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 = 192

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 = 192
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[0.0659220324291838230267892535735, 0.0659220324291838230267892535745]见来源署名packomania.com ↗
本站纪录0.065922032Ἀθηνᾶ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 = 0.065922032

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 = 192 的当前纪录构型,0.065922032
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
    0.065922031→0.065922032
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": [
    [
      "0.922678370",
      "0.069127820"
    ],
    [
      "1.133318484",
      "0.075485083"
    ],
    [
      "0.792384661",
      "0.089287356"
    ],
    [
      "0.663774288",
      "0.128635147"
    ],
    [
      "1.366472038",
      "0.140814400"
    ],
    [
      "1.025607720",
      "0.151519607"
    ],
    [
      "1.235809012",
      "0.158422101"
    ],
    [
      "0.533274229",
      "0.190883998"
    ],
    [
      "0.901399252",
      "0.199243371"
    ],
    [
      "1.483792123",
      "0.200971320"
    ],
    [
      "0.771105543",
      "0.219402906"
    ],
    [
      "1.128098248",
      "0.234456625"
    ],
    [
      "0.645626923",
      "0.259874792"
    ],
    [
      "0.424002468",
      "0.264657568"
    ],
    [
      "1.354163495",
      "0.272082664"
    ],
    [
      "1.591473612",
      "0.277047298"
    ],
    [
      "1.004328601",
      "0.281635157"
    ],
    [
      "1.230424408",
      "0.317596353"
    ],
    [
      "0.871241687",
      "0.327592032"
    ],
    [
      "1.471483580",
      "0.332239584"
    ],
    [
      "0.536355162",
      "0.333648361"
    ],
    [
      "0.741400441",
      "0.350485446"
    ],
    [
      "1.106819130",
      "0.364572176"
    ],
    [
      "1.709419213",
      "0.392360280"
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    [
      "0.401965438",
      "0.394646901"
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      "1.338339548",
      "0.402973688"
    ],
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      "1.578636770",
      "0.409057885"
    ],
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      "0.981927063",
      "0.411562170"
    ],
    [
      "0.271717689",
      "0.415101270"
    ],
    [
      "0.632128680",
      "0.424259016"
    ],
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      "1.209145290",
      "0.447711904"
    ],
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      "0.858856234",
      "0.458853062"
    ],
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      "1.455659633",
      "0.463130608"
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      "0.509057851",
      "0.471549908"
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      "0.567519704"
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      "1.187002788",
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    ],
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      "0.838275933",
      "0.589080972"
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      "0.488477550",
      "0.601777818"
    ],
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      "1.413123109",
      "0.614810596"
    ],
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      "1.063837291",
      "0.624727098"
    ],
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      "1.818303426",
      "0.626589434"
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      "0.715099442",
      "0.636095963"
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      "1.641401227",
      "0.647189599"
    ],
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      "0.365406721",
      "0.649068710"
    ],
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      "0.233566157",
      "0.650029413"
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      "1.290052280",
      "0.662101488"
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      "0.940766462",
      "0.672017991"
    ],
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      "0.590968079",
      "0.684714837"
    ],
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      "1.515613652",
      "0.697747616"
    ],
    [
      "0.113728622",
      "0.705000520"
    ],
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      "1.166422487",
      "0.707911223"
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      "0.817695633",
      "0.719308883"
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      "0.467897250",
      "0.732005729"
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      "1.741241695",
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      "1.895290052",
      "0.733621750"
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      "1.392542808",
      "0.745038506"
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      "1.043256990",
      "0.754955009"
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      "0.779296621"
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      "1.618104171",
      "0.780684647"
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      "0.213517958",
      "0.791168645"
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      "1.269471979",
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      "1.631249501"
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      "1.644170392"
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      "1.666895627"
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      "1.652426562",
      "1.668461839"
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      "0.552775123",
      "1.678264492"
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    [
      "1.522648496",
      "1.691710733"
    ],
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      "1.127727961",
      "1.704270017"
    ],
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      "0.778442143",
      "1.714186519"
    ],
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      "0.421642037",
      "1.733349745"
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      "1.353289318",
      "1.739916143"
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      "1.004098168",
      "1.750079751"
    ],
    [
      "0.655371314",
      "1.761477412"
    ],
    [
      "1.230218489",
      "1.787207035"
    ],
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      "0.880932671",
      "1.797123538"
    ],
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      "0.532194822",
      "1.808492403"
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      "1.464835522",
      "1.810203423"
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      "1.071691680",
      "1.863278579"
    ],
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      "1.332242691",
      "1.872523677"
    ],
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      "0.701345132",
      "1.885046281"
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      "1.193462081",
      "1.913823874"
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      "0.828931986",
      "1.918279578"
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    [
      "0.959927030",
      "1.933217984"
    ]
  ],
  "radius": "0.065922032"
}
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": 192
}

当前第一名的答案

{
  "centers": [
    [
      "0.922678370",
      "0.069127820"
    ],
    [
      "1.133318484",
      "0.075485083"
    ],
    [
      "0.792384661",
      "0.089287356"
    ],
    [
      "0.663774288",
      "0.128635147"
    ],
    [
      "1.366472038",
      "0.140814400"
    ],
    [
      "1.025607720",
      "0.151519607"
    ],
    [
      "1.235809012",
      "0.158422101"
    ],
    [
      "0.533274229",
      "0.190883998"
    ],
    [
      "0.901399252",
      "0.199243371"
    ],
    [
      "1.483792123",
      "0.200971320"
    ],
    [
      "0.771105543",
      "0.219402906"
    ],
    [
      "1.128098248",
      "0.234456625"
    ],
    [
      "0.645626923",
      "0.259874792"
    ],
    [
      "0.424002468",
      "0.264657568"
    ],
    [
      "1.354163495",
      "0.272082664"
    ],
    [
      "1.591473612",
      "0.277047298"
    ],
    [
      "1.004328601",
      "0.281635157"
    ],
    [
      "1.230424408",
      "0.317596353"
    ],
    [
      "0.871241687",
      "0.327592032"
    ],
    [
      "1.471483580",
      "0.332239584"
    ],
    [
      "0.536355162",
      "0.333648361"
    ],
    [
      "0.741400441",
      "0.350485446"
    ],
    [
      "1.106819130",
      "0.364572176"
    ],
    [
      "1.709419213",
      "0.392360280"
    ],
    [
      "0.401965438",
      "0.394646901"
    ],
    [
      "1.338339548",
      "0.402973688"
    ],
    [
      "1.578636770",
      "0.409057885"
    ],
    [
      "0.981927063",
      "0.411562170"
    ],
    [
      "0.271717689",
      "0.415101270"
    ],
    [
      "0.632128680",
      "0.424259016"
    ],
    [
      "1.209145290",
      "0.447711904"
    ],
    [
      "0.858856234",
      "0.458853062"
    ],
    [
      "1.455659633",
      "0.463130608"
    ],
    [
      "0.509057851",
      "0.471549908"
    ],
    [
      "1.084417591",
      "0.494499188"
    ],
    [
      "1.787906140",
      "0.498297336"
    ],
    [
      "0.735679742",
      "0.505868053"
    ],
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      "1.657123697",
      "0.514994941"
    ],
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      "0.354555566",
      "0.517671945"
    ],
    [
      "0.196620528",
      "0.523467658"
    ],
    [
      "1.310632580",
      "0.531873577"
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  "radius": "0.065922032"
}

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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