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 = 134

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 = 134
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[0.0782996679644618640369266705065, 0.0782996679644618640369266705075]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 = 0.078299667

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 = 134 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": [
    [
      "0.94499676",
      "0.079942314"
    ],
    [
      "1.101545812",
      "0.083910512"
    ],
    [
      "0.760513557",
      "0.109956324"
    ],
    [
      "1.32381896",
      "0.137055747"
    ],
    [
      "0.613296036",
      "0.163345026"
    ],
    [
      "1.465664792",
      "0.204666305"
    ],
    [
      "1.187781801",
      "0.21462664"
    ],
    [
      "0.872933259",
      "0.218975417"
    ],
    [
      "0.477241464",
      "0.240885374"
    ],
    [
      "1.036598043",
      "0.270580136"
    ],
    [
      "0.725715738",
      "0.272364119"
    ],
    [
      "1.322978744",
      "0.293652829"
    ],
    [
      "1.594801305",
      "0.295909871"
    ],
    [
      "0.356277318",
      "0.340339018"
    ],
    [
      "0.589661166",
      "0.349904467"
    ],
    [
      "0.914705153",
      "0.369900765"
    ],
    [
      "1.186020984",
      "0.371216076"
    ],
    [
      "1.458244186",
      "0.372561715"
    ],
    [
      "0.766302143",
      "0.424741837"
    ],
    [
      "1.049983825",
      "0.448786968"
    ],
    [
      "0.46869702",
      "0.449358111"
    ],
    [
      "1.321217927",
      "0.450242266"
    ],
    [
      "1.59290519",
      "0.452497727"
    ],
    [
      "0.253895459",
      "0.458835038"
    ],
    [
      "1.749148168",
      "0.463056312"
    ],
    [
      "0.628467996",
      "0.501619272"
    ],
    [
      "0.909217195",
      "0.52640391"
    ],
    [
      "1.185180768",
      "0.527813158"
    ],
    [
      "1.45641487",
      "0.529268455"
    ],
    [
      "0.352234456",
      "0.580707076"
    ],
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      "0.762836494",
      "0.582046001"
    ],
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      "0.173051334",
      "0.592952821"
    ],
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      "1.82923375",
      "0.597628418"
    ],
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      "0.601072916"
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      "1.049143609",
      "0.60538405"
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      "1.617616778",
      "0.610304716"
    ],
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      "0.641872348",
      "0.681499645"
    ],
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      "0.91310645",
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    ],
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      "1.184340552",
      "0.68441024"
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      "1.455574654",
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    ],
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      "0.505078426",
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      "0.760525835"
    ],
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      "0.761981132"
    ],
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      "0.763436429"
    ],
    [
      "1.590688732",
      "0.765033321"
    ],
    [
      "1.893165761",
      "0.772433692"
    ],
    [
      "0.369798029",
      "0.83664143"
    ],
    [
      "0.641032131",
      "0.838096727"
    ],
    [
      "0.912266233",
      "0.839552024"
    ],
    [
      "1.183500336",
      "0.841007322"
    ],
    [
      "1.457028454",
      "0.846631617"
    ],
    [
      "0.215097935",
      "0.86095671"
    ],
    [
      "1.768549592",
      "0.882778278"
    ],
    [
      "0.504994972",
      "0.915667619"
    ],
    [
      "0.776229074",
      "0.917122917"
    ],
    [
      "1.047463176",
      "0.918578214"
    ],
    [
      "1.318697279",
      "0.920033511"
    ],
    [
      "1.615922786",
      "0.92143784"
    ],
    [
      "1.918798646",
      "0.926920931"
    ],
    [
      "0.080220434",
      "0.940526879"
    ],
    [
      "0.368957813",
      "0.993238512"
    ],
    [
      "0.640191915",
      "0.994693809"
    ],
    [
      "0.911426017",
      "0.996149106"
    ],
    [
      "1.182660119",
      "0.997604403"
    ],
    [
      "1.480441191",
      "1.001470875"
    ],
    [
      "0.214774838",
      "1.020642199"
    ],
    [
      "1.767659596",
      "1.039375084"
    ],
    [
      "0.504154756",
      "1.072264701"
    ],
    [
      "0.775388858",
      "1.073719999"
    ],
    [
      "1.04662296",
      "1.075175296"
    ],
    [
      "1.317857062",
      "1.076630593"
    ],
    [
      "1.616265328",
      "1.07941416"
    ],
    [
      "1.91790865",
      "1.083517738"
    ],
    [
      "0.084615554",
      "1.10771638"
    ],
    [
      "0.368117597",
      "1.149835594"
    ],
    [
      "0.639351699",
      "1.151290891"
    ],
    [
      "0.910585801",
      "1.152746188"
    ],
    [
      "1.181819903",
      "1.154201485"
    ],
    [
      "1.453054005",
      "1.155656783"
    ],
    [
      "0.217088055",
      "1.191229186"
    ],
    [
      "1.740272411",
      "1.193560992"
    ],
    [
      "0.50331454",
      "1.228861783"
    ],
    [
      "0.774548642",
      "1.23031708"
    ],
    [
      "1.045782744",
      "1.231772378"
    ],
    [
      "1.317016846",
      "1.233227675"
    ],
    [
      "1.588878143",
      "1.233600068"
    ],
    [
      "1.890521465",
      "1.237703646"
    ],
    [
      "0.328450728",
      "1.30132781"
    ],
    [
      "0.638511483",
      "1.307887973"
    ],
    [
      "0.909745585",
      "1.30934327"
    ],
    [
      "1.180979687",
      "1.310798567"
    ],
    [
      "1.453445806",
      "1.312258451"
    ],
    [
      "0.138046915",
      "1.326431767"
    ],
    [
      "1.683054496",
      "1.358716679"
    ],
    [
      "1.837427674",
      "1.385027783"
    ],
    [
      "0.502474323",
      "1.385458865"
    ],
    [
      "0.773708426",
      "1.386914162"
    ],
    [
      "1.044942528",
      "1.38836946"
    ],
    [
      "1.31617663",
      "1.389824757"
    ],
    [
      "0.361831556",
      "1.454328046"
    ],
    [
      "0.637671266",
      "1.464485055"
    ],
    [
      "0.908905368",
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    ],
    [
      "1.180139471",
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    ],
    [
      "0.205755417",
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    [
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    [
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    ],
    [
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    ],
    [
      "1.044102311",
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    [
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      "1.694070366"
    ],
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    ],
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    ],
    [
      "0.917450986",
      "1.778855245"
    ],
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      "1.35585494",
      "1.850234535"
    ],
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      "0.673771865",
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      "1.911213227"
    ],
    [
      "0.98221633",
      "1.921509517"
    ]
  ],
  "radius": "0.078299667"
}
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": 134
}

