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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 = 69

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 = 69
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[0.1078776433648491252221960975855, 0.1078776433648491252221960975865]见来源署名packomania.com ↗
本站纪录0.107877643Ἀθηνᾶ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 = 0.107877643

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 = 69 的当前纪录构型，0.107877643
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
    0.107877642→0.107877643
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": [
    [
      "0.809343137",
      "0.128488520"
    ],
    [
      "1.190656863",
      "0.128488520"
    ],
    [
      "0.605673733",
      "0.199774018"
    ],
    [
      "1.395181954",
      "0.200386162"
    ],
    [
      "1.000000000",
      "0.229485076"
    ],
    [
      "1.577307337",
      "0.319854032"
    ],
    [
      "0.418364447",
      "0.323551641"
    ],
    [
      "0.817205985",
      "0.344100485"
    ],
    [
      "1.182794015",
      "0.344100485"
    ],
    [
      "0.613581107",
      "0.415425464"
    ],
    [
      "1.385360755",
      "0.418377396"
    ],
    [
      "1.000000000",
      "0.458715894"
    ],
    [
      "1.723707112",
      "0.478338888"
    ],
    [
      "0.378321251",
      "0.535558448"
    ],
    [
      "0.777162790",
      "0.556107292"
    ],
    [
      "1.190656863",
      "0.559712450"
    ],
    [
      "1.531760530",
      "0.576862252"
    ],
    [
      "0.573537912",
      "0.627432271"
    ],
    [
      "0.186374669",
      "0.634081812"
    ],
    [
      "0.959956805",
      "0.670722701"
    ],
    [
      "1.713057557",
      "0.693831187"
    ],
    [
      "1.354587135",
      "0.699987928"
    ],
    [
      "0.367671697",
      "0.751050747"
    ],
    [
      "0.766513235",
      "0.771599590"
    ],
    [
      "1.150613668",
      "0.771719257"
    ],
    [
      "1.535884162",
      "0.816956862"
    ],
    [
      "1.876639240",
      "0.834513015"
    ],
    [
      "0.122322645",
      "0.840110165"
    ],
    [
      "0.562888357",
      "0.842924569"
    ],
    [
      "0.949307250",
      "0.886215000"
    ],
    [
      "1.314543939",
      "0.911994735"
    ],
    [
      "0.303619672",
      "0.957079099"
    ],
    [
      "1.699465845",
      "0.957638690"
    ],
    [
      "0.726996243",
      "0.983705109"
    ],
    [
      "1.123887076",
      "1.012991291"
    ],
    [
      "1.495840967",
      "1.028963669"
    ],
    [
      "0.523371365",
      "1.055030088"
    ],
    [
      "0.109923126",
      "1.057569308"
    ],
    [
      "1.890179163",
      "1.058616426"
    ],
    [
      "0.926805674",
      "1.100793710"
    ],
    [
      "1.305184103",
      "1.129960225"
    ],
    [
      "0.340577350",
      "1.169645497"
    ],
    [
      "1.659422650",
      "1.169645497"
    ],
    [
      "0.734859092",
      "1.199317074"
    ],
    [
      "1.122390089",
      "1.244575634"
    ],
    [
      "0.149920487",
      "1.270642053"
    ],
    [
      "0.531234214",
      "1.270642053"
    ],
    [
      "1.468765786",
      "1.270642053"
    ],
    [
      "1.850079513",
      "1.270642053"
    ],
    [
      "0.918942825",
      "1.316405675"
    ],
    [
      "0.332714502",
      "1.385257462"
    ],
    [
      "1.285971772",
      "1.385257462"
    ],
    [
      "1.667285498",
      "1.385257462"
    ],
    [
      "0.726996243",
      "1.414929039"
    ],
    [
      "1.082524508",
      "1.457087503"
    ],
    [
      "0.523371365",
      "1.486254018"
    ],
    [
      "1.476628635",
      "1.486254018"
    ],
    [
      "0.890577926",
      "1.555610867"
    ],
    [
      "0.340577350",
      "1.600869427"
    ],
    [
      "1.659422650",
      "1.600869427"
    ],
    [
      "1.311019573",
      "1.624543505"
    ],
    [
      "0.686953048",
      "1.626935846"
    ],
    [
      "1.100567774",
      "1.672087001"
    ],
    [
      "0.504147104",
      "1.741609556"
    ],
    [
      "1.494894920",
      "1.742044530"
    ],
    [
      "0.908621192",
      "1.770610365"
    ],
    [
      "1.300370018",
      "1.840035804"
    ],
    [
      "0.704996313",
      "1.841935344"
    ],
    [
      "1.089918219",
      "1.887579299"
    ]
  ],
  "radius": "0.107877643"
}
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": 69
}

