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 = 136

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 = 136
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[0.0777256870114004923035689174495, 0.0777256870114004923035689174505]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 = 0.077725686

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 = 136 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": [
    [
      "1.053159358",
      "0.079258999"
    ],
    [
      "0.837669918",
      "0.092123988"
    ],
    [
      "1.207045056",
      "0.101266306"
    ],
    [
      "0.684045784",
      "0.133534281"
    ],
    [
      "1.355048636",
      "0.1488065"
    ],
    [
      "0.952081528",
      "0.197362352"
    ],
    [
      "0.50298437",
      "0.223174585"
    ],
    [
      "0.795617111",
      "0.241779229"
    ],
    [
      "1.4724761",
      "0.250668761"
    ],
    [
      "1.247696708",
      "0.261237338"
    ],
    [
      "1.092279903",
      "0.264515499"
    ],
    [
      "0.646088573",
      "0.284280359"
    ],
    [
      "1.615006853",
      "0.31271805"
    ],
    [
      "0.379444221",
      "0.317724078"
    ],
    [
      "0.964024384",
      "0.35235428"
    ],
    [
      "1.374541176",
      "0.371391087"
    ],
    [
      "0.522494731",
      "0.378565522"
    ],
    [
      "0.814837004",
      "0.396037861"
    ],
    [
      "1.225353795",
      "0.415074668"
    ],
    [
      "1.721701501",
      "0.425772823"
    ],
    [
      "0.27366917",
      "0.431639697"
    ],
    [
      "1.519235052",
      "0.436542225"
    ],
    [
      "0.665552763",
      "0.439389276"
    ],
    [
      "1.076449165",
      "0.459712552"
    ],
    [
      "0.41671968",
      "0.492481141"
    ],
    [
      "0.927261784",
      "0.503396132"
    ],
    [
      "1.337778576",
      "0.522432939"
    ],
    [
      "0.778074404",
      "0.547079712"
    ],
    [
      "1.630339519",
      "0.551542922"
    ],
    [
      "1.80789272",
      "0.555141302"
    ],
    [
      "0.188529067",
      "0.561702346"
    ],
    [
      "1.188873945",
      "0.567070823"
    ],
    [
      "1.479054005",
      "0.587289571"
    ],
    [
      "0.628790163",
      "0.590431128"
    ],
    [
      "1.039686565",
      "0.610754403"
    ],
    [
      "0.479741201",
      "0.634584686"
    ],
    [
      "0.324314987",
      "0.637381364"
    ],
    [
      "0.890499184",
      "0.654437983"
    ],
    [
      "1.301978197",
      "0.673715377"
    ],
    [
      "1.716530738",
      "0.680911401"
    ],
    [
      "1.87113183",
      "0.697148149"
    ],
    [
      "0.740975711",
      "0.698039365"
    ],
    [
      "0.12644273",
      "0.704216964"
    ],
    [
      "1.152111345",
      "0.718112675"
    ],
    [
      "1.55710285",
      "0.721727317"
    ],
    [
      "0.589089701",
      "0.745074664"
    ],
    [
      "1.002923965",
      "0.761796255"
    ],
    [
      "0.41846933",
      "0.778138734"
    ],
    [
      "0.262228649",
      "0.779895982"
    ],
    [
      "1.413723142",
      "0.78178886"
    ],
    [
      "0.853736584",
      "0.805479835"
    ],
    [
      "1.264535512",
      "0.825471588"
    ],
    [
      "1.669527017",
      "0.82908623"
    ],
    [
      "1.909622217",
      "0.847758972"
    ],
    [
      "0.704548954",
      "0.849162563"
    ],
    [
      "0.089174022",
      "0.855134733"
    ],
    [
      "1.115348132",
      "0.869155168"
    ],
    [
      "0.529990142",
      "0.888853563"
    ],
    [
      "1.526147309",
      "0.889147774"
    ],
    [
      "0.966160751",
      "0.912838749"
    ],
    [
      "0.224959941",
      "0.930813751"
    ],
    [
      "0.380410893",
      "0.931175874"
    ],
    [
      "1.376959679",
      "0.932830502"
    ],
    [
      "1.781951185",
      "0.936445144"
    ],
    [
      "0.816973121",
      "0.956521477"
    ],
    [
      "1.227772299",
      "0.976514082"
    ],
    [
      "1.638571476",
      "0.996506687"
    ],
    [
      "0.667785491",
      "1.000204204"
    ],
    [
      "1.922268229",
      "1.003349977"
    ],
    [
      "0.077781741",
      "1.010168103"
    ],
    [
      "1.078584918",
      "1.020197662"
    ],
    [
      "1.489383846",
      "1.040189415"
    ],
    [
      "0.518597861",
      "1.043886932"
    ],
    [
      "0.929397288",
      "1.06388039"
    ],
    [
      "1.340196466",
      "1.083872996"
    ],
    [
      "0.21356766",
      "1.085847121"
    ],
    [
      "0.369018612",
      "1.086209243"
    ],
    [
      "1.794168421",
      "1.091415686"
    ],
    [
      "0.780209658",
      "1.107563118"
    ],
    [
      "1.191009085",
      "1.127556576"
    ],
    [
      "1.601808014",
      "1.147548329"
    ],
    [
      "0.631022028",
      "1.151245846"
    ],
    [
      "1.041821455",
      "1.171239304"
    ],
    [
      "0.097307386",
      "1.189039555"
    ],
    [
      "1.452620633",
      "1.191231909"
    ],
    [
      "0.48068855",
      "1.194670706"
    ],
    [
      "1.897820544",
      "1.210752193"
    ],
    [
      "0.892633825",
      "1.214922032"
    ],
    [
      "0.290979529",
      "1.220652656"
    ],
    [
      "1.731282979",
      "1.233579503"
    ],
    [
      "1.303433252",
      "1.234915489"
    ],
    [
      "0.743446195",
      "1.25860476"
    ],
    [
      "1.154245622",
      "1.278598217"
    ],
    [
      "1.5650448",
      "1.298590823"
    ],
    [
      "0.594258565",
      "1.302287488"
    ],
    [
      "1.005057992",
      "1.322280945"
    ],
    [
      "0.141879746",
      "1.337963812"
    ],
    [
      "1.41585742",
      "1.342274403"
    ],
    [
      "0.445117029",
      "1.346127332"
    ],
    [
      "1.848632127",
      "1.358647018"
    ],
    [
      "0.855870362",
      "1.365963673"
    ],
    [
      "0.292581715",
      "1.376095773"
    ],
    [
      "1.694520219",
      "1.384621315"
    ],
    [
      "1.26666979",
      "1.385957131"
    ],
    [
      "0.706682732",
      "1.409646401"
    ],
    [
      "1.117820992",
      "1.430814155"
    ],
    [
      "1.555277918",
      "1.453735071"
    ],
    [
      "0.554276531",
      "1.456804036"
    ],
    [
      "0.968294529",
      "1.473322587"
    ],
    [
      "1.382401001",
      "1.494082828"
    ],
    [
      "0.392906058",
      "1.494839849"
    ],
    [
      "0.819106899",
      "1.517005315"
    ],
    [
      "0.2399085",
      "1.522351242"
    ],
    [
      "1.756857369",
      "1.527026404"
    ],
    [
      "1.233213371",
      "1.537765556"
    ],
    [
      "0.669919269",
      "1.560688043"
    ],
    [
      "1.079770159",
      "1.581666085"
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    [
      "1.4978449",
      "1.598187756"
    ],
    [
      "0.503760738",
      "1.603818608"
    ],
    [
      "0.930582529",
      "1.625348813"
    ],
    [
      "1.657590826",
      "1.646656179"
    ],
    [
      "0.348858435",
      "1.649562697"
    ],
    [
      "1.360258276",
      "1.672399724"
    ],
    [
      "0.779385833",
      "1.680845059"
    ],
    [
      "1.193187226",
      "1.687975533"
    ],
    [
      "0.619403476",
      "1.707702614"
    ],
    [
      "1.539642227",
      "1.74791455"
    ],
    [
      "0.471615918",
      "1.75591016"
    ],
    [
      "1.059548558",
      "1.769922407"
    ],
    [
      "0.903253484",
      "1.778379049"
    ],
    [
      "1.406362475",
      "1.827924783"
    ],
    [
      "1.250972339",
      "1.832287731"
    ],
    [
      "0.660663501",
      "1.857578363"
    ],
    [
      "0.809516245",
      "1.902388967"
    ],
    [
      "1.11907395",
      "1.914555249"
    ],
    [
      "0.963780604",
      "1.921562838"
    ]
  ],
  "radius": "0.077725686"
}
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": 136
}

