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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 = 84

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 = 84
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[0.0985267213904549868449913723925, 0.0985267213904549868449913723935]见来源署名packomania.com ↗
本站纪录0.098526721Ἀθηνᾶ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 = 0.098526721

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 = 84 的当前纪录构型，0.098526721
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
    0.098526720→0.098526721
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": [
    [
      "0.838243017",
      "0.113157990"
    ],
    [
      "1.179783991",
      "0.116636095"
    ],
    [
      "0.649413670",
      "0.169492145"
    ],
    [
      "1.367426931",
      "0.176804080"
    ],
    [
      "1.008012430",
      "0.213200011"
    ],
    [
      "0.475503381",
      "0.267219474"
    ],
    [
      "1.540829470",
      "0.279142226"
    ],
    [
      "0.836123702",
      "0.310200036"
    ],
    [
      "1.177664676",
      "0.313678141"
    ],
    [
      "0.647294355",
      "0.366534191"
    ],
    [
      "1.365307616",
      "0.373846126"
    ],
    [
      "0.323994707",
      "0.403618481"
    ],
    [
      "1.005893114",
      "0.410242057"
    ],
    [
      "1.688870767",
      "0.418526752"
    ],
    [
      "0.498311652",
      "0.495507883"
    ],
    [
      "1.512006320",
      "0.505411887"
    ],
    [
      "0.834169632",
      "0.507243790"
    ],
    [
      "1.175710606",
      "0.510721895"
    ],
    [
      "0.322259620",
      "0.600664285"
    ],
    [
      "0.662645994",
      "0.604247403"
    ],
    [
      "1.003939044",
      "0.607285811"
    ],
    [
      "1.345480018",
      "0.610763916"
    ],
    [
      "1.686068504",
      "0.615560269"
    ],
    [
      "0.150735982",
      "0.697667898"
    ],
    [
      "0.491407538",
      "0.701753558"
    ],
    [
      "0.832415406",
      "0.704289424"
    ],
    [
      "1.173956380",
      "0.707767529"
    ],
    [
      "1.514378580",
      "0.712269262"
    ],
    [
      "1.855581089",
      "0.716036847"
    ],
    [
      "0.319883900",
      "0.798757172"
    ],
    [
      "0.661176950",
      "0.801795580"
    ],
    [
      "1.002184818",
      "0.804331445"
    ],
    [
      "1.342854942",
      "0.809272875"
    ],
    [
      "1.683891165",
      "0.812745840"
    ],
    [
      "0.105334653",
      "0.889419766"
    ],
    [
      "0.489653312",
      "0.898799193"
    ],
    [
      "0.830946362",
      "0.901837601"
    ],
    [
      "1.171083380",
      "0.905836791"
    ],
    [
      "1.896840275",
      "0.908722443"
    ],
    [
      "1.512367527",
      "0.909749453"
    ],
    [
      "0.274482571",
      "0.990509039"
    ],
    [
      "0.659422724",
      "0.998841214"
    ],
    [
      "0.999844924",
      "1.003342947"
    ],
    [
      "1.725150351",
      "1.005431436"
    ],
    [
      "1.340595965",
      "1.006313369"
    ],
    [
      "0.103734267",
      "1.094028395"
    ],
    [
      "0.487899086",
      "1.095844827"
    ],
    [
      "0.828321286",
      "1.100346560"
    ],
    [
      "1.169357509",
      "1.103819525"
    ],
    [
      "1.510108551",
      "1.106789947"
    ],
    [
      "1.894017825",
      "1.114041462"
    ],
    [
      "0.311674645",
      "1.184020819"
    ],
    [
      "0.656797648",
      "1.197350173"
    ],
    [
      "1.684637834",
      "1.198275402"
    ],
    [
      "0.997833871",
      "1.200823138"
    ],
    [
      "1.338870094",
      "1.204296103"
    ],
    [
      "0.484442159",
      "1.292867945"
    ],
    [
      "0.147441525",
      "1.292913158"
    ],
    [
      "0.826310233",
      "1.297826751"
    ],
    [
      "1.167346457",
      "1.301299717"
    ],
    [
      "1.508932130",
      "1.303839878"
    ],
    [
      "1.845976982",
      "1.311411331"
    ],
    [
      "0.315955549",
      "1.395055622"
    ],
    [
      "0.995822819",
      "1.398303329"
    ],
    [
      "1.675162471",
      "1.409658275"
    ],
    [
      "0.674002486",
      "1.422856385"
    ],
    [
      "1.316798076",
      "1.429729747"
    ],
    [
      "0.488695978",
      "1.489875468"
    ],
    [
      "1.500531247",
      "1.500948993"
    ],
    [
      "0.843515072",
      "1.523332963"
    ],
    [
      "1.145274438",
      "1.526733360"
    ],
    [
      "0.320209368",
      "1.592063146"
    ],
    [
      "1.666906939",
      "1.606538710"
    ],
    [
      "0.633580819",
      "1.623436100"
    ],
    [
      "1.352808945",
      "1.631364409"
    ],
    [
      "0.992966691",
      "1.651762995"
    ],
    [
      "0.807017211",
      "1.716976878"
    ],
    [
      "1.177399404",
      "1.721150555"
    ],
    [
      "0.465094209",
      "1.725623777"
    ],
    [
      "1.519184637",
      "1.736954126"
    ],
    [
      "0.641054272",
      "1.826677925"
    ],
    [
      "1.343024956",
      "1.832747919"
    ],
    [
      "0.891636459",
      "1.894936542"
    ],
    [
      "1.088678013",
      "1.897101043"
    ]
  ],
  "radius": "0.098526721"
}
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": 84
}

