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 = 178

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 = 178
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[0.0682317779785, 0.0682317779795]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 = 0.068231777

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 = 178 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": [
    [
      "1.058461791",
      "0.070067616"
    ],
    [
      "0.922005771",
      "0.071501793"
    ],
    [
      "1.211001729",
      "0.09243717"
    ],
    [
      "0.787222691",
      "0.092851811"
    ],
    [
      "0.657003581",
      "0.133659723"
    ],
    [
      "1.447430051",
      "0.182688328"
    ],
    [
      "1.310991574",
      "0.185304457"
    ],
    [
      "1.118393166",
      "0.192666688"
    ],
    [
      "0.53414158",
      "0.193050217"
    ],
    [
      "0.981845462",
      "0.194208703"
    ],
    [
      "0.846666007",
      "0.215688264"
    ],
    [
      "0.661839108",
      "0.270037579"
    ],
    [
      "0.413373922",
      "0.276078827"
    ],
    [
      "1.624638111",
      "0.308610638"
    ],
    [
      "1.206326859",
      "0.312701763"
    ],
    [
      "1.488242176",
      "0.312906117"
    ],
    [
      "1.350741502",
      "0.315850409"
    ],
    [
      "1.067571069",
      "0.319313506"
    ],
    [
      "0.931153411",
      "0.322852534"
    ],
    [
      "0.538977107",
      "0.329428074"
    ],
    [
      "0.781522067",
      "0.335598936"
    ],
    [
      "0.313927001",
      "0.369527273"
    ],
    [
      "1.718925657",
      "0.407262352"
    ],
    [
      "0.439530186",
      "0.422876519"
    ],
    [
      "1.559515894",
      "0.42927776"
    ],
    [
      "1.277305583",
      "0.430869833"
    ],
    [
      "1.422015219",
      "0.432222052"
    ],
    [
      "1.140887925",
      "0.434408861"
    ],
    [
      "1.002427129",
      "0.439224177"
    ],
    [
      "0.866009471",
      "0.442763205"
    ],
    [
      "0.715854431",
      "0.455223731"
    ],
    [
      "0.579461563",
      "0.459748106"
    ],
    [
      "0.229196017",
      "0.476499056"
    ],
    [
      "1.797800818",
      "0.519330428"
    ],
    [
      "1.654299557",
      "0.527452906"
    ],
    [
      "1.3485793",
      "0.547241476"
    ],
    [
      "1.493288936",
      "0.548593695"
    ],
    [
      "1.212161642",
      "0.550780504"
    ],
    [
      "0.343569802",
      "0.550936542"
    ],
    [
      "0.480014642",
      "0.553196552"
    ],
    [
      "1.075743984",
      "0.554319532"
    ],
    [
      "0.937283188",
      "0.559134849"
    ],
    [
      "0.800340781",
      "0.562389685"
    ],
    [
      "0.649242526",
      "0.577020948"
    ],
    [
      "0.160998407",
      "0.594699683"
    ],
    [
      "1.860699959",
      "0.643086005"
    ],
    [
      "1.724245112",
      "0.644627654"
    ],
    [
      "1.587802681",
      "0.64702872"
    ],
    [
      "1.419853018",
      "0.663613119"
    ],
    [
      "1.28343536",
      "0.667152147"
    ],
    [
      "0.409834996",
      "0.670231245"
    ],
    [
      "0.549795605",
      "0.670469393"
    ],
    [
      "1.147017702",
      "0.670691175"
    ],
    [
      "0.27336751",
      "0.672222812"
    ],
    [
      "1.010600044",
      "0.674230204"
    ],
    [
      "0.872139247",
      "0.67904552"
    ],
    [
      "0.735721589",
      "0.682584548"
    ],
    [
      "0.149989393",
      "0.730718445"
    ],
    [
      "1.903601209",
      "0.772630534"
    ],
    [
      "1.766592211",
      "0.774354387"
    ],
    [
      "1.628859473",
      "0.777169575"
    ],
    [
      "1.492384888",
      "0.779204779"
    ],
    [
      "1.354709077",
