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 = 101

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 = 101
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[0.0897107705211865789013955836435, 0.0897107705211865789013955836445]见来源署名packomania.com ↗
本站纪录0.089710770Ἀθηνᾶ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 = 0.089710770

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 = 101 的当前纪录构型,0.089710770
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
    0.089710769→0.089710770
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": [
    [
      "0.945563224",
      "0.091339932"
    ],
    [
      "0.768392107",
      "0.119668094"
    ],
    [
      "1.244443225",
      "0.123145399"
    ],
    [
      "0.600218943",
      "0.182197097"
    ],
    [
      "1.411685010",
      "0.188124434"
    ],
    [
      "1.084631503",
      "0.204707049"
    ],
    [
      "0.916918560",
      "0.268460153"
    ],
    [
      "0.447577249",
      "0.276497695"
    ],
    [
      "1.566511426",
      "0.287513917"
    ],
    [
      "0.739747443",
      "0.296788315"
    ],
    [
      "1.223852862",
      "0.317886121"
    ],
    [
      "1.396450436",
      "0.366898026"
    ],
    [
      "1.056226231",
      "0.381865819"
    ],
    [
      "0.586479697",
      "0.390067934"
    ],
    [
      "1.706634833",
      "0.426158825"
    ],
    [
      "0.888513288",
      "0.445618924"
    ],
    [
      "0.418673761",
      "0.453575864"
    ],
    [
      "1.535552678",
      "0.480223468"
    ],
    [
      "0.243721804",
      "0.493374131"
    ],
    [
      "1.195128679",
      "0.495436058"
    ],
    [
      "0.720707352",
      "0.509126853"
    ],
    [
      "1.367845877",
      "0.544024728"
    ],
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      "1.027415737",
      "0.559189163"
    ],
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      "0.552901415",
      "0.572634782"
    ],
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      "1.805464184",
      "0.575908112"
    ],
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      "0.859609800",
      "0.622697092"
    ],
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      "0.382679540",
      "0.629349882"
    ],
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      "1.623619363",
      "0.636544767"
    ],
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      "0.159040677",
      "0.651555028"
    ],
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      "1.166495140",
      "0.672558110"
    ],
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      "0.691803864",
      "0.686205021"
    ],
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      "1.455922585",
      "0.700340380"
    ],
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      "0.998512249",
      "0.736267331"
    ],
    [
      "0.516891453",
      "0.748432569"
    ],
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      "1.762521811",
      "0.750115006"
    ],
    [
      "0.284680446",
      "0.779643817"
    ],
    [
      "1.301161403",
      "0.791120696"
    ],
    [
      "0.830706312",
      "0.799775261"
    ],
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      "1.594825034",
      "0.813910619"
    ],
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      "0.107030770",
      "0.823272991"
    ],
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      "1.133355467",
      "0.854628625"
    ],
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      "0.656721585",
      "0.862163321"
    ],
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      "1.900240654",
      "0.865117657"
    ],
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      "1.440063851",
      "0.904690935"
    ],
    [
      "0.965549530",
      "0.918136555"
    ],
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      "0.488915649",
      "0.925671250"
    ],
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      "1.732543876",
      "0.928913270"
    ],
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      "0.312239139",
      "0.956936253"
    ],
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      "1.272257915",
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      "0.795624034",
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      "0.138059999",
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      "1.581731109",
      "1.026112029"
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      "1.031706794"
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      "1.039241489"
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      "1.909209917",
      "1.044314872"
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      "1.081769104"
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      "0.936646042",
      "1.095214723"
    ],
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      "0.460012161",
      "1.102749419"
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      "1.741513140",
      "1.108110485"
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      "0.283335651",
      "1.134014421"
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      "1.145277033"
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      "1.176708488"
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      "0.231615714",
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      "1.215527769",
      "1.325276824"
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      "0.737817058",
      "1.329889897"
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      "1.506968066"
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      "1.564351145"
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      "1.645642044"
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      "0.999462611",
      "1.741429313"
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      "0.538780541",
      "1.784744252"
    ],
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      "1.410606497",
      "1.812421557"
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      "0.712158468",
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      "1.877178442"
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      "0.887136506",
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    ],
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    ]
  ],
  "radius": "0.089710770"
}
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": 101
}

当前第一名的答案

{
  "centers": [
    [
      "0.945563224",
      "0.091339932"
    ],
    [
      "0.768392107",
      "0.119668094"
    ],
    [
      "1.244443225",
      "0.123145399"
    ],
    [
      "0.600218943",
      "0.182197097"
    ],
    [
      "1.411685010",
      "0.188124434"
    ],
    [
      "1.084631503",
      "0.204707049"
    ],
    [
      "0.916918560",
      "0.268460153"
    ],
    [
      "0.447577249",
      "0.276497695"
    ],
    [
      "1.566511426",
      "0.287513917"
    ],
    [
      "0.739747443",
      "0.296788315"
    ],
    [
      "1.223852862",
      "0.317886121"
    ],
    [
      "1.396450436",
      "0.366898026"
    ],
    [
      "1.056226231",
      "0.381865819"
    ],
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      "0.586479697",
      "0.390067934"
    ],
    [
      "1.706634833",
      "0.426158825"
    ],
    [
      "0.888513288",
      "0.445618924"
    ],
    [
      "0.418673761",
      "0.453575864"
    ],
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      "1.535552678",
      "0.480223468"
    ],
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      "0.243721804",
      "0.493374131"
    ],
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      "1.195128679",
      "0.495436058"
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      "0.720707352",
      "0.509126853"
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      "1.367845877",
      "0.544024728"
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      "0.575908112"
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      "0.622697092"
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      "0.629349882"
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      "1.623619363",
      "0.636544767"
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      "0.651555028"
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      "0.672558110"
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      "0.736267331"
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      "1.108110485"
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      "1.134014421"
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      "1.145277033"
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      "1.272292892"
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      "1.285149853"
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      "0.414363482",
      "1.289599342"
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      "0.231615714",
      "1.305819943"
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      "1.215527769",
      "1.325276824"
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      "1.329889897"
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      "0.570011121",
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      "1.356361035",
      "1.436443728"
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      "1.794202484",
      "1.444824567"
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      "0.878839066",
      "1.449371060"
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      "1.532763142"
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      "1.549322629"
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      "1.564351145"
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      "1.591962203"
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      "1.603283472"
    ],
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      "0.860560163",
      "1.627859074"
    ],
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      "0.356316878",
      "1.643660252"
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      "1.344449990",
      "1.645642044"
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      "0.690634666",
      "1.685456079"
    ],
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      "1.176269463",
      "1.710664116"
    ],
    [
      "1.561982386",
      "1.716102143"
    ],
    [
      "0.999462611",
      "1.741429313"
    ],
    [
      "0.538780541",
      "1.784744252"
    ],
    [
      "1.410606497",
      "1.812421557"
    ],
    [
      "0.712158468",
      "1.863581921"
    ],
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      "1.243278570",
      "1.877178442"
    ],
    [
      "0.887136506",
      "1.903265361"
    ],
    [
      "1.066499285",
      "1.907856997"
    ]
  ],
  "radius": "0.089710770"
}

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

讨论区

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。

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