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 = 231

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 = 231
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[0.0601361383685, 0.0601361383695]见来源署名packomania.com ↗
本站纪录0.060136138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 = 0.060136138

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 = 231 的当前纪录构型,0.060136138
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
    0.060136137→0.060136138
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

{
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      "0.062437424",
      "0.934269532"
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    [
      "0.078508214",
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      "0.061719923",
      "1.054539668"
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    [
      "0.109669122",
      "0.698910302"
    ],
    [
      "0.101545850",
      "1.168026743"
    ],
    [
      "0.161222613",
      "0.581662992"
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    [
      "0.105425332",
      "1.288236435"
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    [
      "0.219826874",
      "0.475906524"
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      "0.180389124",
      "0.957781595"
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    [
      "0.196459914",
      "0.838587844"
    ],
    [
      "0.149559390",
      "1.400118541"
    ],
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      "0.181355650",
      "1.078049988"
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      "0.229007107",
      "0.683948155"
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    [
      "0.293144478",
      "0.380565298"
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      "0.223031026",
      "1.190871012"
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      "0.284558311",
      "0.577273489"
    ],
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      "0.230245445",
      "1.310926719"
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      "0.228705890",
      "1.490679305"
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      "0.779943627"
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  "radius": "0.060136138"
}
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": 231
}

当前第一名的答案

{
  "centers": [
    [
      "0.062437424",
      "0.934269532"
    ],
    [
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  "radius": "0.060136138"
}

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?