起步布局答案

{
  "centers": [
    [
      "1.053159358",
      "0.079258999"
    ],
    [
      "0.837669918",
      "0.092123988"
    ],
    [
      "1.207045056",
      "0.101266306"
    ],
    [
      "0.684045784",
      "0.133534281"
    ],
    [
      "1.355048636",
      "0.1488065"
    ],
    [
      "0.952081528",
      "0.197362352"
    ],
    [
      "0.50298437",
      "0.223174585"
    ],
    [
      "0.795617111",
      "0.241779229"
    ],
    [
      "1.4724761",
      "0.250668761"
    ],
    [
      "1.247696708",
      "0.261237338"
    ],
    [
      "1.092279903",
      "0.264515499"
    ],
    [
      "0.646088573",
      "0.284280359"
    ],
    [
      "1.615006853",
      "0.31271805"
    ],
    [
      "0.379444221",
      "0.317724078"
    ],
    [
      "0.964024384",
      "0.35235428"
    ],
    [
      "1.374541176",
      "0.371391087"
    ],
    [
      "0.522494731",
      "0.378565522"
    ],
    [
      "0.814837004",
      "0.396037861"
    ],
    [
      "1.225353795",
      "0.415074668"
    ],
    [
      "1.721701501",
      "0.425772823"
    ],
    [
      "0.27366917",
      "0.431639697"
    ],
    [
      "1.519235052",
      "0.436542225"
    ],
    [
      "0.665552763",
      "0.439389276"
    ],
    [
      "1.076449165",
      "0.459712552"
    ],
    [
      "0.41671968",
      "0.492481141"
    ],
    [
      "0.927261784",
      "0.503396132"
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    [
      "1.337778576",
      "0.522432939"
    ],
    [
      "0.778074404",
      "0.547079712"
    ],
    [
      "1.630339519",
      "0.551542922"
    ],
    [
      "1.80789272",
      "0.555141302"
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      "0.188529067",
      "0.561702346"
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      "1.188873945",
      "0.567070823"
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      "0.587289571"
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      "0.590431128"
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      "0.610754403"
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      "0.634584686"
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      "0.680911401"
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      "0.697148149"
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      "1.020197662"
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      "1.107563118"
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      "1.234915489"
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      "1.25860476"
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      "1.278598217"
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      "1.298590823"
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      "1.302287488"
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      "1.322280945"
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      "0.141879746",
      "1.337963812"
    ],
    [
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      "1.342274403"
    ],
    [
      "0.445117029",
      "1.346127332"
    ],
    [
      "1.848632127",
      "1.358647018"
    ],
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      "1.365963673"
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  "radius": "0.077725686"
}

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?