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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 = 82

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 = 82
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[0.0994943278051571223443808233385, 0.0994943278051571223443808233395]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 = 0.099494327

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 = 82 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": [
    [
      "0.911169064",
      "0.10388643"
    ],
    [
      "1.254313995",
      "0.136151137"
    ],
    [
      "0.716532099",
      "0.14527407"
    ],
    [
      "1.437824698",
      "0.213094669"
    ],
    [
      "1.073427511",
      "0.219076193"
    ],
    [
      "0.535736776",
      "0.22839769"
    ],
    [
      "0.892541027",
      "0.302001248"
    ],
    [
      "1.599956568",
      "0.328462524"
    ],
    [
      "1.235685958",
      "0.334265956"
    ],
    [
      "0.697957692",
      "0.343640302"
    ],
    [
      "0.377611269",
      "0.349198391"
    ],
    [
      "1.419097105",
      "0.411446497"
    ],
    [
      "1.054799473",
      "0.417191011"
    ],
    [
      "0.539832185",
      "0.464441003"
    ],
    [
      "1.732792761",
      "0.476621327"
    ],
    [
      "0.873912989",
      "0.500116067"
    ],
    [
      "1.21705792",
      "0.532380774"
    ],
    [
      "0.358221227",
      "0.547240082"
    ],
    [
      "1.551933298",
      "0.5596053"
    ],
    [
      "0.695547481",
      "0.588333019"
    ],
    [
      "1.036171436",
      "0.61530583"
    ],
    [
      "0.178152864",
      "0.631926976"
    ],
    [
      "1.37585314",
      "0.652299761"
    ],
    [
      "0.514359896",
      "0.670598102"
    ],
    [
      "1.714927026",
      "0.674806343"
    ],
    [
      "0.857805928",
      "0.703522782"
    ],
    [
      "1.194966656",
      "0.735224816"
    ],
    [
      "0.325515002",
      "0.765646403"
    ],
    [
      "1.538530444",
      "0.766897228"
    ],
    [
      "0.676919444",
      "0.786447838"
    ],
    [
      "1.876053258",
      "0.791574584"
    ],
    [
      "0.117381394",
      "0.821408672"
    ],
    [
      "1.016601148",
      "0.823441768"
    ],
    [
      "1.357225102",
      "0.850414579"
    ],
    [
      "0.495731858",
      "0.868712921"
    ],
    [
      "1.706348029",
      "0.89548137"
    ],
    [
      "0.835714663",
      "0.906366824"
    ],
    [
      "1.178859595",
      "0.938631531"
    ],
    [
      "0.264743532",
      "0.9551281"
    ],
    [
      "1.519902407",
      "0.965012047"
    ],
    [
      "1.900439257",
      "0.989063338"
    ],
    [
      "0.654500827",
      "0.989139237"
    ],
    [
      "0.99797311",
      "1.021556587"
    ],
    [
      "1.341464018",
      "1.053709184"
    ],
    [
      "0.466384057",
      "1.065525501"
    ],
    [
      "0.102252215",
      "1.069993725"
    ],
    [
      "1.730734027",
      "1.092970125"
    ],
    [
      "0.817086626",
      "1.104481642"
    ],
    [
      "1.160231557",
      "1.13674635"
    ],
    [
      "0.287557759",
      "1.152804594"
    ],
    [
      "1.512049165",
      "1.163845675"
    ],
    [
      "0.636200141",
      "1.187406698"
    ],
    [
      "0.979345073",
      "1.219671405"
    ],
    [
      "1.868880319",
      "1.236399114"
    ],
    [
      "1.322490004",
      "1.251936113"
    ],
    [
      "0.45255682",
      "1.264033165"
    ],
    [
      "1.671471821",
      "1.282929267"
    ],
    [
      "0.149607125",
      "1.296213477"
    ],
    [
      "0.798458588",
      "1.302596461"
    ],
    [
      "1.141603519",
      "1.334861168"
    ],
    [
      "1.488631077",
      "1.361451541"
    ],
    [
      "0.617763307",
      "1.38593733"
    ],
    [
      "0.960717035",
      "1.417786224"
    ],
    [
      "1.780272217",
      "1.449539468"
    ],
    [
      "1.303594378",
      "1.450426937"
    ],
    [
      "0.433947776",
      "1.462149768"
    ],
    [
      "0.235424156",
      "1.475745987"
    ],
    [
      "0.780021754",
      "1.501127093"
    ],
    [
      "1.121931853",
      "1.53443213"
    ],
    [
      "1.469983876",
      "1.559564556"
    ],
    [
      "0.59913527",
      "1.584052148"
    ],
    [
      "1.662493495",
      "1.609928549"
    ],
    [
      "1.284952628",
      "1.648540465"
    ],
    [
      "0.38090936",
      "1.653939787"
    ],
    [
      "0.834203932",
      "1.692597134"
    ],
    [
      "1.032238499",
      "1.712059795"
    ],
    [
      "1.486342333",
      "1.757879674"
    ],
    [
      "0.539643555",
      "1.773939539"
    ],
    [
      "1.308021738",
      "1.846187376"
    ],
    [
      "0.720856801",
      "1.856148083"
    ],
    [
      "1.114660544",
      "1.893176033"
    ],
    [
      "0.915700515",
      "1.896551205"
    ]
  ],
  "radius": "0.099494327"
}
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": 82
}

