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 = 169

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 = 169
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[0.0699527765624092938598222631035, 0.0699527765624092938598222631045]见来源署名packomania.com ↗
本站纪录0.069952776Ἀθηνᾶ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 = 0.069952776

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 = 169 的当前纪录构型,0.069952776
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
    0.069952775→0.069952776
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": [
    [
      "1.011631065",
      "0.070025508"
    ],
    [
      "0.872001120",
      "0.078802885"
    ],
    [
      "1.150997813",
      "0.082292259"
    ],
    [
      "0.735267626",
      "0.108425770"
    ],
    [
      "1.286947675",
      "0.115325557"
    ],
    [
      "0.604524685",
      "0.158223834"
    ],
    [
      "1.416404280",
      "0.168377902"
    ],
    [
      "0.971716599",
      "0.206049741"
    ],
    [
      "0.829288544",
      "0.212029007"
    ],
    [
      "1.118292632",
      "0.218321428"
    ],
    [
      "0.482730843",
      "0.227070212"
    ],
    [
      "1.536438191",
      "0.240248787"
    ],
    [
      "1.299509134",
      "0.254666052"
    ],
    [
      "0.623919215",
      "0.296778575"
    ],
    [
      "0.372642137",
      "0.313406999"
    ],
    [
      "1.422473112",
      "0.321400032"
    ],
    [
      "1.644333194",
      "0.329311866"
    ],
    [
      "0.758942079",
      "0.333417215"
    ],
    [
      "0.898847632",
      "0.333417215"
    ],
    [
      "1.038753185",
      "0.333417215"
    ],
    [
      "1.188245408",
      "0.339483191"
    ],
    [
      "1.530292411",
      "0.411394023"
    ],
    [
      "0.276749734",
      "0.415980031"
    ],
    [
      "0.553966438",
      "0.417940338"
    ],
    [
      "1.737647765",
      "0.433551756"
    ],
    [
      "0.416927915",
      "0.446118467"
    ],
    [
      "0.688989302",
      "0.454578978"
    ],
    [
      "0.828894855",
      "0.454578978"
    ],
    [
      "0.968800408",
      "0.454578978"
    ],
    [
      "1.108705961",
      "0.454578978"
    ],
    [
      "1.398095915",
      "0.459165471"
    ],
    [
      "1.258198185",
      "0.460644954"
    ],
    [
      "1.629328920",
      "0.522098872"
    ],
    [
      "0.323267389",
      "0.550047624"
    ],
    [
      "1.814270314",
      "0.550609642"
    ],
    [
      "0.183441902",
      "0.554780156"
    ],
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      "1.494745666",
      "0.560320744"
    ],
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      "0.573231303"
    ],
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      "0.474954335",
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    ],
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      "0.758942079",
      "0.575740741"
    ],
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      "0.898847632",
      "0.575740741"
    ],
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      "0.575740741"
    ],
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    ],
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      "1.328150961",
      "0.581806717"
    ],
    [
      "1.705951470",
      "0.639156758"
    ],
    [
      "1.571368216",
      "0.678777686"
    ],
    [
      "0.270351897",
      "0.679560227"
    ],
    [
      "1.425760587",
      "0.682036078"
    ],
    [
      "0.125896715",
      "0.682303155"
    ],
    [
      "1.840319088",
      "0.688068819"
    ],
    [
      "0.549083881",
      "0.696710443"
    ],
    [
      "0.409178459",
      "0.696902353"
    ],
    [
      "0.688989302",
      "0.696902505"
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      "0.828894855",
      "0.696902505"
    ],
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      "0.968800408",
      "0.696902505"
    ],
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      "1.108705961",
      "0.696902505"
    ],
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      "1.248611515",
      "0.696902505"
    ],
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      "1.732000244",
      "0.776615935"
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      "0.200398892",
      "0.800721859"
    ],
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      "1.910271864",
      "0.809230582"
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      "0.339225419",
      "0.818064268"
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      "0.818064268"
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      "0.818064268"
    ],
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      "1.038753185",
      "0.818064268"
    ],
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      "1.178658738",
      "0.818064268"
    ],
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      "1.318564291",
      "0.818064268"
    ],
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      "1.458469844",
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      "1.598375397",
      "0.818064268"
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      "0.079037059",
      "0.870326954"
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      "1.009940714"
    ],
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      "1.771035906",
      "1.035623431"
    ],
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      "0.339225419",
      "1.060387794"
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      "1.060387794"
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      "1.060387794"
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      "0.082019319",
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      "1.172717848"
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      "0.409178196",
      "1.181549557"
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      "1.181549557"
    ],
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      "0.688989302",
      "1.181549557"
    ],
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      "0.828894855",
      "1.181549557"
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      "1.545034847"
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      "1.744906189",
      "1.556868572"
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      "0.433863099",
      "1.561597455"
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      "0.306703979",
      "1.619942308"
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      "1.658391490"
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      "1.679227641"
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      "0.678644596",
      "1.680298718"
    ],
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      "0.407540973",
      "1.716924082"
    ],
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      "1.529567075",
      "1.764556441"
    ],
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      "1.185393823",
      "1.783211719"
    ],
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      "0.769341324",
      "1.792794164"
    ],
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      "0.909246877",
      "1.792794164"
    ],
    [
      "0.521784553",
      "1.797682784"
    ],
    [
      "1.048085132",
      "1.810042433"
    ],
    [
      "1.408890111",
      "1.835342274"
    ],
    [
      "0.646849536",
      "1.860390950"
    ],
    [
      "1.278960489",
      "1.887225385"
    ],
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      "0.836449986",
      "1.915554057"
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      "1.141319294",
      "1.919031724"
    ],
    [
      "0.975635697",
      "1.929728035"
    ]
  ],
  "radius": "0.069952776"
}
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": 169
}

