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 = 150

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 = 150
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[0.0742897544501232235132018213205, 0.0742897544501232235132018213215]见来源署名packomania.com ↗
本站纪录0.074289754Ἀθηνᾶ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 = 0.074289754

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 = 150 的当前纪录构型,0.074289754
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Ἀθηνᾶ
解题方式AI · Anthropic: Claude Opus 5
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. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.074289753→0.074289754
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": [
    [
      "1.001031578",
      "0.074290329"
    ],
    [
      "1.149118485",
      "0.086379107"
    ],
    [
      "0.748320637",
      "0.109159410"
    ],
    [
      "1.297604938",
      "0.123452743"
    ],
    [
      "0.885091299",
      "0.167208154"
    ],
    [
      "1.437723006",
      "0.184317446"
    ],
    [
      "0.512303875",
      "0.213176037"
    ],
    [
      "1.044390214",
      "0.216402606"
    ],
    [
      "1.192477121",
      "0.228491383"
    ],
    [
      "0.660071075",
      "0.228691352"
    ],
    [
      "1.329448566",
      "0.286064761"
    ],
    [
      "0.796841737",
      "0.286740096"
    ],
    [
      "0.392705233",
      "0.301335184"
    ],
    [
      "1.626895085",
      "0.318867112"
    ],
    [
      "0.940920660",
      "0.323032441"
    ],
    [
      "1.478548202",
      "0.327178126"
    ],
    [
      "0.549293259",
      "0.357077587"
    ],
    [
      "1.084999584",
      "0.359324786"
    ],
    [
      "0.693372182",
      "0.393369932"
    ],
    [
      "1.229078507",
      "0.395617131"
    ],
    [
      "0.283849365",
      "0.413544383"
    ],
    [
      "0.837451106",
      "0.429662276"
    ],
    [
      "1.373157430",
      "0.431909476"
    ],
    [
      "0.429694617",
      "0.445236734"
    ],
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      "1.667504454",
      "0.461789292"
    ],
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      "0.981530029",
      "0.465954621"
    ],
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      "1.519157571",
      "0.470100306"
    ],
    [
      "0.589902628",
      "0.499999767"
    ],
    [
      "1.125608953",
      "0.502246966"
    ],
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      "1.798566632",
      "0.531779119"
    ],
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      "0.733981551",
      "0.536292112"
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      "1.269687876",
      "0.538539311"
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      "0.538623349"
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      "0.717753836"
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      "0.753450460"
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      "1.602897812",
      "0.755204055"
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      "1.768115107",
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      "0.788091327"
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      "0.815200290",
      "0.822136472"
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      "0.824546431"
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      "1.073935687"
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      "0.896419028",
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  ],
  "radius": "0.074289754"
}
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": 150
}

当前第一名的答案

{
  "centers": [
    [
      "1.001031578",
      "0.074290329"
    ],
    [
      "1.149118485",
      "0.086379107"
    ],
    [
      "0.748320637",
      "0.109159410"
    ],
    [
      "1.297604938",
      "0.123452743"
    ],
    [
      "0.885091299",
      "0.167208154"
    ],
    [
      "1.437723006",
      "0.184317446"
    ],
    [
      "0.512303875",
      "0.213176037"
    ],
    [
      "1.044390214",
      "0.216402606"
    ],
    [
      "1.192477121",
      "0.228491383"
    ],
    [
      "0.660071075",
      "0.228691352"
    ],
    [
      "1.329448566",
      "0.286064761"
    ],
    [
      "0.796841737",
      "0.286740096"
    ],
    [
      "0.392705233",
      "0.301335184"
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      "0.318867112"
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      "0.940920660",
      "0.323032441"
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      "0.327178126"
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      "0.357077587"
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      "0.395617131"
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      "0.413544383"
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  "radius": "0.074289754"
}

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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