当前第一名的答案

{
  "centers": [
    [
      "0.838243017",
      "0.113157990"
    ],
    [
      "1.179783991",
      "0.116636095"
    ],
    [
      "0.649413670",
      "0.169492145"
    ],
    [
      "1.367426931",
      "0.176804080"
    ],
    [
      "1.008012430",
      "0.213200011"
    ],
    [
      "0.475503381",
      "0.267219474"
    ],
    [
      "1.540829470",
      "0.279142226"
    ],
    [
      "0.836123702",
      "0.310200036"
    ],
    [
      "1.177664676",
      "0.313678141"
    ],
    [
      "0.647294355",
      "0.366534191"
    ],
    [
      "1.365307616",
      "0.373846126"
    ],
    [
      "0.323994707",
      "0.403618481"
    ],
    [
      "1.005893114",
      "0.410242057"
    ],
    [
      "1.688870767",
      "0.418526752"
    ],
    [
      "0.498311652",
      "0.495507883"
    ],
    [
      "1.512006320",
      "0.505411887"
    ],
    [
      "0.834169632",
      "0.507243790"
    ],
    [
      "1.175710606",
      "0.510721895"
    ],
    [
      "0.322259620",
      "0.600664285"
    ],
    [
      "0.662645994",
      "0.604247403"
    ],
    [
      "1.003939044",
      "0.607285811"
    ],
    [
      "1.345480018",
      "0.610763916"
    ],
    [
      "1.686068504",
      "0.615560269"
    ],
    [
      "0.150735982",
      "0.697667898"
    ],
    [
      "0.491407538",
      "0.701753558"
    ],
    [
      "0.832415406",
      "0.704289424"
    ],
    [
      "1.173956380",
      "0.707767529"
    ],
    [
      "1.514378580",
      "0.712269262"
    ],
    [
      "1.855581089",
      "0.716036847"
    ],
    [
      "0.319883900",
      "0.798757172"
    ],
    [
      "0.661176950",
      "0.801795580"
    ],
    [
      "1.002184818",
      "0.804331445"
    ],
    [
      "1.342854942",
      "0.809272875"
    ],
    [
      "1.683891165",
      "0.812745840"
    ],
    [
      "0.105334653",
      "0.889419766"
    ],
    [
      "0.489653312",
      "0.898799193"
    ],
    [
      "0.830946362",
      "0.901837601"
    ],
    [
      "1.171083380",
      "0.905836791"
    ],
    [
      "1.896840275",
      "0.908722443"
    ],
    [
      "1.512367527",
      "0.909749453"
    ],
    [
      "0.274482571",
      "0.990509039"
    ],
    [
      "0.659422724",
      "0.998841214"
    ],
    [
      "0.999844924",
      "1.003342947"
    ],
    [
      "1.725150351",
      "1.005431436"
    ],
    [
      "1.340595965",
      "1.006313369"
    ],
    [
      "0.103734267",
      "1.094028395"
    ],
    [
      "0.487899086",
      "1.095844827"
    ],
    [
      "0.828321286",
      "1.100346560"
    ],
    [
      "1.169357509",
      "1.103819525"
    ],
    [
      "1.510108551",
      "1.106789947"
    ],
    [
      "1.894017825",
      "1.114041462"
    ],
    [
      "0.311674645",
      "1.184020819"
    ],
    [
      "0.656797648",
      "1.197350173"
    ],
    [
      "1.684637834",
      "1.198275402"
    ],
    [
      "0.997833871",
      "1.200823138"
    ],
    [
      "1.338870094",
      "1.204296103"
    ],
    [
      "0.484442159",
      "1.292867945"
    ],
    [
      "0.147441525",
      "1.292913158"
    ],
    [
      "0.826310233",
      "1.297826751"
    ],
    [
      "1.167346457",
      "1.301299717"
    ],
    [
      "1.508932130",
      "1.303839878"
    ],
    [
      "1.845976982",
      "1.311411331"
    ],
    [
      "0.315955549",
      "1.395055622"
    ],
    [
      "0.995822819",
      "1.398303329"
    ],
    [
      "1.675162471",
      "1.409658275"
    ],
    [
      "0.674002486",
      "1.422856385"
    ],
    [
      "1.316798076",
      "1.429729747"
    ],
    [
      "0.488695978",
      "1.489875468"
    ],
    [
      "1.500531247",
      "1.500948993"
    ],
    [
      "0.843515072",
      "1.523332963"
    ],
    [
      "1.145274438",
      "1.526733360"
    ],
    [
      "0.320209368",
      "1.592063146"
    ],
    [
      "1.666906939",
      "1.606538710"
    ],
    [
      "0.633580819",
      "1.623436100"
    ],
    [
      "1.352808945",
      "1.631364409"
    ],
    [
      "0.992966691",
      "1.651762995"
    ],
    [
      "0.807017211",
      "1.716976878"
    ],
    [
      "1.177399404",
      "1.721150555"
    ],
    [
      "0.465094209",
      "1.725623777"
    ],
    [
      "1.519184637",
      "1.736954126"
    ],
    [
      "0.641054272",
      "1.826677925"
    ],
    [
      "1.343024956",
      "1.832747919"
    ],
    [
      "0.891636459",
      "1.894936542"
    ],
    [
      "1.088678013",
      "1.897101043"
    ]
  ],
  "radius": "0.098526721"
}

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？