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 = 123

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 = 123
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[0.0814571052924587183547572882375, 0.0814571052924587183547572882385]见来源署名packomania.com ↗
本站纪录0.081457105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 = 0.081457105

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 = 123 的当前纪录构型,0.081457105
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
    0.081457104→0.081457105
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": [
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      "0.960215190",
      "0.082319109"
    ],
    [
      "1.122961008",
      "0.089724416"
    ],
    [
      "1.281838826",
      "0.125764377"
    ],
    [
      "0.683224320",
      "0.137808480"
    ],
    [
      "1.431850805",
      "0.189305279"
    ],
    [
      "0.837665776",
      "0.189662861"
    ],
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      "0.535889592",
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      "0.995137628",
      "0.241446301"
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      "1.157883446",
      "0.248851608"
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      "1.568277998",
      "0.278348307"
    ],
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      "0.715537837",
      "0.297485894"
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      "1.312496822",
      "0.300191106"
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      "0.402891582",
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      "0.872208597",
      "0.348872891"
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      "0.568203109",
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  "radius": "0.081457105"
}
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": 123
}

当前第一名的答案

{
  "centers": [
    [
      "0.960215190",
      "0.082319109"
    ],
    [
      "1.122961008",
      "0.089724416"
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      "0.837665776",
      "0.189662861"
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      "1.648526045"
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    [
      "1.366018177",
      "1.698396508"
    ],
    [
      "0.404075827",
      "1.698996015"
    ],
    [
      "0.726430566",
      "1.705219703"
    ],
    [
      "1.210381731",
      "1.746545617"
    ],
    [
      "0.883785932",
      "1.747413644"
    ],
    [
      "1.517203597",
      "1.759092543"
    ],
    [
      "1.046120424",
      "1.761145098"
    ],
    [
      "0.536935440",
      "1.793279435"
    ],
    [
      "1.329898304",
      "1.857256180"
    ],
    [
      "0.684361733",
      "1.862608563"
    ],
    [
      "1.166258632",
      "1.903370973"
    ],
    [
      "0.841717100",
      "1.904802504"
    ],
    [
      "1.004051592",
      "1.918533958"
    ]
  ],
  "radius": "0.081457105"
}

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?