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 = 145

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 = 145
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[0.0753200993320499990516714286545, 0.0753200993320499990516714286555]见来源署名packomania.com ↗
本站纪录0.075320099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 = 0.075320099

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 = 145 的当前纪录构型,0.075320099
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
    0.075320098→0.075320099
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": [
    [
      "0.925403940",
      "0.078415007"
    ],
    [
      "1.076119456",
      "0.078458495"
    ],
    [
      "0.775372723",
      "0.103018671"
    ],
    [
      "1.224739397",
      "0.103046756"
    ],
    [
      "0.556148822",
      "0.188810102"
    ],
    [
      "1.443929867",
      "0.188853163"
    ],
    [
      "1.000799358",
      "0.208916734"
    ],
    [
      "0.852046250",
      "0.232686090"
    ],
    [
      "0.700020332",
      "0.233458261"
    ],
    [
      "1.149419299",
      "0.233504995"
    ],
    [
      "1.300059497",
      "0.233504995"
    ],
    [
      "0.479926295",
      "0.318743145"
    ],
    [
      "1.520095239",
      "0.318819718"
    ],
    [
      "0.334496624",
      "0.358020113"
    ],
    [
      "1.665536351",
      "0.358054298"
    ],
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      "0.623398029",
      "0.364659710"
    ],
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      "0.774038227",
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      "0.924678426",
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    ],
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      "0.364659710"
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      "1.225958822",
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      "0.239089994",
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      "0.548077929",
      "0.495117949"
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      "0.698718127",
      "0.495117949"
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  "radius": "0.075320099"
}
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": 145
}

当前第一名的答案

{
  "centers": [
    [
      "0.925403940",
      "0.078415007"
    ],
    [
      "1.076119456",
      "0.078458495"
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      "0.103018671"
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      "1.224739397",
      "0.103046756"
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      "0.556148822",
      "0.188810102"
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    [
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      "0.208916734"
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      "0.232686090"
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      "0.233458261"
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    [
      "1.149419299",
      "0.233504995"
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  "radius": "0.075320099"
}

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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