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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 = 81

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 = 81
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[0.0998914754916361976475653395475, 0.0998914754916361976475653395485]见来源署名packomania.com ↗
本站纪录0.099891475Ἀθηνᾶ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 = 0.099891475

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 = 81 的当前纪录构型，0.099891475
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
    0.099891474→0.099891475
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": [
    [
      "0.941015916",
      "0.101826167"
    ],
    [
      "1.140590927",
      "0.110938952"
    ],
    [
      "0.714956872",
      "0.146216789"
    ],
    [
      "1.358651791",
      "0.174430955"
    ],
    [
      "0.483489871",
      "0.262877046"
    ],
    [
      "1.531924253",
      "0.273878836"
    ],
    [
      "0.859439298",
      "0.284195251"
    ],
    [
      "1.203670276",
      "0.300502195"
    ],
    [
      "0.667705232",
      "0.340331472"
    ],
    [
      "1.026770413",
      "0.393344350"
    ],
    [
      "0.332807739",
      "0.395806452"
    ],
    [
      "1.377250408",
      "0.401822603"
    ],
    [
      "1.678992155",
      "0.409098139"
    ],
    [
      "0.517477634",
      "0.472031497"
    ],
    [
      "0.848578963",
      "0.483682796"
    ],
    [
      "1.202115509",
      "0.500279096"
    ],
    [
      "1.792610385",
      "0.573427459"
    ],
    [
      "1.572512858",
      "0.578140698"
    ],
    [
      "0.679100788",
      "0.589467367"
    ],
    [
      "1.025215645",
      "0.593121251"
    ],
    [
      "0.327794026",
      "0.595526482"
    ],
    [
      "1.374143488",
      "0.601864537"
    ],
    [
      "0.505087924",
      "0.687613946"
    ],
    [
      "0.851104650",
      "0.691093639"
    ],
    [
      "0.153683030",
      "0.693498870"
    ],
    [
      "1.199906588",
      "0.700049835"
    ],
    [
      "1.667457624",
      "0.753921013"
    ],
    [
      "1.867181689",
      "0.758771323"
    ],
    [
      "0.677091786",
      "0.789240217"
    ],
    [
      "1.023006724",
      "0.792891990"
    ],
    [
      "0.325585105",
      "0.795297221"
    ],
    [
      "1.371934567",
      "0.801635276"
    ],
    [
      "0.502879003",
      "0.887384685"
    ],
    [
      "0.106920364",
      "0.887731927"
    ],
    [
      "0.848993861",
      "0.891038568"
    ],
    [
      "1.191860321",
      "0.899670688"
    ],
    [
      "1.540788164",
      "0.908413975"
    ],
    [
      "1.899032407",
      "0.955999007"
    ],
    [
      "0.674882865",
      "0.989010956"
    ],
    [
      "0.278822439",
      "0.989530278"
    ],
    [
      "1.017847457",
      "0.997817267"
    ],
    [
      "1.363888300",
      "1.001256129"
    ],
    [
      "1.708689230",
      "1.016684282"
    ],
    [
      "0.493645055",
      "1.086954126"
    ],
    [
      "0.104181256",
      "1.087530315"
    ],
    [
      "0.843736462",
      "1.095789655"
    ],
    [
      "1.192893189",
      "1.105419243"
    ],
    [
      "1.531789366",
      "1.109526437"
    ],
    [
      "1.886593455",
      "1.155394343"
    ],
    [
      "0.319632191",
      "1.185100704"
    ],
    [
      "0.665648917",
      "1.188580397"
    ],
    [
      "1.015638536",
      "1.197588006"
    ],
    [
      "1.362378172",
      "1.215418243"
    ],
    [
      "1.697784502",
      "1.220696768"
    ],
    [
      "0.145619328",
      "1.283247283"
    ],
    [
      "0.491636053",
      "1.286726975"
    ],
    [
      "0.841527541",
      "1.295560394"
    ],
    [
      "1.183930557",
      "1.305249620"
    ],
    [
      "1.534302226",
      "1.335530444"
    ],
    [
      "1.814702288",
      "1.382695097"
    ],
    [
      "0.317623190",
      "1.384873554"
    ],
    [
      "0.663639915",
      "1.388353246"
    ],
    [
      "1.010387938",
      "1.404225308"
    ],
    [
      "1.350950619",
      "1.414874099"
    ],
    [
      "0.489627052",
      "1.486499825"
    ],
    [
      "0.832489184",
      "1.495138788"
    ],
    [
      "1.182806791",
      "1.522767012"
    ],
    [
      "1.511340044",
      "1.533989421"
    ],
    [
      "1.710218524",
      "1.552978304"
    ],
    [
      "0.315614189",
      "1.584646403"
    ],
    [
      "1.000171945",
      "1.603746888"
    ],
    [
      "0.635447960",
      "1.623062962"
    ],
    [
      "1.343196216",
      "1.641882334"
    ],
    [
      "0.821966908",
      "1.694644451"
    ],
    [
      "0.461435097",
      "1.721209540"
    ],
    [
      "1.160561369",
      "1.722862210"
    ],
    [
      "1.521690587",
      "1.733506176"
    ],
    [
      "0.642672480",
      "1.824031694"
    ],
    [
      "1.322009422",
      "1.840538689"
    ],
    [
      "0.864489620",
      "1.889849590"
    ],
    [
      "1.064113426",
      "1.897822268"
    ]
  ],
  "radius": "0.099891475"
}
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": 81
}