当前第一名的答案

{
  "centers": [
    [
      "0.809343137",
      "0.128488520"
    ],
    [
      "1.190656863",
      "0.128488520"
    ],
    [
      "0.605673733",
      "0.199774018"
    ],
    [
      "1.395181954",
      "0.200386162"
    ],
    [
      "1.000000000",
      "0.229485076"
    ],
    [
      "1.577307337",
      "0.319854032"
    ],
    [
      "0.418364447",
      "0.323551641"
    ],
    [
      "0.817205985",
      "0.344100485"
    ],
    [
      "1.182794015",
      "0.344100485"
    ],
    [
      "0.613581107",
      "0.415425464"
    ],
    [
      "1.385360755",
      "0.418377396"
    ],
    [
      "1.000000000",
      "0.458715894"
    ],
    [
      "1.723707112",
      "0.478338888"
    ],
    [
      "0.378321251",
      "0.535558448"
    ],
    [
      "0.777162790",
      "0.556107292"
    ],
    [
      "1.190656863",
      "0.559712450"
    ],
    [
      "1.531760530",
      "0.576862252"
    ],
    [
      "0.573537912",
      "0.627432271"
    ],
    [
      "0.186374669",
      "0.634081812"
    ],
    [
      "0.959956805",
      "0.670722701"
    ],
    [
      "1.713057557",
      "0.693831187"
    ],
    [
      "1.354587135",
      "0.699987928"
    ],
    [
      "0.367671697",
      "0.751050747"
    ],
    [
      "0.766513235",
      "0.771599590"
    ],
    [
      "1.150613668",
      "0.771719257"
    ],
    [
      "1.535884162",
      "0.816956862"
    ],
    [
      "1.876639240",
      "0.834513015"
    ],
    [
      "0.122322645",
      "0.840110165"
    ],
    [
      "0.562888357",
      "0.842924569"
    ],
    [
      "0.949307250",
      "0.886215000"
    ],
    [
      "1.314543939",
      "0.911994735"
    ],
    [
      "0.303619672",
      "0.957079099"
    ],
    [
      "1.699465845",
      "0.957638690"
    ],
    [
      "0.726996243",
      "0.983705109"
    ],
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      "1.123887076",
      "1.012991291"
    ],
    [
      "1.495840967",
      "1.028963669"
    ],
    [
      "0.523371365",
      "1.055030088"
    ],
    [
      "0.109923126",
      "1.057569308"
    ],
    [
      "1.890179163",
      "1.058616426"
    ],
    [
      "0.926805674",
      "1.100793710"
    ],
    [
      "1.305184103",
      "1.129960225"
    ],
    [
      "0.340577350",
      "1.169645497"
    ],
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      "1.659422650",
      "1.169645497"
    ],
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      "0.734859092",
      "1.199317074"
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      "1.244575634"
    ],
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      "0.149920487",
      "1.270642053"
    ],
    [
      "0.531234214",
      "1.270642053"
    ],
    [
      "1.468765786",
      "1.270642053"
    ],
    [
      "1.850079513",
      "1.270642053"
    ],
    [
      "0.918942825",
      "1.316405675"
    ],
    [
      "0.332714502",
      "1.385257462"
    ],
    [
      "1.285971772",
      "1.385257462"
    ],
    [
      "1.667285498",
      "1.385257462"
    ],
    [
      "0.726996243",
      "1.414929039"
    ],
    [
      "1.082524508",
      "1.457087503"
    ],
    [
      "0.523371365",
      "1.486254018"
    ],
    [
      "1.476628635",
      "1.486254018"
    ],
    [
      "0.890577926",
      "1.555610867"
    ],
    [
      "0.340577350",
      "1.600869427"
    ],
    [
      "1.659422650",
      "1.600869427"
    ],
    [
      "1.311019573",
      "1.624543505"
    ],
    [
      "0.686953048",
      "1.626935846"
    ],
    [
      "1.100567774",
      "1.672087001"
    ],
    [
      "0.504147104",
      "1.741609556"
    ],
    [
      "1.494894920",
      "1.742044530"
    ],
    [
      "0.908621192",
      "1.770610365"
    ],
    [
      "1.300370018",
      "1.840035804"
    ],
    [
      "0.704996313",
      "1.841935344"
    ],
    [
      "1.089918219",
      "1.887579299"
    ]
  ],
  "radius": "0.107877643"
}

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？