起步布局答案

{
  "centers": [
    [
      "0.94499676",
      "0.079942314"
    ],
    [
      "1.101545812",
      "0.083910512"
    ],
    [
      "0.760513557",
      "0.109956324"
    ],
    [
      "1.32381896",
      "0.137055747"
    ],
    [
      "0.613296036",
      "0.163345026"
    ],
    [
      "1.465664792",
      "0.204666305"
    ],
    [
      "1.187781801",
      "0.21462664"
    ],
    [
      "0.872933259",
      "0.218975417"
    ],
    [
      "0.477241464",
      "0.240885374"
    ],
    [
      "1.036598043",
      "0.270580136"
    ],
    [
      "0.725715738",
      "0.272364119"
    ],
    [
      "1.322978744",
      "0.293652829"
    ],
    [
      "1.594801305",
      "0.295909871"
    ],
    [
      "0.356277318",
      "0.340339018"
    ],
    [
      "0.589661166",
      "0.349904467"
    ],
    [
      "0.914705153",
      "0.369900765"
    ],
    [
      "1.186020984",
      "0.371216076"
    ],
    [
      "1.458244186",
      "0.372561715"
    ],
    [
      "0.766302143",
      "0.424741837"
    ],
    [
      "1.049983825",
      "0.448786968"
    ],
    [
      "0.46869702",
      "0.449358111"
    ],
    [
      "1.321217927",
      "0.450242266"
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      "0.452497727"
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      "0.52640391"
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      "1.307887973"
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      "1.386914162"
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      "1.38836946"
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  "radius": "0.078299667"
}

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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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?