当前第一名的答案

{
  "centers": [
    [
      "0.941015916",
      "0.101826167"
    ],
    [
      "1.140590927",
      "0.110938952"
    ],
    [
      "0.714956872",
      "0.146216789"
    ],
    [
      "1.358651791",
      "0.174430955"
    ],
    [
      "0.483489871",
      "0.262877046"
    ],
    [
      "1.531924253",
      "0.273878836"
    ],
    [
      "0.859439298",
      "0.284195251"
    ],
    [
      "1.203670276",
      "0.300502195"
    ],
    [
      "0.667705232",
      "0.340331472"
    ],
    [
      "1.026770413",
      "0.393344350"
    ],
    [
      "0.332807739",
      "0.395806452"
    ],
    [
      "1.377250408",
      "0.401822603"
    ],
    [
      "1.678992155",
      "0.409098139"
    ],
    [
      "0.517477634",
      "0.472031497"
    ],
    [
      "0.848578963",
      "0.483682796"
    ],
    [
      "1.202115509",
      "0.500279096"
    ],
    [
      "1.792610385",
      "0.573427459"
    ],
    [
      "1.572512858",
      "0.578140698"
    ],
    [
      "0.679100788",
      "0.589467367"
    ],
    [
      "1.025215645",
      "0.593121251"
    ],
    [
      "0.327794026",
      "0.595526482"
    ],
    [
      "1.374143488",
      "0.601864537"
    ],
    [
      "0.505087924",
      "0.687613946"
    ],
    [
      "0.851104650",
      "0.691093639"
    ],
    [
      "0.153683030",
      "0.693498870"
    ],
    [
      "1.199906588",
      "0.700049835"
    ],
    [
      "1.667457624",
      "0.753921013"
    ],
    [
      "1.867181689",
      "0.758771323"
    ],
    [
      "0.677091786",
      "0.789240217"
    ],
    [
      "1.023006724",
      "0.792891990"
    ],
    [
      "0.325585105",
      "0.795297221"
    ],
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      "1.371934567",
      "0.801635276"
    ],
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      "0.887384685"
    ],
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      "0.106920364",
      "0.887731927"
    ],
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      "0.848993861",
      "0.891038568"
    ],
    [
      "1.191860321",
      "0.899670688"
    ],
    [
      "1.540788164",
      "0.908413975"
    ],
    [
      "1.899032407",
      "0.955999007"
    ],
    [
      "0.674882865",
      "0.989010956"
    ],
    [
      "0.278822439",
      "0.989530278"
    ],
    [
      "1.017847457",
      "0.997817267"
    ],
    [
      "1.363888300",
      "1.001256129"
    ],
    [
      "1.708689230",
      "1.016684282"
    ],
    [
      "0.493645055",
      "1.086954126"
    ],
    [
      "0.104181256",
      "1.087530315"
    ],
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      "0.843736462",
      "1.095789655"
    ],
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      "1.192893189",
      "1.105419243"
    ],
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      "1.531789366",
      "1.109526437"
    ],
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      "1.155394343"
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      "0.319632191",
      "1.185100704"
    ],
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      "0.665648917",
      "1.188580397"
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      "1.197588006"
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      "1.362378172",
      "1.215418243"
    ],
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      "1.697784502",
      "1.220696768"
    ],
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      "0.145619328",
      "1.283247283"
    ],
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      "0.491636053",
      "1.286726975"
    ],
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      "0.841527541",
      "1.295560394"
    ],
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      "1.183930557",
      "1.305249620"
    ],
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      "1.534302226",
      "1.335530444"
    ],
    [
      "1.814702288",
      "1.382695097"
    ],
    [
      "0.317623190",
      "1.384873554"
    ],
    [
      "0.663639915",
      "1.388353246"
    ],
    [
      "1.010387938",
      "1.404225308"
    ],
    [
      "1.350950619",
      "1.414874099"
    ],
    [
      "0.489627052",
      "1.486499825"
    ],
    [
      "0.832489184",
      "1.495138788"
    ],
    [
      "1.182806791",
      "1.522767012"
    ],
    [
      "1.511340044",
      "1.533989421"
    ],
    [
      "1.710218524",
      "1.552978304"
    ],
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      "0.315614189",
      "1.584646403"
    ],
    [
      "1.000171945",
      "1.603746888"
    ],
    [
      "0.635447960",
      "1.623062962"
    ],
    [
      "1.343196216",
      "1.641882334"
    ],
    [
      "0.821966908",
      "1.694644451"
    ],
    [
      "0.461435097",
      "1.721209540"
    ],
    [
      "1.160561369",
      "1.722862210"
    ],
    [
      "1.521690587",
      "1.733506176"
    ],
    [
      "0.642672480",
      "1.824031694"
    ],
    [
      "1.322009422",
      "1.840538689"
    ],
    [
      "0.864489620",
      "1.889849590"
    ],
    [
      "1.064113426",
      "1.897822268"
    ]
  ],
  "radius": "0.099891475"
}

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。

还没有帖子。第一个聊聊这道题的思路？