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 = 207

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 = 207
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[0.0634107746264077939409908109485, 0.0634107746264077939409908109495]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 = 0.063410774

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 = 207 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": [
    [
      "1.445565489",
      "1.006775445"
    ],
    [
      "1.772058896",
      "0.904002653"
    ],
    [
      "0.365344126",
      "1.688775216"
    ],
    [
      "1.530420901",
      "1.458565449"
    ],
    [
      "0.51818805",
      "1.35473681"
    ],
    [
      "0.984956508",
      "0.518735368"
    ],
    [
      "1.85317555",
      "1.386381752"
    ],
    [
      "0.414064143",
      "1.1234512"
    ],
    [
      "0.622311956",
      "1.586022421"
    ],
    [
      "1.785016966",
      "0.649343754"
    ],
    [
      "1.163199045",
      "0.647112602"
    ],
    [
      "1.276985755",
      "1.774268438"
    ],
    [
      "1.157109271",
      "0.076682041"
    ],
    [
      "0.287883559",
      "1.110716772"
    ],
    [
      "1.237317675",
      "0.544204225"
    ],
    [
      "0.418539932",
      "0.316252992"
    ],
    [
      "0.910837878",
      "0.621643745"
    ],
    [
      "1.70680398",
      "0.446488705"
    ],
    [
      "1.28040659",
      "0.10637171"
    ],
    [
      "0.466126096",
      "1.239094005"
    ],
    [
      "0.339945513",
      "1.226359577"
    ],
    [
      "0.161251802",
      "1.10378111"
    ],
    [
      "0.940843154",
      "0.955837732"
    ],
    [
      "0.488182773",
      "1.020542824"
    ],
    [
      "0.814662571",
      "0.943103304"
    ],
    [
      "0.809005271",
      "1.719987537"
    ],
    [
      "0.688481987",
      "0.930368875"
    ],
    [
      "0.710538664",
      "0.711817693"
    ],
    [
      "0.8967298",
      "1.392940095"
    ],
    [
      "0.391863673",
      "1.342067151"
    ],
    [
      "1.912653871",
      "1.210386047"
    ],
    [
      "0.317641987",
      "1.447654705"
    ],
    [
      "1.593802749",
      "0.800958692"
    ],
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      "0.406115543",
      "0.570706032"
    ],
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      "0.562301403",
      "0.917634447"
    ],
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      "0.784657294",
      "0.608909317"
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    ],
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      "1.022757213"
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      "0.105798364"
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      "1.453514088",
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    ],
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    ],
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      "1.270738947"
    ],
    [
      "0.82261117",
      "1.495848472"
    ],
    [
      "0.932894555",
      "0.403092563"
    ],
    [
      "1.332468543",
      "0.222014515"
    ],
    [
      "0.680533387",
      "0.377623706"
    ],
    [
      "1.393503535",
      "0.89113264"
    ],
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      "0.644368633",
      "1.367471239"
    ],
    [
      "1.697926656",
      "1.032244302"
    ],
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      "1.022910384",
      "1.405674524"
    ],
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      "0.888781201",
      "0.840194927"
    ],
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      "0.918786477",
      "1.174388914"
    ],
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      "0.384058866",
      "0.789257213"
    ],
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      "0.66642531",
      "1.148920057"
    ],
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      "0.06919664",
      "0.896055388"
    ],
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      "0.122602903",
      "1.32767928"
    ],
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      "1.601751349",
      "1.35370386"
    ],
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      "1.067023738",
      "0.96857216"
    ],
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      "0.687641011",
      "1.756790007"
    ],
    [
      "1.497627442",
      "1.12241825"
    ],
    [
      "0.166654885",
      "0.668905426"
    ],
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      "0.948791754",
      "1.508582901"
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      "1.489678843",
      "0.569673082"
    ],
    [
      "1.726994966",
      "1.373647324"
    ],
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      "0.606414757",
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    ],
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      "0.175916131",
      "1.445066737"
    ],
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      "0.263042766",
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    ],
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      "0.289297596",
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      "1.935663715",
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      "1.374803433",
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      "1.402320301",
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      "0.572879394",
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      "1.501319138",
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    [
      "0.495457949",
      "1.592012338"
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    [
      "0.741330078",
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      "1.587160278",
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      "1.806858296",
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      "0.689432214",
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      "1.014571412",
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      "0.844667847",
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      "0.827460498"
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      "1.778674781",
      "1.141399466"
    ],
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      "1.223209598",
      "1.315500575"
    ],
    [
      "1.767476192",
      "0.776645961"
    ],
    [
      "1.341441582",
      "0.775489835"
    ],
    [
      "1.141142368",
      "0.865663783"
    ],
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      "1.363498259",
      "0.556938653"
    ],
    [
      "1.126266644",
      "1.928038852"
    ],
    [
      "0.458177497",
      "0.686348837"
    ],
    [
      "1.082990641",
      "0.179590417"
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    [
      "1.185255722",
      "0.42856142"
    ],
    [
      "1.030928391",
      "0.063921962"
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      "0.858775924",
      "0.50600094"
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    [
      "0.086378025",
      "1.20614137"
    ],
    [
      "1.401452135",
      "1.443877809"
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    [
      "1.014961785",
      "0.852929355"
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    [
      "1.007013185",
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      "0.304812078",
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      "1.089080415",
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      "0.51023945",
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      "0.994041017"
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      "1.437616889",
      "0.454030277"
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      "1.666974275",
      "1.485366713"
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      "1.717529497",
      "1.601787902"
    ],
    [
      "0.416731831",
      "0.443322511"
    ],
    [
      "1.074972337",
      "1.521317329"
    ],
    [
      "1.571746073",
      "1.019509874"
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    [
      "0.614363357",
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    [
      "1.133193769",
      "0.312918615"
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      "1.645864703",
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    [
      "0.464214007",
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      "1.541740796",
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      "0.43612082",
      "0.904900018"
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      "1.846796437"
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      "1.219423915"
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      "0.068324937"
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      "1.126998168",
      "1.637313377"
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      "1.275271551",
      "1.431143381"
    ],
    [
      "1.297328228",
      "1.212592199"
    ],
    [
      "0.992905108",
      "1.071480537"
    ],
    [
      "1.655132156",
      "0.330671056"
    ],
    [
      "0.570250003",
      "1.470379616"
    ],
    [
      "0.866724524",
      "1.058746109"
    ]
  ],
  "radius": "0.063410774"
}
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": 207
}

