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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 = 90

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 = 90
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[0.0948220595869476841614774768655, 0.0948220595869476841614774768665]见来源署名packomania.com ↗
本站纪录0.094822059Ἀθηνᾶ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 = 0.094822059

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 = 90 的当前纪录构型，0.094822059
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
    0.094822058→0.094822059
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": [
    [
      "1.054032600",
      "0.096436177"
    ],
    [
      "0.864582339",
      "0.105008849"
    ],
    [
      "1.241111122",
      "0.127525054"
    ],
    [
      "0.681076179",
      "0.152866775"
    ],
    [
      "1.417606175",
      "0.196910848"
    ],
    [
      "0.510991910",
      "0.238279453"
    ],
    [
      "1.042606457",
      "0.285735768"
    ],
    [
      "0.853156197",
      "0.294308440"
    ],
    [
      "1.575770583",
      "0.301547897"
    ],
    [
      "1.229684980",
      "0.316824646"
    ],
    [
      "0.669650036",
      "0.342166367"
    ],
    [
      "0.363014044",
      "0.356885706"
    ],
    [
      "1.406133019",
      "0.386329906"
    ],
    [
      "1.708661786",
      "0.436843204"
    ],
    [
      "0.521672171",
      "0.460772619"
    ],
    [
      "1.071333552",
      "0.473191486"
    ],
    [
      "0.881883291",
      "0.481764157"
    ],
    [
      "0.242996405",
      "0.503721186"
    ],
    [
      "1.258276229",
      "0.505087059"
    ],
    [
      "1.539024223",
      "0.521625213"
    ],
    [
      "0.698377131",
      "0.529622084"
    ],
    [
      "1.810446582",
      "0.596858038"
    ],
    [
      "0.401654532",
      "0.607608100"
    ],
    [
      "1.391167432",
      "0.640382366"
    ],
    [
      "1.020609981",
      "0.655926297"
    ],
    [
      "0.831183106",
      "0.665001052"
    ],
    [
      "0.156207116",
      "0.672340614"
    ],
    [
      "0.578359492",
      "0.676457564"
    ],
    [
      "1.640809019",
      "0.681640047"
    ],
    [
      "1.207552658",
      "0.687821870"
    ],
    [
      "1.876657171",
      "0.774568617"
    ],
    [
      "0.314865243",
      "0.776227527"
    ],
    [
      "1.492952228",
      "0.800397200"
    ],
    [
      "0.709395265",
      "0.813550665"
    ],
    [
      "1.086489122",
      "0.833844798"
    ],
    [
      "0.896746506",
      "0.842951417"
    ],
    [
      "0.491570204",
      "0.845076992"
    ],
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      "1.309337454",
      "0.847836705"
    ],
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      "0.106455757",
      "0.855342510"
    ],
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      "1.707019608",
      "0.859350626"
    ],
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      "0.265113883",
      "0.959229424"
    ],
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      "1.904387267",
      "0.962174410"
    ],
    [
      "1.559162817",
      "0.978107779"
    ],
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      "0.622605976",
      "0.982170093"
    ],
    [
      "1.188850092",
      "0.994286995"
    ],
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      "0.999380570",
      "1.002422859"
    ],
    [
      "0.809957218",
      "1.011570844"
    ],
    [
      "1.375761434",
      "1.026365678"
    ],
    [
      "0.439869307",
      "1.032886969"
    ],
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      "0.095937760",
      "1.044928497"
    ],
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      "1.735377825",
      "1.048201735"
    ],
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      "1.891711814",
      "1.155554313"
    ],
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      "1.160757486"
    ],
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      "1.162569823"
    ],
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      "1.099759469",
      "1.163323352"
    ],
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      "1.505605305",
      "1.164588184"
    ],
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      "0.910336117",
      "1.172471337"
    ],
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      "1.286920255",
      "1.193913094"
    ],
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      "0.507473792",
      "1.210071973"
    ],
    [
      "1.683396247",
      "1.230582676"
    ],
    [
      "0.125140667",
      "1.232310679"
    ],
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      "0.794286087",
      "1.322462278"
    ],
    [
      "1.416764125",
      "1.332135601"
    ],
    [
      "1.839730236",
      "1.337935253"
    ],
    [
      "0.369100206",
      "1.339754827"
    ],
    [
      "1.161698466",
      "1.342567466"
    ],
    [
      "0.610512384",
      "1.369282338"
    ],
    [
      "0.975401712",
      "1.381923353"
    ],
    [
      "1.594555067",
      "1.398130092"
    ],
    [
      "0.192745152",
      "1.409495683"
    ],
    [
      "1.294962991",
      "1.477495071"
    ],
    [
      "0.472138798",
      "1.498965192"
    ],
    [
      "1.750889056",
      "1.505482670"
    ],
    [
      "0.746794160",
      "1.506063500"
    ],
    [
      "1.111479995",
      "1.525441732"
    ],
    [
      "1.472753933",
      "1.543489562"
    ],
    [
      "0.926419979",
      "1.566886662"
    ],
    [
      "0.295783744",
      "1.568706048"
    ],
    [
      "0.606088663",
      "1.633212448"
    ],
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      "1.629087921",
      "1.650842140"
    ],
    [
      "1.319883968",
      "1.665494637"
    ],
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      "0.785714482",
      "1.694035610"
    ],
    [
      "0.429733609",
      "1.702953304"
    ],
    [
      "1.136400973",
      "1.713441298"
    ],
    [
      "1.479673240",
      "1.767633172"
    ],
    [
      "0.957102113",
      "1.775221642"
    ],
    [
      "0.621473312",
      "1.822231507"
    ],
    [
      "1.290209298",
      "1.857268142"
    ],
    [
      "0.801099131",
      "1.883054669"
    ],
    [
      "1.100254890",
      "1.899608838"
    ]
  ],
  "radius": "0.094822059"
}
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": 90
}

