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 = 154

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 = 154
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[0.0733304615283166121355343277035, 0.0733304615283166121355343277045]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

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:起步布局

2y0
0x2
已验证构造r = 0.073330461

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 = 154 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。
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"。

起步布局答案

{
  "centers": [
    [
      "1.283346443",
      "0.566048894"
    ],
    [
      "0.722245676",
      "1.13404191"
    ],
    [
      "0.482200208",
      "0.965576868"
    ],
    [
      "1.802119385",
      "0.535973142"
    ],
    [
      "0.27834697",
      "1.465857568"
    ],
    [
      "0.997245118",
      "0.862211241"
    ],
    [
      "0.438891929",
      "1.55573343"
    ],
    [
      "1.4324178",
      "1.819592142"
    ],
    [
      "1.124059582",
      "0.212726993"
    ],
    [
      "1.512668729",
      "0.747500895"
    ],
    [
      "0.370440829",
      "0.737458214"
    ],
    [
      "1.780045667",
      "1.075187895"
    ],
    [
      "1.419412638",
      "0.496938988"
    ],
    [
      "1.010635888",
      "0.711019064"
    ],
    [
      "1.887136134",
      "1.267779597"
    ],
    [
      "0.546054818",
      "1.655861057"
    ],
    [
      "0.947442091",
      "1.593822687"
    ],
    [
      "1.484232047",
      "1.189403972"
    ],
    [
      "0.382962895",
      "0.308637834"
    ],
    [
      "1.833777801",
      "1.404389679"
    ],
    [
      "0.878931522",
      "0.775543372"
    ],
    [
      "0.933548932",
      "1.739843738"
    ],
    [
      "0.499767242",
      "0.219946398"
    ],
    [
      "0.744968736",
      "0.835237577"
    ],
    [
      "1.297694765",
      "1.877550147"
    ],
    [
      "0.087077585",
      "1.159025463"
    ],
    [
      "0.800157516",
      "1.800804023"
    ],
    [
      "1.2861945",
      "0.273408781"
    ],
    [
      "0.613904574",
      "0.90105256"
    ],
    [
      "1.152312748",
      "0.500173275"
    ],
    [
      "1.128949484",
      "0.797686933"
    ],
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      "0.905034537",
      "0.336368826"
    ],
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      "0.629101593",
      "0.150794015"
    ],
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      "0.519885646",
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      "0.205050411",
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      "1.409405913"
    ],
    [
      "1.609740044",
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      "0.341389257",
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    ],
    [
      "1.655493369",
      "0.539172782"
    ],
    [
      "1.352527681",
      "1.253928279"
    ],
    [
      "1.089118948",
      "1.382976895"
    ],
    [
      "1.212807169",
      "1.468860163"
    ],
    [
      "1.687886225",
      "1.389387197"
    ],
    [
      "1.486638271",
      "0.211393871"
    ],
    [
      "1.666266535",
      "1.644053831"
    ],
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      "0.210325096",
      "0.51509797"
    ],
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      "1.663259433",
      "0.685627945"
    ],
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      "0.702300689",
      "1.426684871"
    ],
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      "0.665751935"
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      "1.012903582"
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      "1.477813393",
      "1.339393357"
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      "0.754941229",
      "0.688916096"
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      "0.692328196",
      "1.573006352"
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      "1.913526972"
    ],
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      "1.163627693",
      "0.35394948"
    ],
    [
      "1.197312697",
      "1.616167044"
    ],
    [
      "1.911339796",
      "0.832142291"
    ],
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      "0.787844739",
      "0.248187323"
    ],
    [
      "1.020608382",
      "0.564697583"
    ],
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      "1.068687263",
      "0.076233701"
    ],
    [
      "1.456211887",
      "1.633913771"
    ],
    [
      "0.957414582",
      "1.447501203"
    ],
    [
      "0.770072215",
      "0.393767417"
    ],
    [
      "1.142340255",
      "0.646494756"
    ],
    [
      "1.556309484",
      "1.74110471"
    ],
    [
      "0.570596323",
      "1.491209179"
    ],
    [
      "1.63911992",
      "1.121591954"
    ],
    [
      "0.649219996",
      "0.296068501"
    ],
    [
      "1.918273175",
      "1.124462079"
    ],
    [
      "1.197597169",
      "1.770359207"
    ],
    [
      "0.378975462",
      "0.591045829"
    ],
    [
      "1.513611923",
      "1.045715944"
    ],
    [
      "1.109063935",
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    [
      "0.888904016",
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    [
      "0.360468335",
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    [
      "1.362500174",
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    [
      "0.335957021",
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      "0.217643425",
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      "0.975291526"
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      "0.402284101"
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      "0.899285985"
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      "1.294661388",
      "0.419825099"
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      "1.331437387",
      "1.556837507"
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      "1.07269297",
      "1.693493301"
    ],
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      "1.773309606",
      "0.782573216"
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    [
      "0.580568817",
      "1.344887698"
    ],
    [
      "0.633209356",
      "0.607118923"
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      "0.639520879",
      "0.46059387"
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    [
      "1.748867464",
      "1.218879687"
    ],
    [
      "1.240768301",
      "1.025809626"
    ],
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      "1.079146455",
      "1.529298376"
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    [
      "0.814060069",
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    ]
  ],
  "radius": "0.073330461"
}
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": 154
}

起步布局答案

{
  "centers": [
    [
      "1.283346443",
      "0.566048894"
    ],
    [
      "0.722245676",
      "1.13404191"
    ],
    [
      "0.482200208",
      "0.965576868"
    ],
    [
      "1.802119385",
      "0.535973142"
    ],
    [
      "0.27834697",
      "1.465857568"
    ],
    [
      "0.997245118",
      "0.862211241"
    ],
    [
      "0.438891929",
      "1.55573343"
    ],
    [
      "1.4324178",
      "1.819592142"
    ],
    [
      "1.124059582",
      "0.212726993"
    ],
    [
      "1.512668729",
      "0.747500895"
    ],
    [
      "0.370440829",
      "0.737458214"
    ],
    [
      "1.780045667",
      "1.075187895"
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  "radius": "0.073330461"
}

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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