起步布局答案

{
  "centers": [
    [
      "1.445565489",
      "1.006775445"
    ],
    [
      "1.772058896",
      "0.904002653"
    ],
    [
      "0.365344126",
      "1.688775216"
    ],
    [
      "1.530420901",
      "1.458565449"
    ],
    [
      "0.51818805",
      "1.35473681"
    ],
    [
      "0.984956508",
      "0.518735368"
    ],
    [
      "1.85317555",
      "1.386381752"
    ],
    [
      "0.414064143",
      "1.1234512"
    ],
    [
      "0.622311956",
      "1.586022421"
    ],
    [
      "1.785016966",
      "0.649343754"
    ],
    [
      "1.163199045",
      "0.647112602"
    ],
    [
      "1.276985755",
      "1.774268438"
    ],
    [
      "1.157109271",
      "0.076682041"
    ],
    [
      "0.287883559",
      "1.110716772"
    ],
    [
      "1.237317675",
      "0.544204225"
    ],
    [
      "0.418539932",
      "0.316252992"
    ],
    [
      "0.910837878",
      "0.621643745"
    ],
    [
      "1.70680398",
      "0.446488705"
    ],
    [
      "1.28040659",
      "0.10637171"
    ],
    [
      "0.466126096",
      "1.239094005"
    ],
    [
      "0.339945513",
      "1.226359577"
    ],
    [
      "0.161251802",
      "1.10378111"
    ],
    [
      "0.940843154",
      "0.955837732"
    ],
    [
      "0.488182773",
      "1.020542824"
    ],
    [
      "0.814662571",
      "0.943103304"
    ],
    [
      "0.809005271",
      "1.719987537"
    ],
    [
      "0.688481987",
      "0.930368875"
    ],
    [
      "0.710538664",
      "0.711817693"
    ],
    [
      "0.8967298",
      "1.392940095"
    ],
    [
      "0.391863673",
      "1.342067151"
    ],
    [
      "1.912653871",
      "1.210386047"
    ],
    [
      "0.317641987",
      "1.447654705"
    ],
    [
      "1.593802749",
      "0.800958692"
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  "radius": "0.063410774"
}

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?