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 = 166

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 = 166
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[0.0704936324071078040818401487435, 0.0704936324071078040818401487445]见来源署名packomania.com ↗
本站纪录0.070493632Ἀθηνᾶ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 = 0.070493632

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 = 166 的当前纪录构型,0.070493632
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
    0.070493631→0.070493632
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": [
    [
      "0.938487827",
      "0.072531219"
    ],
    [
      "1.161366641",
      "0.084607792"
    ],
    [
      "1.297956938",
      "0.119543442"
    ],
    [
      "0.680293673",
      "0.127205665"
    ],
    [
      "0.819674129",
      "0.148430586"
    ],
    [
      "1.045265958",
      "0.164595542"
    ],
    [
      "1.428798869",
      "0.177085993"
    ],
    [
      "0.546955715",
      "0.188536205"
    ],
    [
      "1.171136863",
      "0.228108722"
    ],
    [
      "0.925179371",
      "0.241950919"
    ],
    [
      "1.553577922",
      "0.253318293"
    ],
    [
      "1.308437135",
      "0.260140648"
    ],
    [
      "0.663975743",
      "0.267245423"
    ],
    [
      "0.408037573",
      "0.283367199"
    ],
    [
      "1.051050276",
      "0.305464100"
    ],
    [
      "0.801156713",
      "0.311410251"
    ],
    [
      "1.432943394",
      "0.326288969"
    ],
    [
      "0.539691902",
      "0.333810697"
    ],
    [
      "1.660140205",
      "0.345632369"
    ],
    [
      "1.176921181",
      "0.368977280"
    ],
    [
      "0.306461524",
      "0.381140994"
    ],
    [
      "0.923682063",
      "0.382930233"
    ],
    [
      "1.313992859",
      "0.401973704"
    ],
    [
      "0.674841733",
      "0.407813340"
    ],
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      "1.539505677",
      "0.418603045"
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      "0.438115853",
      "0.431584492"
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      "0.446443413"
    ],
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      "1.751514816",
      "0.453001309"
    ],
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      "0.805742583",
      "0.460181044"
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      "0.564039546",
      "0.494992948"
    ],
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      "1.419072282",
      "0.495972235"
    ],
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      "1.175423873",
      "0.509956594"
    ],
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      "0.280367202",
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      "0.523694225"
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      "0.635756940"
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      "0.636982954"
    ],
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      "0.326488748",
      "0.652922332"
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      "0.946682206",
      "0.674387013"
    ],
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      "0.702871820",
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      "0.454031491",
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      "1.309226203",
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      "1.850471843"
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      "1.898863451"
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      "1.911347437"
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      "0.901970186",
      "1.924322586"
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    [
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  ],
  "radius": "0.070493632"
}
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": 166
}

当前第一名的答案

{
  "centers": [
    [
      "0.938487827",
      "0.072531219"
    ],
    [
      "1.161366641",
      "0.084607792"
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    [
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      "0.119543442"
    ],
    [
      "0.680293673",
      "0.127205665"
    ],
    [
      "0.819674129",
      "0.148430586"
    ],
    [
      "1.045265958",
      "0.164595542"
    ],
    [
      "1.428798869",
      "0.177085993"
    ],
    [
      "0.546955715",
      "0.188536205"
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      "1.171136863",
      "0.228108722"
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      "0.241950919"
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      "0.253318293"
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      "0.260140648"
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      "0.267245423"
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    [
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  "radius": "0.070493632"
}

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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