当前第一名的答案

{
  "centers": [
    [
      "1.011631065",
      "0.070025508"
    ],
    [
      "0.872001120",
      "0.078802885"
    ],
    [
      "1.150997813",
      "0.082292259"
    ],
    [
      "0.735267626",
      "0.108425770"
    ],
    [
      "1.286947675",
      "0.115325557"
    ],
    [
      "0.604524685",
      "0.158223834"
    ],
    [
      "1.416404280",
      "0.168377902"
    ],
    [
      "0.971716599",
      "0.206049741"
    ],
    [
      "0.829288544",
      "0.212029007"
    ],
    [
      "1.118292632",
      "0.218321428"
    ],
    [
      "0.482730843",
      "0.227070212"
    ],
    [
      "1.536438191",
      "0.240248787"
    ],
    [
      "1.299509134",
      "0.254666052"
    ],
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      "0.623919215",
      "0.296778575"
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    [
      "0.372642137",
      "0.313406999"
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    [
      "1.422473112",
      "0.321400032"
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      "0.329311866"
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      "0.333417215"
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      "0.333417215"
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      "0.333417215"
    ],
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      "1.188245408",
      "0.339483191"
    ],
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      "1.530292411",
      "0.411394023"
    ],
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      "0.276749734",
      "0.415980031"
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      "0.553966438",
      "0.417940338"
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      "0.433551756"
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      "0.446118467"
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      "0.454578978"
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      "0.454578978"
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      "0.454578978"
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      "0.459165471"
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      "0.460644954"
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      "1.629328920",
      "0.522098872"
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      "0.323267389",
      "0.550047624"
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      "0.550609642"
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      "0.898847632",
      "0.575740741"
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      "0.575740741"
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      "0.639156758"
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      "0.678777686"
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      "0.679560227"
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      "0.696710443"
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      "0.696902505"
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      "0.696902505"
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      "0.696902505"
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      "0.696902505"
    ],
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      "0.776615935"
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    [
      "0.200398892",
      "0.800721859"
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    [
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      "0.809230582"
    ],
    [
      "0.339225419",
      "0.818064268"
    ],
    [
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      "0.818064268"
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    [
      "0.619036526",
      "0.818064268"
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  "radius": "0.069952776"
}

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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