当前第一名的答案

{
  "centers": [
    [
      "1.054032600",
      "0.096436177"
    ],
    [
      "0.864582339",
      "0.105008849"
    ],
    [
      "1.241111122",
      "0.127525054"
    ],
    [
      "0.681076179",
      "0.152866775"
    ],
    [
      "1.417606175",
      "0.196910848"
    ],
    [
      "0.510991910",
      "0.238279453"
    ],
    [
      "1.042606457",
      "0.285735768"
    ],
    [
      "0.853156197",
      "0.294308440"
    ],
    [
      "1.575770583",
      "0.301547897"
    ],
    [
      "1.229684980",
      "0.316824646"
    ],
    [
      "0.669650036",
      "0.342166367"
    ],
    [
      "0.363014044",
      "0.356885706"
    ],
    [
      "1.406133019",
      "0.386329906"
    ],
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      "1.708661786",
      "0.436843204"
    ],
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      "0.521672171",
      "0.460772619"
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      "1.071333552",
      "0.473191486"
    ],
    [
      "0.881883291",
      "0.481764157"
    ],
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      "0.242996405",
      "0.503721186"
    ],
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      "1.258276229",
      "0.505087059"
    ],
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      "1.539024223",
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    ],
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      "0.698377131",
      "0.529622084"
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    ],
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      "0.672340614"
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      "0.578359492",
      "0.676457564"
    ],
    [
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      "0.681640047"
    ],
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      "1.207552658",
      "0.687821870"
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      "0.774568617"
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      "0.709395265",
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      "1.409495683"
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      "1.477495071"
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      "0.472138798",
      "1.498965192"
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      "1.750889056",
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      "1.633212448"
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      "1.650842140"
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      "1.319883968",
      "1.665494637"
    ],
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      "0.785714482",
      "1.694035610"
    ],
    [
      "0.429733609",
      "1.702953304"
    ],
    [
      "1.136400973",
      "1.713441298"
    ],
    [
      "1.479673240",
      "1.767633172"
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      "0.957102113",
      "1.775221642"
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      "0.621473312",
      "1.822231507"
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      "1.290209298",
      "1.857268142"
    ],
    [
      "0.801099131",
      "1.883054669"
    ],
    [
      "1.100254890",
      "1.899608838"
    ]
  ],
  "radius": "0.094822059"
}

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？