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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 = 80

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 = 80
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[0.1003194994161765797066344701745, 0.1003194994161765797066344701755]见来源署名packomania.com ↗
本站纪录0.100319499Ἀθηνᾶ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 = 0.100319499

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 = 80 的当前纪录构型，0.100319499
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
    0.100319498→0.100319499
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": [
    [
      "1.042689873",
      "0.101332888"
    ],
    [
      "0.842465118",
      "0.114219122"
    ],
    [
      "1.246624254",
      "0.134782409"
    ],
    [
      "0.650075214",
      "0.171158853"
    ],
    [
      "1.432241519",
      "0.210954835"
    ],
    [
      "0.475088498",
      "0.269320236"
    ],
    [
      "1.116819811",
      "0.287775266"
    ],
    [
      "0.916648419",
      "0.303291196"
    ],
    [
      "1.631217599",
      "0.358924938"
    ],
    [
      "0.689407482",
      "0.367904833"
    ],
    [
      "0.326207773",
      "0.403821304"
    ],
    [
      "1.436888843",
      "0.411540005"
    ],
    [
      "1.239346793",
      "0.446656185"
    ],
    [
      "0.516014612",
      "0.468854923"
    ],
    [
      "1.039813711",
      "0.473048222"
    ],
    [
      "0.840917641",
      "0.499436880"
    ],
    [
      "1.780004472",
      "0.551649661"
    ],
    [
      "1.579368793",
      "0.552803842"
    ],
    [
      "0.210837484",
      "0.567972772"
    ],
    [
      "1.368529322",
      "0.600174529"
    ],
    [
      "1.162340693",
      "0.631929142"
    ],
    [
      "0.963482289",
      "0.658600161"
    ],
    [
      "0.565112754",
      "0.663393814"
    ],
    [
      "0.764586220",
      "0.684988819"
    ],
    [
      "0.366843516",
      "0.694139827"
    ],
    [
      "1.680686183",
      "0.725982347"
    ],
    [
      "0.134715467",
      "0.753610716"
    ],
    [
      "1.483763745",
      "0.764421404"
    ],
    [
      "1.873088634",
      "0.782879664"
    ],
    [
      "1.284867675",
      "0.790810061"
    ],
    [
      "1.086009271",
      "0.817481080"
    ],
    [
      "0.887150868",
      "0.844152100"
    ],
    [
      "0.488312843",
      "0.853830811"
    ],
    [
      "0.688254798",
      "0.870540757"
    ],
    [
      "0.289550926",
      "0.881211611"
    ],
    [
      "1.611885884",
      "0.921376273"
    ],
    [
      "1.407168957",
      "0.949864782"
    ],
    [
      "0.101627580",
      "0.951502604"
    ],
    [
      "1.208310554",
      "0.976535801"
    ],
    [
      "1.899495853",
      "0.981773270"
    ],
    [
      "1.009452151",
      "1.003206820"
    ],
    [
      "0.810556081",
      "1.029595478"
    ],
    [
      "0.411623920",
      "1.040441608"
    ],
    [
      "0.611660011",
      "1.055984135"
    ],
    [
      "1.728960345",
      "1.087478750"
    ],
    [
      "1.529470240",
      "1.108919502"
    ],
    [
      "1.330611837",
      "1.135590521"
    ],
    [
      "0.113219418",
      "1.151806466"
    ],
    [
      "1.131753433",
      "1.162261540"
    ],
    [
      "0.932857364",
      "1.188650198"
    ],
    [
      "0.733961294",
      "1.215038856"
    ],
    [
      "0.328615402",
      "1.223104121"
    ],
    [
      "1.869643610",
      "1.230531981"
    ],
    [
      "0.528651493",
      "1.238646649"
    ],
    [
      "1.675414246",
      "1.280840647"
    ],
    [
      "1.453138819",
      "1.294471441"
    ],
    [
      "1.254280415",
      "1.321142460"
    ],
    [
      "0.168914474",
      "1.344560373"
    ],
    [
      "1.055158646",
      "1.347704919"
    ],
    [
      "0.856262576",
      "1.374093576"
    ],
    [
      "0.650952776",
      "1.397701369"
    ],
    [
      "0.449474627",
      "1.425763466"
    ],
    [
      "1.574492731",
      "1.454250150"
    ],
    [
      "1.775477089",
      "1.455953046"
    ],
    [
      "1.377971766",
      "1.494691691"
    ],
    [
      "1.177685628",
      "1.506585838"
    ],
    [
      "0.265942803",
      "1.520177887"
    ],
    [
      "0.980746500",
      "1.544939297"
    ],
    [
      "0.780669617",
      "1.559947585"
    ],
    [
      "0.581732052",
      "1.586021573"
    ],
    [
      "1.623191396",
      "1.648889426"
    ],
    [
      "0.399478792",
      "1.669924833"
    ],
    [
      "1.288129331",
      "1.674091648"
    ],
    [
      "0.893705617",
      "1.725715104"
    ],
    [
      "1.094344274",
      "1.726085491"
    ],
    [
      "1.463887177",
      "1.770865546"
    ],
    [
      "0.562881150",
      "1.786353683"
    ],
    [
      "0.748023213",
      "1.863673956"
    ],
    [
      "1.236312421",
      "1.868058739"
    ],
    [
      "0.993704181",
      "1.899658472"
    ]
  ],
  "radius": "0.100319499"
}
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": 80
}

