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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 = 83

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 = 83
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[0.0988449192768667154957425485445, 0.0988449192768667154957425485455]见来源署名packomania.com ↗
本站纪录0.098844919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 = 0.098844919

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 = 83 的当前纪录构型，0.098844919
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
    0.098844918→0.098844919
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": [
    [
      "0.227827890",
      "0.535424159"
    ],
    [
      "0.347709450",
      "0.378231017"
    ],
    [
      "0.110372238",
      "0.856323541"
    ],
    [
      "0.498995185",
      "0.250950833"
    ],
    [
      "0.100450175",
      "1.053764228"
    ],
    [
      "0.262480939",
      "0.730053135"
    ],
    [
      "0.413708052",
      "0.602728226"
    ],
    [
      "0.674380878",
      "0.159730599"
    ],
    [
      "0.533648234",
      "0.445579809"
    ],
    [
      "0.294247350",
      "0.928926679"
    ],
    [
      "0.137921862",
      "1.259309451"
    ],
    [
      "0.284325287",
      "1.126367366"
    ],
    [
      "0.446356051",
      "0.802656274"
    ],
    [
      "0.709033927",
      "0.354359575"
    ],
    [
      "0.597583164",
      "0.675331365"
    ],
    [
      "0.221481065",
      "1.453859832"
    ],
    [
      "0.860261040",
      "0.227034666"
    ],
    [
      "0.479657530",
      "0.997529219"
    ],
    [
      "0.357212504",
      "1.310130055"
    ],
    [
      "0.748810277",
      "0.548006455"
    ],
    [
      "1.010810977",
      "0.098909770"
    ],
    [
      "0.632236213",
      "0.869960341"
    ],
    [
      "0.900037390",
      "0.420681546"
    ],
    [
      "0.515662149",
      "1.191914226"
    ],
    [
      "0.783463326",
      "0.742635432"
    ],
    [
      "0.406851367",
      "1.522555707"
    ],
    [
      "1.050587327",
      "0.292556650"
    ],
    [
      "0.666889262",
      "1.064589317"
    ],
    [
      "0.934690440",
      "0.615310523"
    ],
    [
      "0.550315198",
      "1.386543203"
    ],
    [
      "1.207033713",
      "0.122949533"
    ],
    [
      "0.818116376",
      "0.937264408"
    ],
    [
      "0.451635416",
      "1.715106120"
    ],
    [
      "1.086268832",
      "0.487007917"
    ],
    [
      "0.701542312",
      "1.259218294"
    ],
    [
      "0.969343489",
      "0.809939499"
    ],
    [
      "1.246810063",
      "0.316596413"
    ],
    [
      "0.852769425",
      "1.131893384"
    ],
    [
      "0.622397637",
      "1.615499212"
    ],
    [
      "1.120570602",
      "0.682614590"
    ],
    [
      "0.736195361",
      "1.453847270"
    ],
    [
      "1.003996538",
      "1.004568475"
    ],
    [
      "1.410981621",
      "0.203222996"
    ],
    [
      "0.620759413",
      "1.817469912"
    ],
    [
      "1.271797715",
      "0.555289681"
    ],
    [
      "0.887422474",
      "1.326522361"
    ],
    [
      "1.155223651",
      "0.877243566"
    ],
    [
      "1.415553704",
      "0.419586004"
    ],
    [
      "1.040900717",
      "1.201920189"
    ],
    [
      "0.808277800",
      "1.682803279"
    ],
    [
      "1.306450765",
      "0.749918657"
    ],
    [
      "0.922075523",
      "1.521151337"
    ],
    [
      "1.189852124",
      "1.071940380"
    ],
    [
      "1.580658366",
      "0.310858258"
    ],
    [
      "0.808134316",
      "1.880493066"
    ],
    [
      "1.457677878",
      "0.622593748"
    ],
    [
      "1.075553766",
      "1.396549165"
    ],
    [
      "1.341079237",
      "0.944615471"
    ],
    [
      "1.004886215",
      "1.703452814"
    ],
    [
      "1.226780879",
      "1.269224256"
    ],
    [
      "1.616755671",
      "0.505224553"
    ],
    [
      "1.492306350",
      "0.817290562"
    ],
    [
      "1.375732286",
      "1.139244447"
    ],
    [
      "1.004742731",
      "1.901142600"
    ],
    [
      "1.261433928",
      "1.463853232"
    ],
    [
      "1.526959399",
      "1.011919538"
    ],
    [
      "1.671633921",
      "0.734086627"
    ],
    [
      "1.201266387",
      "1.680734973"
    ],
    [
      "1.413430575",
      "1.334377844"
    ],
    [
      "1.798667748",
      "0.582614917"
    ],
    [
      "1.201122903",
      "1.878424759"
    ],
    [
      "1.706286970",
      "0.928715603"
    ],
    [
      "1.387967544",
      "1.615743049"
    ],
    [
      "1.865871566",
      "0.770940975"
    ],
    [
      "1.692112825",
      "1.125896652"
    ],
    [
      "1.538536509",
      "1.487640516"
    ],
    [
      "1.635284384",
      "1.315242369"
    ],
    [
      "1.387824061",
      "1.813432835"
    ],
    [
      "1.900499463",
      "0.965631452"
    ],
    [
      "1.584424658",
      "1.680430509"
    ],
    [
      "1.886325318",
      "1.162812500"
    ],
    [
      "1.732749002",
      "1.524556364"
    ],
    [
      "1.829496877",
      "1.352158217"
    ]
  ],
  "radius": "0.098844919"
}
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": 83
}

