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 = 121

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 = 121
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[0.0824755662624009582930285215995, 0.0824755662624009582930285216005]见来源署名packomania.com ↗
本站纪录0.082475566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 = 0.082475566

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 = 121 的当前纪录构型,0.082475566
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
    0.082475565→0.082475566
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": [
    [
      "0.213796856",
      "0.526992915"
    ],
    [
      "0.311194255",
      "0.393866573"
    ],
    [
      "0.106165907",
      "0.792848604"
    ],
    [
      "0.436553465",
      "0.276707318"
    ],
    [
      "0.235529335",
      "0.690506142"
    ],
    [
      "0.083519800",
      "0.956237802"
    ],
    [
      "0.579659075",
      "0.184423766"
    ],
    [
      "0.366269704",
      "0.589928649"
    ],
    [
      "0.463667103",
      "0.456802307"
    ],
    [
      "0.091896170",
      "1.126357633"
    ],
    [
      "0.253291808",
      "0.867433089"
    ],
    [
      "0.588629967",
      "0.349130776"
    ],
    [
      "0.732481325",
      "0.122341271"
    ],
    [
      "0.388002183",
      "0.753441876"
    ],
    [
      "0.230645701",
      "1.030822287"
    ],
    [
      "0.523320613",
      "0.640365622"
    ],
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      "0.741452217",
      "0.287048281"
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      "0.130670181",
      "1.293456560"
    ],
    [
      "0.624544062",
      "0.510124732"
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    [
      "0.410673711",
      "0.916827549"
    ],
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      "0.261410550",
      "1.192879068"
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      "0.885794211",
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      "1.444368353"
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      "0.671664658",
      "0.712496894"
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      "0.907526690",
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  "radius": "0.082475566"
}
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": 121
}

当前第一名的答案

{
  "centers": [
    [
      "0.213796856",
      "0.526992915"
    ],
    [
      "0.311194255",
      "0.393866573"
    ],
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      "0.106165907",
      "0.792848604"
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      "0.436553465",
      "0.276707318"
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      "0.690506142"
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      "0.083519800",
      "0.956237802"
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      "0.579659075",
      "0.184423766"
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    [
      "0.366269704",
      "0.589928649"
    ],
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      "0.456802307"
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      "1.126357633"
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    [
      "1.536332897",
      "1.543197693"
    ],
    [
      "1.633730296",
      "1.410071351"
    ],
    [
      "1.420340925",
      "1.815576234"
    ],
    [
      "1.916480200",
      "1.043762198"
    ],
    [
      "1.764470665",
      "1.309493858"
    ],
    [
      "1.563447167",
      "1.723289327"
    ],
    [
      "1.893834093",
      "1.207151396"
    ],
    [
      "1.688805745",
      "1.606133427"
    ],
    [
      "1.786203144",
      "1.473007085"
    ]
  ],
  "radius": "0.082475566"
}

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

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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?