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 = 137

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 = 137
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[0.0774325239613626498818290418955, 0.0774325239613626498818290418965]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 = 0.077432523

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 = 137 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": [
    [
      "1.496336502",
      "0.383493362"
    ],
    [
      "0.380266476",
      "0.642915271"
    ],
    [
      "1.084417273",
      "1.032424702"
    ],
    [
      "1.0228412",
      "1.768564235"
    ],
    [
      "0.436458987",
      "1.730446627"
    ],
    [
      "1.160548694",
      "1.908490432"
    ],
    [
      "1.314217427",
      "0.894072115"
    ],
    [
      "1.645108418",
      "0.340480571"
    ],
    [
      "0.937211128",
      "1.304919754"
    ],
    [
      "1.168359191",
      "0.092924518"
    ],
    [
      "0.769822437",
      "0.769913045"
    ],
    [
      "0.614983408",
      "0.772751748"
    ],
    [
      "0.238895694",
      "1.19754776"
    ],
    [
      "0.520395141",
      "0.378300229"
    ],
    [
      "0.280315872",
      "1.346770922"
    ],
    [
      "1.327432966",
      "1.431514349"
    ],
    [
      "0.542707162",
      "1.440164704"
    ],
    [
      "0.944314503",
      "1.574295524"
    ],
    [
      "1.309300648",
      "0.62588305"
    ],
    [
      "1.079500494",
      "0.764235638"
    ],
    [
      "1.326305909",
      "1.713145079"
    ],
    [
      "1.347602356",
      "0.339700005"
    ],
    [
      "0.702494196",
      "1.175083277"
    ],
    [
      "1.171928061",
      "1.434756231"
    ],
    [
      "1.695956502",
      "1.605619762"
    ],
    [
      "0.636498181",
      "0.152062988"
    ],
    [
      "1.090182145",
      "1.626312965"
    ],
    [
      "1.826536911",
      "0.590161639"
    ],
    [
      "0.380386179",
      "0.908337627"
    ],
    [
      "0.090520376",
      "0.845153103"
    ],
    [
      "0.706597",
      "1.874668753"
    ],
    [
      "1.901928372",
      "1.194051436"
    ],
    [
      "0.705997201",
      "1.445284243"
    ],
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      "0.079647248"
    ],
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      "1.623758624",
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      "0.93756"
    ],
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    ],
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      "1.782067051"
    ],
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      "1.074583716",
      "0.496046573"
    ],
    [
      "0.547655167",
      "1.17792198"
    ],
    [
      "0.627533071",
      "1.310597161"
    ],
    [
      "0.540022283",
      "0.908265632"
    ],
    [
      "1.401640866",
      "1.293317786"
    ],
    [
      "0.090341108",
      "1.153790271"
    ],
    [
      "0.322126835",
      "1.62577415"
    ],
    [
      "1.246889186",
      "1.299242347"
    ],
    [
      "0.225977416",
      "0.920216898"
    ],
    [
      "0.680798304",
      "0.364036673"
    ],
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      "0.129235412",
      "0.695205348"
    ],
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      "1.370489365",
      "0.155092681"
    ],
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      "1.396724088",
      "1.025128721"
    ],
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      "0.999622591",
      "0.631560457"
    ],
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      "0.772028878",
      "0.226993711"
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      "1.410530058",
      "0.743082836"
    ],
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      "1.915460966",
      "0.885711036"
    ],
    [
      "0.689944533",
      "0.637237864"
    ],
    [
      "0.415326734",
      "0.492071108"
    ],
    [
      "1.236161189",
      "0.232158393"
    ],
    [
      "0.92681751",
      "0.222129282"
    ],
    [
      "0.619900187",
      "1.040940813"
    ],
    [
      "0.924661466",
      "0.767074341"
    ],
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      "0.849700341",
      "0.902588225"
    ],
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      "1.017089032",
      "1.437594935"
    ],
    [
      "0.392810197",
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    ],
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    ],
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      "0.862250003",
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    ],
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    ],
    [
      "0.877843248",
      "1.714169593"
    ],
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      "0.789475474",
      "1.577134227"
    ],
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      "0.535105504",
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    ],
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      "0.377165807",
      "0.31940591"
    ],
    [
      "1.239256302",
      "1.029585999"
    ],
    [
      "1.081680116",
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    ],
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      "1.551563117",
      "1.022290017"
    ],
    [
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    ],
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      "1.883445818",
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    ],
    [
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      "0.500693147"
    ],
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      "0.764905658",
      "0.50172398"
    ],
    [
      "0.2720974",
      "0.433176789"
    ],
    [
      "1.506597869",
      "0.228968647"
    ],
    [
      "0.929578244",
      "1.035263406"
    ],
    [
      "0.844783562",
      "0.63439916"
    ],
    [
      "0.499784451",
      "0.224812827"
    ],
    [
      "1.012172253",
      "1.16940587"
    ],
    [
      "1.310250653",
      "1.868835589"
    ],
    [
      "0.07743268",
      "0.999464137"
    ],
    [
      "0.861324838",
      "1.868151167"
    ],
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      "1.787453671",
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    ],
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    ],
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    ],
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      "0.460144379",
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    ],
    [
      "1.159378398",
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    ],
    [
      "0.301016115",
      "1.055687842"
    ],
    [
      "0.635105349",
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    [
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    ],
    [
      "1.236482482",
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    [
      "1.391319666",
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      "1.342183733"
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      "1.307758457"
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      "0.490369163"
    ],
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      "0.237037141",
      "0.584020951"
    ],
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    ],
    [
      "1.482184646",
      "1.425589788"
    ]
  ],
  "radius": "0.077432523"
}
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": 137
}

起步布局答案

{
  "centers": [
    [
      "1.496336502",
      "0.383493362"
    ],
    [
      "0.380266476",
      "0.642915271"
    ],
    [
      "1.084417273",
      "1.032424702"
    ],
    [
      "1.0228412",
      "1.768564235"
    ],
    [
      "0.436458987",
      "1.730446627"
    ],
    [
      "1.160548694",
      "1.908490432"
    ],
    [
      "1.314217427",
      "0.894072115"
    ],
    [
      "1.645108418",
      "0.340480571"
    ],
    [
      "0.937211128",
      "1.304919754"
    ],
    [
      "1.168359191",
      "0.092924518"
    ],
    [
      "0.769822437",
      "0.769913045"
    ],
    [
      "0.614983408",
      "0.772751748"
    ],
    [
      "0.238895694",
      "1.19754776"
    ],
    [
      "0.520395141",
      "0.378300229"
    ],
    [
      "0.280315872",
      "1.346770922"
    ],
    [
      "1.327432966",
      "1.431514349"
    ],
    [
      "0.542707162",
      "1.440164704"
    ],
    [
      "0.944314503",
      "1.574295524"
    ],
    [
      "1.309300648",
      "0.62588305"
    ],
    [
      "1.079500494",
      "0.764235638"
    ],
    [
      "1.326305909",
      "1.713145079"
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    [
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  "radius": "0.077432523"
}

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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还没有帖子。第一个聊聊这道题的思路?