当前第一名的答案

{
  "centers": [
    [
      "1.042689873",
      "0.101332888"
    ],
    [
      "0.842465118",
      "0.114219122"
    ],
    [
      "1.246624254",
      "0.134782409"
    ],
    [
      "0.650075214",
      "0.171158853"
    ],
    [
      "1.432241519",
      "0.210954835"
    ],
    [
      "0.475088498",
      "0.269320236"
    ],
    [
      "1.116819811",
      "0.287775266"
    ],
    [
      "0.916648419",
      "0.303291196"
    ],
    [
      "1.631217599",
      "0.358924938"
    ],
    [
      "0.689407482",
      "0.367904833"
    ],
    [
      "0.326207773",
      "0.403821304"
    ],
    [
      "1.436888843",
      "0.411540005"
    ],
    [
      "1.239346793",
      "0.446656185"
    ],
    [
      "0.516014612",
      "0.468854923"
    ],
    [
      "1.039813711",
      "0.473048222"
    ],
    [
      "0.840917641",
      "0.499436880"
    ],
    [
      "1.780004472",
      "0.551649661"
    ],
    [
      "1.579368793",
      "0.552803842"
    ],
    [
      "0.210837484",
      "0.567972772"
    ],
    [
      "1.368529322",
      "0.600174529"
    ],
    [
      "1.162340693",
      "0.631929142"
    ],
    [
      "0.963482289",
      "0.658600161"
    ],
    [
      "0.565112754",
      "0.663393814"
    ],
    [
      "0.764586220",
      "0.684988819"
    ],
    [
      "0.366843516",
      "0.694139827"
    ],
    [
      "1.680686183",
      "0.725982347"
    ],
    [
      "0.134715467",
      "0.753610716"
    ],
    [
      "1.483763745",
      "0.764421404"
    ],
    [
      "1.873088634",
      "0.782879664"
    ],
    [
      "1.284867675",
      "0.790810061"
    ],
    [
      "1.086009271",
      "0.817481080"
    ],
    [
      "0.887150868",
      "0.844152100"
    ],
    [
      "0.488312843",
      "0.853830811"
    ],
    [
      "0.688254798",
      "0.870540757"
    ],
    [
      "0.289550926",
      "0.881211611"
    ],
    [
      "1.611885884",
      "0.921376273"
    ],
    [
      "1.407168957",
      "0.949864782"
    ],
    [
      "0.101627580",
      "0.951502604"
    ],
    [
      "1.208310554",
      "0.976535801"
    ],
    [
      "1.899495853",
      "0.981773270"
    ],
    [
      "1.009452151",
      "1.003206820"
    ],
    [
      "0.810556081",
      "1.029595478"
    ],
    [
      "0.411623920",
      "1.040441608"
    ],
    [
      "0.611660011",
      "1.055984135"
    ],
    [
      "1.728960345",
      "1.087478750"
    ],
    [
      "1.529470240",
      "1.108919502"
    ],
    [
      "1.330611837",
      "1.135590521"
    ],
    [
      "0.113219418",
      "1.151806466"
    ],
    [
      "1.131753433",
      "1.162261540"
    ],
    [
      "0.932857364",
      "1.188650198"
    ],
    [
      "0.733961294",
      "1.215038856"
    ],
    [
      "0.328615402",
      "1.223104121"
    ],
    [
      "1.869643610",
      "1.230531981"
    ],
    [
      "0.528651493",
      "1.238646649"
    ],
    [
      "1.675414246",
      "1.280840647"
    ],
    [
      "1.453138819",
      "1.294471441"
    ],
    [
      "1.254280415",
      "1.321142460"
    ],
    [
      "0.168914474",
      "1.344560373"
    ],
    [
      "1.055158646",
      "1.347704919"
    ],
    [
      "0.856262576",
      "1.374093576"
    ],
    [
      "0.650952776",
      "1.397701369"
    ],
    [
      "0.449474627",
      "1.425763466"
    ],
    [
      "1.574492731",
      "1.454250150"
    ],
    [
      "1.775477089",
      "1.455953046"
    ],
    [
      "1.377971766",
      "1.494691691"
    ],
    [
      "1.177685628",
      "1.506585838"
    ],
    [
      "0.265942803",
      "1.520177887"
    ],
    [
      "0.980746500",
      "1.544939297"
    ],
    [
      "0.780669617",
      "1.559947585"
    ],
    [
      "0.581732052",
      "1.586021573"
    ],
    [
      "1.623191396",
      "1.648889426"
    ],
    [
      "0.399478792",
      "1.669924833"
    ],
    [
      "1.288129331",
      "1.674091648"
    ],
    [
      "0.893705617",
      "1.725715104"
    ],
    [
      "1.094344274",
      "1.726085491"
    ],
    [
      "1.463887177",
      "1.770865546"
    ],
    [
      "0.562881150",
      "1.786353683"
    ],
    [
      "0.748023213",
      "1.863673956"
    ],
    [
      "1.236312421",
      "1.868058739"
    ],
    [
      "0.993704181",
      "1.899658472"
    ]
  ],
  "radius": "0.100319499"
}

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？