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 = 188

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 = 188
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[0.0665391302775, 0.0665391302785]见来源署名packomania.com ↗
本站纪录0.066539130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 = 0.066539130

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 = 188 的当前纪录构型,0.066539130
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
    0.066539129→0.066539130
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": [
    [
      "1.017435686",
      "0.066701981"
    ],
    [
      "0.847225422",
      "0.079125892"
    ],
    [
      "1.275938136",
      "0.108256012"
    ],
    [
      "1.142890220",
      "0.111097739"
    ],
    [
      "0.712949388",
      "0.115193909"
    ],
    [
      "1.399941492",
      "0.156557058"
    ],
    [
      "0.575340605",
      "0.168727726"
    ],
    [
      "0.939764623",
      "0.174762285"
    ],
    [
      "0.810671476",
      "0.207085387"
    ],
    [
      "1.197464266",
      "0.232471117"
    ],
    [
      "0.461447780",
      "0.237562657"
    ],
    [
      "1.548585949",
      "0.244749941"
    ],
    [
      "0.683493041",
      "0.246270449"
    ],
    [
      "1.064238277",
      "0.249239016"
    ],
    [
      "1.422625001",
      "0.287687839"
    ],
    [
      "0.940928666",
      "0.307835454"
    ],
    [
      "0.564942849",
      "0.321220257"
    ],
    [
      "0.358500813",
      "0.321893822"
    ],
    [
      "1.294475072",
      "0.323568465"
    ],
    [
      "1.651518147",
      "0.336144643"
    ],
    [
      "0.811835520",
      "0.340158557"
    ],
    [
      "1.165521752",
      "0.361658962"
    ],
    [
      "1.522790785",
      "0.375304271"
    ],
    [
      "0.684657085",
      "0.379343618"
    ],
    [
      "1.037810269",
      "0.399070197"
    ],
    [
      "1.389623570",
      "0.416609245"
    ],
    [
      "0.474779048",
      "0.419099327"
    ],
    [
      "1.742662743",
      "0.434490278"
    ],
    [
      "0.909000900",
      "0.437026944"
    ],
    [
      "1.261776574",
      "0.453554733"
    ],
    [
      "0.346475769",
      "0.454427673"
    ],
    [
      "1.614815706",
      "0.471435625"
    ],
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      "0.780508281",
      "0.471660312"
    ],
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      "0.218220782",
      "0.489930937"
    ],
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      "1.134065091",
      "0.490965968"
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    ],
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      "1.582887940",
      "0.600627115"
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      "0.747998910",
      "0.601330566"
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      "0.613703318"
    ],
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      "1.102137325",
      "0.620157458"
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      "1.454815204",
      "0.636782303"
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      "0.619077687",
      "0.637289606"
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      "0.137370315",
      "0.643321963"
    ],
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      "0.973190671",
      "0.657261949"
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      "0.490473555",
      "0.671506571"
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      "0.675107741"
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      "0.362170275",
      "0.706834917"
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      "1.198056554",
      "0.712403460"
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      "0.227890991",
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      "1.874943396"
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  ],
  "radius": "0.066539130"
}
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": 188
}

当前第一名的答案

{
  "centers": [
    [
      "1.017435686",
      "0.066701981"
    ],
    [
      "0.847225422",
      "0.079125892"
    ],
    [
      "1.275938136",
      "0.108256012"
    ],
    [
      "1.142890220",
      "0.111097739"
    ],
    [
      "0.712949388",
      "0.115193909"
    ],
    [
      "1.399941492",
      "0.156557058"
    ],
    [
      "0.575340605",
      "0.168727726"
    ],
    [
      "0.939764623",
      "0.174762285"
    ],
    [
      "0.810671476",
      "0.207085387"
    ],
    [
      "1.197464266",
      "0.232471117"
    ],
    [
      "0.461447780",
      "0.237562657"
    ],
    [
      "1.548585949",
      "0.244749941"
    ],
    [
      "0.683493041",
      "0.246270449"
    ],
    [
      "1.064238277",
      "0.249239016"
    ],
    [
      "1.422625001",
      "0.287687839"
    ],
    [
      "0.940928666",
      "0.307835454"
    ],
    [
      "0.564942849",
      "0.321220257"
    ],
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      "0.358500813",
      "0.321893822"
    ],
    [
      "1.294475072",
      "0.323568465"
    ],
    [
      "1.651518147",
      "0.336144643"
    ],
    [
      "0.811835520",
      "0.340158557"
    ],
    [
      "1.165521752",
      "0.361658962"
    ],
    [
      "1.522790785",
      "0.375304271"
    ],
    [
      "0.684657085",
      "0.379343618"
    ],
    [
      "1.037810269",
      "0.399070197"
    ],
    [
      "1.389623570",
      "0.416609245"
    ],
    [
      "0.474779048",
      "0.419099327"
    ],
    [
      "1.742662743",
      "0.434490278"
    ],
    [
      "0.909000900",
      "0.437026944"
    ],
    [
      "1.261776574",
      "0.453554733"
    ],
    [
      "0.346475769",
      "0.454427673"
    ],
    [
      "1.614815706",
      "0.471435625"
    ],
    [
      "0.780508281",
      "0.471660312"
    ],
    [
      "0.218220782",
      "0.489930937"
    ],
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      "1.134065091",
      "0.490965968"
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      "1.486742970",
      "0.507590813"
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      "0.508511534"
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      "0.528261687"
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      "0.542728499"
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      "0.545800735"
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      "1.710734977",
      "0.563681769"
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      "0.876309067",
      "0.566027206"
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      "0.395726669",
      "0.578056845"
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      "0.583211970"
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    [
      "1.840388303",
      "0.593679075"
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  "radius": "0.066539130"
}

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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