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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 = 96

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 = 96
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[0.0918847164826246234413915370045, 0.0918847164826246234413915370055]见来源署名packomania.com ↗
本站纪录0.091884716Ἀθηνᾶ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 = 0.091884716

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 = 96 的当前纪录构型，0.091884716
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
    0.091884715→0.091884716
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

{
  "centers": [
    [
      "0.908115284",
      "0.096545205"
    ],
    [
      "1.091884716",
      "0.096545205"
    ],
    [
      "1.275441534",
      "0.135767454"
    ],
    [
      "0.663623528",
      "0.156481039"
    ],
    [
      "1.439032222",
      "0.225192252"
    ],
    [
      "0.497513635",
      "0.243616404"
    ],
    [
      "0.817750167",
      "0.256561924"
    ],
    [
      "1.149585806",
      "0.271020959"
    ],
    [
      "0.654014259",
      "0.339999067"
    ],
    [
      "0.976728792",
      "0.350272304"
    ],
    [
      "1.304710074",
      "0.369548448"
    ],
    [
      "1.667225054",
      "0.383977196"
    ],
    [
      "1.484797966",
      "0.406148389"
    ],
    [
      "0.309701051",
      "0.409948537"
    ],
    [
      "0.492597567",
      "0.427839011"
    ],
    [
      "0.811286386",
      "0.440217645"
    ],
    [
      "1.129300096",
      "0.453667323"
    ],
    [
      "0.648396943",
      "0.525295456"
    ],
    [
      "1.777659609",
      "0.531044884"
    ],
    [
      "0.966410653",
      "0.538745135"
    ],
    [
      "1.284424363",
      "0.552194813"
    ],
    [
      "1.595212327",
      "0.553049286"
    ],
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      "0.205043152",
      "0.561004581"
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      "0.470986860",
      "0.610333345"
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      "0.803521211",
      "0.623822946"
    ],
    [
      "1.121534921",
      "0.637272624"
    ],
    [
      "1.439548630",
      "0.650722302"
    ],
    [
      "1.856149031",
      "0.697209306"
    ],
    [
      "0.315323164",
      "0.708006362"
    ],
    [
      "0.640631768",
      "0.708900757"
    ],
    [
      "0.958645478",
      "0.722350436"
    ],
    [
      "0.132896075",
      "0.730177555"
    ],
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      "1.276659188",
      "0.735800114"
    ],
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      "1.601967792",
      "0.736694509"
    ],
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      "0.477742326",
      "0.793978569"
    ],
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      "0.795756035",
      "0.807428247"
    ],
    [
      "1.113769745",
      "0.820877925"
    ],
    [
      "1.431783455",
      "0.834327603"
    ],
    [
      "1.749207168",
      "0.846657097"
    ],
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      "0.243310436",
      "0.877078451"
    ],
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      "0.632866593",
      "0.892506058"
    ],
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      "0.950880302",
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    ],
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      "1.268894013",
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      "0.469977150",
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      "1.106004570",
      "1.004483226"
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      "1.116460394"
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      "1.161189171"
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      "1.273166338"
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      "1.253363662",
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      "0.446959506",
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      "0.171642838",
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      "1.828357162",
      "1.372152899"
    ],
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      "1.408487929",
      "1.385143506"
    ],
    [
      "0.609571067",
      "1.443321961"
    ],
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      "0.927584777",
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      "0.447288656",
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      "1.531301896"
    ],
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    ],
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      "1.082709044",
      "1.555299129"
    ],
    [
      "1.393632023",
      "1.579112440"
    ],
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      "0.601415296",
      "1.629632907"
    ],
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      "0.927045348",
      "1.652972146"
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      "1.612166322",
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      "0.758195265",
      "1.725503892"
    ],
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      "1.089464510",
      "1.738944353"
    ],
    [
      "1.438083006",
      "1.795418163"
    ],
    [
      "0.596299090",
      "1.813387333"
    ],
    [
      "1.199878871",
      "1.885845249"
    ],
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      "0.834149918",
      "1.892842158"
    ],
    [
      "1.017297239",
      "1.907950535"
    ]
  ],
  "radius": "0.091884716"
}
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": 96
}

当前第一名的答案

{
  "centers": [
    [
      "0.908115284",
      "0.096545205"
    ],
    [
      "1.091884716",
      "0.096545205"
    ],
    [
      "1.275441534",
      "0.135767454"
    ],
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      "0.663623528",
      "0.156481039"
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      "1.439032222",
      "0.225192252"
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      "0.497513635",
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    ],
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      "0.339999067"
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      "0.976728792",
      "0.350272304"
    ],
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      "0.369548448"
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      "1.089464510",
      "1.738944353"
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      "1.892842158"
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    [
      "1.017297239",
      "1.907950535"
    ]
  ],
  "radius": "0.091884716"
}

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

DISCUSSION

讨论区

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？