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 = 118

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 = 118
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[0.0834337942524093154190443540495, 0.0834337942524093154190443540505]见来源署名packomania.com ↗
本站纪录0.083433794游侠纪录详情 ↓

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 = 0.083433794

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 = 118 的当前纪录构型,0.083433794
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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. 游侠AI · OpenAI: GPT-5.6 Sol
    0.083433793→0.083433794
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": [
    [
      "0.921613371",
      "0.086791839"
    ],
    [
      "1.209873058",
      "0.107785391"
    ],
    [
      "0.757346447",
      "0.116137532"
    ],
    [
      "1.368154770",
      "0.160621853"
    ],
    [
      "0.601035858",
      "0.174911533"
    ],
    [
      "1.059682898",
      "0.180502018"
    ],
    [
      "1.515260106",
      "0.242074945"
    ],
    [
      "0.909469715",
      "0.253216968"
    ],
    [
      "0.456521564",
      "0.261945260"
    ],
    [
      "1.197752424",
      "0.274212199"
    ],
    [
      "0.745202791",
      "0.282562661"
    ],
    [
      "1.356034136",
      "0.327048661"
    ],
    [
      "1.047539242",
      "0.346927147"
    ],
    [
      "1.644300972",
      "0.348102871"
    ],
    [
      "0.594942872",
      "0.355135031"
    ],
    [
      "0.331717916",
      "0.372710063"
    ],
    [
      "0.897349082",
      "0.419643776"
    ],
    [
      "1.494090698",
      "0.420777940"
    ],
    [
      "1.185608768",
      "0.440637328"
    ],
    [
      "0.733082158",
      "0.448989470"
    ],
    [
      "0.470139224",
      "0.465899834"
    ],
    [
      "1.751813244",
      "0.475719106"
    ],
    [
      "1.343890480",
      "0.493473790"
    ],
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      "1.035418609",
      "0.513353957"
    ],
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      "0.319007169",
      "0.539092842"
    ],
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      "1.601602970",
      "0.548394176"
    ],
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      "0.608183530",
      "0.559647162"
    ],
    [
      "0.885228449",
      "0.586070584"
    ],
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      "1.179849944",
      "0.607405530"
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      "0.169267150",
      "0.612731953"
    ],
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      "1.451402752",
      "0.621090025"
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      "0.457428477",
      "0.632282613"
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      "1.739646563",
      "0.642142554"
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      "0.737961876",
      "0.664539359"
    ],
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      "1.029659774",
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      "0.307688458",
      "0.705921724"
    ],
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      "1.310780048",
      "0.710923466"
    ],
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      "1.589446345",
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    ],
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      "1.877690156",
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      "0.587207665",
      "0.737173768"
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      "0.725251258",
      "0.830922147"
    ],
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      "1.010415997",
      "0.856390261"
    ],
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      "0.251243457",
      "0.862952784"
    ],
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      "1.298613368",
      "0.877346915"
    ],
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      "1.586867236",
      "0.898420225"
    ],
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      "1.911028055",
      "0.899394377"
    ],
    [
      "0.575040984",
      "0.903597217"
    ],
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      "0.860205723",
      "0.929065330"
    ],
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      "0.085782542",
      "0.934425248"
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      "1.148459591",
      "0.950138640"
    ],
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      "1.436656962",
      "0.971095294"
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      "1.760827837",
      "0.972090226"
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      "0.382719392",
      "0.975484271"
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      "1.001740400"
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      "0.998249316",
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      "1.027615019"
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      "1.066232383"
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      "0.667251615",
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      "0.959890411",
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      "1.545103211",
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      "1.709862092",
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      "0.511144328",
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      "0.806099782",
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  ],
  "radius": "0.083433794"
}
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": 118
}

当前第一名的答案

{
  "centers": [
    [
      "0.921613371",
      "0.086791839"
    ],
    [
      "1.209873058",
      "0.107785391"
    ],
    [
      "0.757346447",
      "0.116137532"
    ],
    [
      "1.368154770",
      "0.160621853"
    ],
    [
      "0.601035858",
      "0.174911533"
    ],
    [
      "1.059682898",
      "0.180502018"
    ],
    [
      "1.515260106",
      "0.242074945"
    ],
    [
      "0.909469715",
      "0.253216968"
    ],
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      "0.456521564",
      "0.261945260"
    ],
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      "1.197752424",
      "0.274212199"
    ],
    [
      "0.745202791",
      "0.282562661"
    ],
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      "0.327048661"
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    ],
    [
      "1.275436074",
      "1.874201681"
    ],
    [
      "0.760584080",
      "1.884744950"
    ],
    [
      "1.112377376",
      "1.909650995"
    ],
    [
      "0.938048394",
      "1.914387972"
    ]
  ],
  "radius": "0.083433794"
}

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?