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 = 180

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 = 180
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[0.0678470475313352011570895134595, 0.0678470475313352011570895134605]见来源署名packomania.com ↗
本站纪录0.067847047Ἀθηνᾶ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 = 0.067847047

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 = 180 的当前纪录构型,0.067847047
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
    0.067847046→0.067847047
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": [
    [
      "1.686106618",
      "0.368995376"
    ],
    [
      "1.651478654",
      "1.607574255"
    ],
    [
      "0.184599332",
      "0.939177641"
    ],
    [
      "0.440702510",
      "0.254281927"
    ],
    [
      "0.895548531",
      "1.926282364"
    ],
    [
      "1.006948570",
      "1.565013754"
    ],
    [
      "0.996323059",
      "0.717554149"
    ],
    [
      "0.244011708",
      "0.817181506"
    ],
    [
      "1.337705699",
      "0.266241721"
    ],
    [
      "0.078979331",
      "1.143631664"
    ],
    [
      "1.089129227",
      "0.618559691"
    ],
    [
      "0.532292834",
      "1.212527013"
    ],
    [
      "0.250041676",
      "0.446401193"
    ],
    [
      "1.764724354",
      "0.771371005"
    ],
    [
      "0.195693840",
      "1.074417425"
    ],
    [
      "0.863958553",
      "0.686464030"
    ],
    [
      "0.928696618",
      "1.305152247"
    ],
    [
      "1.587224933",
      "0.276069061"
    ],
    [
      "1.925459540",
      "0.888493202"
    ],
    [
      "1.353707181",
      "0.680600215"
    ],
    [
      "1.231889536",
      "1.496894566"
    ],
    [
      "0.717904556",
      "1.014537522"
    ],
    [
      "0.808957315",
      "0.087633944"
    ],
    [
      "1.458790007",
      "0.188568638"
    ],
    [
      "0.678575925",
      "0.884667794"
    ],
    [
      "1.260592658",
      "0.779304688"
    ],
    [
      "0.850039355",
      "1.045412792"
    ],
    [
      "0.319957272",
      "0.929632200"
    ],
    [
      "1.460374042",
      "0.324253487"
    ],
    [
      "0.942845523",
      "0.946418334"
    ],
    [
      "0.409666647",
      "0.824028525"
    ],
    [
      "0.222497664",
      "1.514197671"
    ],
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      "1.536319606",
      "0.436704181"
    ],
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      "1.246443753",
      "1.138038602"
    ],
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      "0.375619204",
      "0.690391364"
    ],
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      "1.104099003",
      "1.817223637"
    ],
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      "0.585769757",
      "0.983662252"
    ],
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      "1.485841980",
      "0.711475485"
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      "0.156082071",
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      "1.202019447",
      "0.267700657"
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      "1.378578743",
      "1.168913053"
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      "1.024037066"
    ],
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      "0.889367987",
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      "1.390654897",
      "1.546890160"
    ],
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      "1.764790909",
      "1.532920250"
    ],
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      "0.177613808",
      "0.561149139"
    ],
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      "0.429013999",
      "1.565686349"
    ],
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      "0.305389161",
      "1.621630846"
    ],
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      "1.074980322",
      "0.977293604"
    ],
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      "0.687300919",
      "1.732506796"
    ],
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      "1.400961666",
      "0.446249622"
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      "0.786893138",
      "0.421538686"
    ],
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      "1.834059413",
      "1.416237939"
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      "0.853380489"
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      "1.076288062"
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      "1.279081062",
      "1.624118178"
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      "0.067884823",
      "1.008391880"
    ],
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      "0.956993623",
      "0.587684664"
    ],
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      "0.762173371",
      "1.901303290"
    ],
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      "0.781508976",
      "1.633284157"
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      "0.959316673",
      "1.806333552"
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      "1.235942522"
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      "1.107163332"
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      "0.731920710",
      "0.655176724"
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      "0.395286629"
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      "0.069564390"
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      "1.070208580"
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      "1.736568873"
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      "1.128457858",
      "0.748429418"
    ],
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      "0.110480657",
      "0.721316551"
    ],
    [
      "0.874718052",
      "1.534536562"
    ],
    [
      "0.771382093",
      "0.785673336"
    ],
    [
      "1.525670147",
      "0.841194852"
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    [
      "0.665156993",
      "0.260071773"
    ],
    [
      "0.649421515",
      "1.602206995"
    ]
  ],
  "radius": "0.067847047"
}
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": 180
}

当前第一名的答案

{
  "centers": [
    [
      "1.686106618",
      "0.368995376"
    ],
    [
      "1.651478654",
      "1.607574255"
    ],
    [
      "0.184599332",
      "0.939177641"
    ],
    [
      "0.440702510",
      "0.254281927"
    ],
    [
      "0.895548531",
      "1.926282364"
    ],
    [
      "1.006948570",
      "1.565013754"
    ],
    [
      "0.996323059",
      "0.717554149"
    ],
    [
      "0.244011708",
      "0.817181506"
    ],
    [
      "1.337705699",
      "0.266241721"
    ],
    [
      "0.078979331",
      "1.143631664"
    ],
    [
      "1.089129227",
      "0.618559691"
    ],
    [
      "0.532292834",
      "1.212527013"
    ],
    [
      "0.250041676",
      "0.446401193"
    ],
    [
      "1.764724354",
      "0.771371005"
    ],
    [
      "0.195693840",
      "1.074417425"
    ],
    [
      "0.863958553",
      "0.686464030"
    ],
    [
      "0.928696618",
      "1.305152247"
    ],
    [
      "1.587224933",
      "0.276069061"
    ],
    [
      "1.925459540",
      "0.888493202"
    ],
    [
      "1.353707181",
      "0.680600215"
    ],
    [
      "1.231889536",
      "1.496894566"
    ],
    [
      "0.717904556",
      "1.014537522"
    ],
    [
      "0.808957315",
      "0.087633944"
    ],
    [
      "1.458790007",
      "0.188568638"
    ],
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      "0.678575925",
      "0.884667794"
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      "1.260592658",
      "0.779304688"
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      "0.850039355",
      "1.045412792"
    ],
    [
      "0.319957272",
      "0.929632200"
    ],
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      "1.460374042",
      "0.324253487"
    ],
    [
      "0.942845523",
      "0.946418334"
    ],
    [
      "0.409666647",
      "0.824028525"
    ],
    [
      "0.222497664",
      "1.514197671"
    ],
    [
      "1.536319606",
      "0.436704181"
    ],
    [
      "1.246443753",
      "1.138038602"
    ],
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      "0.375619204",
      "0.690391364"
    ],
    [
      "1.104099003",
      "1.817223637"
    ],
    [
      "0.585769757",
      "0.983662252"
    ],
    [
      "1.485841980",
      "0.711475485"
    ],
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      "0.156082071",
      "1.395868231"
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      "1.202019447",
      "0.267700657"
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      "1.378578743",
      "1.168913053"
    ],
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      "1.931842984",
      "1.024037066"
    ],
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      "0.889367987",
      "1.175282520"
    ],
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      "1.390654897",
      "1.546890160"
    ],
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      "1.764790909",
      "1.532920250"
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      "0.177613808",
      "0.561149139"
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      "0.429013999",
      "1.565686349"
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      "0.305389161",
      "1.621630846"
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    [
      "1.074980322",
      "0.977293604"
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      "0.687300919",
      "1.732506796"
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    [
      "1.400961666",
      "0.446249622"
    ],
    [
      "0.786893138",
      "0.421538686"
    ],
    [
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  "radius": "0.067847047"
}

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

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