当前第一名的答案

{
  "centers": [
    [
      "0.227827890",
      "0.535424159"
    ],
    [
      "0.347709450",
      "0.378231017"
    ],
    [
      "0.110372238",
      "0.856323541"
    ],
    [
      "0.498995185",
      "0.250950833"
    ],
    [
      "0.100450175",
      "1.053764228"
    ],
    [
      "0.262480939",
      "0.730053135"
    ],
    [
      "0.413708052",
      "0.602728226"
    ],
    [
      "0.674380878",
      "0.159730599"
    ],
    [
      "0.533648234",
      "0.445579809"
    ],
    [
      "0.294247350",
      "0.928926679"
    ],
    [
      "0.137921862",
      "1.259309451"
    ],
    [
      "0.284325287",
      "1.126367366"
    ],
    [
      "0.446356051",
      "0.802656274"
    ],
    [
      "0.709033927",
      "0.354359575"
    ],
    [
      "0.597583164",
      "0.675331365"
    ],
    [
      "0.221481065",
      "1.453859832"
    ],
    [
      "0.860261040",
      "0.227034666"
    ],
    [
      "0.479657530",
      "0.997529219"
    ],
    [
      "0.357212504",
      "1.310130055"
    ],
    [
      "0.748810277",
      "0.548006455"
    ],
    [
      "1.010810977",
      "0.098909770"
    ],
    [
      "0.632236213",
      "0.869960341"
    ],
    [
      "0.900037390",
      "0.420681546"
    ],
    [
      "0.515662149",
      "1.191914226"
    ],
    [
      "0.783463326",
      "0.742635432"
    ],
    [
      "0.406851367",
      "1.522555707"
    ],
    [
      "1.050587327",
      "0.292556650"
    ],
    [
      "0.666889262",
      "1.064589317"
    ],
    [
      "0.934690440",
      "0.615310523"
    ],
    [
      "0.550315198",
      "1.386543203"
    ],
    [
      "1.207033713",
      "0.122949533"
    ],
    [
      "0.818116376",
      "0.937264408"
    ],
    [
      "0.451635416",
      "1.715106120"
    ],
    [
      "1.086268832",
      "0.487007917"
    ],
    [
      "0.701542312",
      "1.259218294"
    ],
    [
      "0.969343489",
      "0.809939499"
    ],
    [
      "1.246810063",
      "0.316596413"
    ],
    [
      "0.852769425",
      "1.131893384"
    ],
    [
      "0.622397637",
      "1.615499212"
    ],
    [
      "1.120570602",
      "0.682614590"
    ],
    [
      "0.736195361",
      "1.453847270"
    ],
    [
      "1.003996538",
      "1.004568475"
    ],
    [
      "1.410981621",
      "0.203222996"
    ],
    [
      "0.620759413",
      "1.817469912"
    ],
    [
      "1.271797715",
      "0.555289681"
    ],
    [
      "0.887422474",
      "1.326522361"
    ],
    [
      "1.155223651",
      "0.877243566"
    ],
    [
      "1.415553704",
      "0.419586004"
    ],
    [
      "1.040900717",
      "1.201920189"
    ],
    [
      "0.808277800",
      "1.682803279"
    ],
    [
      "1.306450765",
      "0.749918657"
    ],
    [
      "0.922075523",
      "1.521151337"
    ],
    [
      "1.189852124",
      "1.071940380"
    ],
    [
      "1.580658366",
      "0.310858258"
    ],
    [
      "0.808134316",
      "1.880493066"
    ],
    [
      "1.457677878",
      "0.622593748"
    ],
    [
      "1.075553766",
      "1.396549165"
    ],
    [
      "1.341079237",
      "0.944615471"
    ],
    [
      "1.004886215",
      "1.703452814"
    ],
    [
      "1.226780879",
      "1.269224256"
    ],
    [
      "1.616755671",
      "0.505224553"
    ],
    [
      "1.492306350",
      "0.817290562"
    ],
    [
      "1.375732286",
      "1.139244447"
    ],
    [
      "1.004742731",
      "1.901142600"
    ],
    [
      "1.261433928",
      "1.463853232"
    ],
    [
      "1.526959399",
      "1.011919538"
    ],
    [
      "1.671633921",
      "0.734086627"
    ],
    [
      "1.201266387",
      "1.680734973"
    ],
    [
      "1.413430575",
      "1.334377844"
    ],
    [
      "1.798667748",
      "0.582614917"
    ],
    [
      "1.201122903",
      "1.878424759"
    ],
    [
      "1.706286970",
      "0.928715603"
    ],
    [
      "1.387967544",
      "1.615743049"
    ],
    [
      "1.865871566",
      "0.770940975"
    ],
    [
      "1.692112825",
      "1.125896652"
    ],
    [
      "1.538536509",
      "1.487640516"
    ],
    [
      "1.635284384",
      "1.315242369"
    ],
    [
      "1.387824061",
      "1.813432835"
    ],
    [
      "1.900499463",
      "0.965631452"
    ],
    [
      "1.584424658",
      "1.680430509"
    ],
    [
      "1.886325318",
      "1.162812500"
    ],
    [
      "1.732749002",
      "1.524556364"
    ],
    [
      "1.829496877",
      "1.352158217"
    ]
  ],
  "radius": "0.098844919"
}

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？