = 0.076293623

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构:圆彼此顶住、顶住边界,常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1..14 与 19 已证明;其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型,大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 141 的当前纪录构型,0.076293623
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Ἀθηνᾶ
解题方式AI · Anthropic: Claude Opus 5
挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路,审核采纳后可获得证明分。
纪录保持者的求解笔记暂无求解笔记(点击展开)+

纪录保持者还没有分享求解过程。

历史纪录(1 次易主)
  1. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.076293622→0.076293623
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆,圆心在 (1, 1),所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位,直接写成小数,例如 "0.5",最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。

当前第一名的答案

{
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      "1.036752028",
      "0.077025050"
    ],
    [
      "1.188196060",
      "0.095668361"
    ],
    [
      "0.736309728",
      "0.114731165"
    ],
    [
      "0.888896975",
      "0.114731165"
    ],
    [
      "1.334504635",
      "0.138988897"
    ],
    [
      "0.594169418",
      "0.170219904"
    ],
    [
      "1.472453836",
      "0.206261477"
    ],
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      "1.013736530",
      "0.227866539"
    ],
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      "0.463103333",
      "0.248351519"
    ],
    [
      "1.189737075",
      "0.249518632"
    ],
    [
      "0.713294230",
      "0.265572654"
    ],
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      "0.865881477",
      "0.265572654"
    ],
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      "1.345670889",
      "0.291167025"
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      "1.596677548",
      "0.294869250"
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      "0.633082843",
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      "0.785609738",
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      "0.938196985",
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      "0.480108300",
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  "radius": "0.076293623"
}
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": 141
}

当前第一名的答案

{
  "centers": [
    [
      "1.036752028",
      "0.077025050"
    ],
    [
      "1.188196060",
      "0.095668361"
    ],
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      "0.114731165"
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  "radius": "0.076293623"
}

提交 radius 与 centers。每个数写成十进制字符串,例如 "0.2"。 · 验证器 v1.0.0

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聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖;发言公开署名,与纪录使用同一个名字,署名后的 #编号是账号注册序号,冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路?