起步布局答案

{
  "centers": [
    [
      "0.911169064",
      "0.10388643"
    ],
    [
      "1.254313995",
      "0.136151137"
    ],
    [
      "0.716532099",
      "0.14527407"
    ],
    [
      "1.437824698",
      "0.213094669"
    ],
    [
      "1.073427511",
      "0.219076193"
    ],
    [
      "0.535736776",
      "0.22839769"
    ],
    [
      "0.892541027",
      "0.302001248"
    ],
    [
      "1.599956568",
      "0.328462524"
    ],
    [
      "1.235685958",
      "0.334265956"
    ],
    [
      "0.697957692",
      "0.343640302"
    ],
    [
      "0.377611269",
      "0.349198391"
    ],
    [
      "1.419097105",
      "0.411446497"
    ],
    [
      "1.054799473",
      "0.417191011"
    ],
    [
      "0.539832185",
      "0.464441003"
    ],
    [
      "1.732792761",
      "0.476621327"
    ],
    [
      "0.873912989",
      "0.500116067"
    ],
    [
      "1.21705792",
      "0.532380774"
    ],
    [
      "0.358221227",
      "0.547240082"
    ],
    [
      "1.551933298",
      "0.5596053"
    ],
    [
      "0.695547481",
      "0.588333019"
    ],
    [
      "1.036171436",
      "0.61530583"
    ],
    [
      "0.178152864",
      "0.631926976"
    ],
    [
      "1.37585314",
      "0.652299761"
    ],
    [
      "0.514359896",
      "0.670598102"
    ],
    [
      "1.714927026",
      "0.674806343"
    ],
    [
      "0.857805928",
      "0.703522782"
    ],
    [
      "1.194966656",
      "0.735224816"
    ],
    [
      "0.325515002",
      "0.765646403"
    ],
    [
      "1.538530444",
      "0.766897228"
    ],
    [
      "0.676919444",
      "0.786447838"
    ],
    [
      "1.876053258",
      "0.791574584"
    ],
    [
      "0.117381394",
      "0.821408672"
    ],
    [
      "1.016601148",
      "0.823441768"
    ],
    [
      "1.357225102",
      "0.850414579"
    ],
    [
      "0.495731858",
      "0.868712921"
    ],
    [
      "1.706348029",
      "0.89548137"
    ],
    [
      "0.835714663",
      "0.906366824"
    ],
    [
      "1.178859595",
      "0.938631531"
    ],
    [
      "0.264743532",
      "0.9551281"
    ],
    [
      "1.519902407",
      "0.965012047"
    ],
    [
      "1.900439257",
      "0.989063338"
    ],
    [
      "0.654500827",
      "0.989139237"
    ],
    [
      "0.99797311",
      "1.021556587"
    ],
    [
      "1.341464018",
      "1.053709184"
    ],
    [
      "0.466384057",
      "1.065525501"
    ],
    [
      "0.102252215",
      "1.069993725"
    ],
    [
      "1.730734027",
      "1.092970125"
    ],
    [
      "0.817086626",
      "1.104481642"
    ],
    [
      "1.160231557",
      "1.13674635"
    ],
    [
      "0.287557759",
      "1.152804594"
    ],
    [
      "1.512049165",
      "1.163845675"
    ],
    [
      "0.636200141",
      "1.187406698"
    ],
    [
      "0.979345073",
      "1.219671405"
    ],
    [
      "1.868880319",
      "1.236399114"
    ],
    [
      "1.322490004",
      "1.251936113"
    ],
    [
      "0.45255682",
      "1.264033165"
    ],
    [
      "1.671471821",
      "1.282929267"
    ],
    [
      "0.149607125",
      "1.296213477"
    ],
    [
      "0.798458588",
      "1.302596461"
    ],
    [
      "1.141603519",
      "1.334861168"
    ],
    [
      "1.488631077",
      "1.361451541"
    ],
    [
      "0.617763307",
      "1.38593733"
    ],
    [
      "0.960717035",
      "1.417786224"
    ],
    [
      "1.780272217",
      "1.449539468"
    ],
    [
      "1.303594378",
      "1.450426937"
    ],
    [
      "0.433947776",
      "1.462149768"
    ],
    [
      "0.235424156",
      "1.475745987"
    ],
    [
      "0.780021754",
      "1.501127093"
    ],
    [
      "1.121931853",
      "1.53443213"
    ],
    [
      "1.469983876",
      "1.559564556"
    ],
    [
      "0.59913527",
      "1.584052148"
    ],
    [
      "1.662493495",
      "1.609928549"
    ],
    [
      "1.284952628",
      "1.648540465"
    ],
    [
      "0.38090936",
      "1.653939787"
    ],
    [
      "0.834203932",
      "1.692597134"
    ],
    [
      "1.032238499",
      "1.712059795"
    ],
    [
      "1.486342333",
      "1.757879674"
    ],
    [
      "0.539643555",
      "1.773939539"
    ],
    [
      "1.308021738",
      "1.846187376"
    ],
    [
      "0.720856801",
      "1.856148083"
    ],
    [
      "1.114660544",
      "1.893176033"
    ],
    [
      "0.915700515",
      "1.896551205"
    ]
  ],
  "radius": "0.099494327"
}

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

DISCUSSION

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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。

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