      "0.78352379"
    ],
    [
      "1.218291419",
      "0.787062819"
    ],
    [
      "1.081873761",
      "0.790601847"
    ],
    [
      "0.945456103",
      "0.794140875"
    ],
    [
      "0.806995307",
      "0.798956191"
    ],
    [
      "0.670577649",
      "0.802495219"
    ],
    [
      "0.534159991",
      "0.806034248"
    ],
    [
      "0.397696487",
      "0.806153865"
    ],
    [
      "0.261278829",
      "0.809692893"
    ],
    [
      "0.080669618",
      "0.848264482"
    ],
    [
      "1.836581524",
      "0.891503003"
    ],
    [
      "1.70013319",
      "0.893541218"
    ],
    [
      "1.562400452",
      "0.896356405"
    ],
    [
      "1.425982794",
      "0.899895434"
    ],
    [
      "1.289565136",
      "0.903434462"
    ],
    [
      "1.153147478",
      "0.90697349"
    ],
    [
      "1.01672982",
      "0.910512518"
    ],
    [
      "0.880312163",
      "0.914051546"
    ],
    [
      "0.741851366",
      "0.918866863"
    ],
    [
      "0.605433708",
      "0.922405891"
    ],
    [
      "0.46901605",
      "0.925944919"
    ],
    [
      "0.332552547",
      "0.926064536"
    ],
    [
      "0.192542429",
      "0.927581016"
    ],
    [
      "0.068366219",
      "0.984172276"
    ],
    [
      "1.931707318",
      "0.989346681"
    ],
    [
      "1.770122503",
      "1.010689834"
    ],
    [
      "1.633674169",
      "1.012728049"
    ],
    [
      "1.497256511",
      "1.016267077"
    ],
    [
      "1.360838853",
      "1.019806105"
    ],
    [
      "1.224421196",
      "1.023345133"
    ],
    [
      "1.088003538",
      "1.026884161"
    ],
    [
      "0.95158588",
      "1.030423189"
    ],
    [
      "0.815168222",
      "1.033962218"
    ],
    [
      "0.676707425",
      "1.038777534"
    ],
    [
      "0.540289767",
      "1.042316562"
    ],
    [
      "0.263816147",
      "1.043952659"
    ],
    [
      "0.40088789",
      "1.044185588"
    ],
    [
      "1.865248297",
      "1.108533512"
    ],
    [
      "0.076045917",
      "1.120419567"
    ],
    [
      "1.704978562",
      "1.130600505"
    ],
    [
      "1.568530229",
      "1.13263872"
    ],
    [
      "1.432112571",
      "1.136177748"
    ],
    [
      "1.295694913",
      "1.139716776"
    ],
    [
      "1.159277255",
      "1.143255804"
    ],
    [
      "1.022859597",
      "1.146794833"
    ],
    [
      "0.886441939",
      "1.150333861"
    ],
    [
      "0.750024281",
      "1.153872889"
    ],
    [
      "0.613690352",
      "1.159819471"
    ],
    [
      "0.477272694",
      "1.163358499"
    ],
    [
      "0.340850866",
      "1.166732958"
    ],
    [
      "0.204387325",
      "1.166796125"
    ],
    [
      "1.900210824",
      "1.240442283"
    ],
    [
      "1.764369795",
      "1.253462149"
    ],
    [
      "1.627921461",
      "1.255500364"
    ],
    [
      "1.491503804",
      "1.259039392"
    ],
    [
      "0.105478982",
      "1.260814431"
    ],
    [
      "1.355086146",
      "1.26257842"
    ],
    [
      "1.218668488",
      "1.266117448"
    ],
    [
      "1.08225083",
      "1.269656477"
    ],
    [
      "0.945833172",
      "1.273195505"
    ],
    [
      "0.809415514",
      "1.276734533"
    ],
    [
      "0.673081585",
      "1.282681115"
    ],
    [
      "0.536663927",
      "1.286220143"
    ],
    [
      "0.388539817",
      "1.294592514"
    ],
    [
      "0.252076275",
      "1.294655682"
    ],
    [
      "1.853737027",
      "1.371271762"
    ],
    [
      "1.562777521",
      "1.375411035"
    ],
    [
      "1.426359863",
      "1.378950063"
    ],
    [
      "1.716486246",
      "1.381248955"
    ],
    [
      "1.289942205",
      "1.382489092"
    ],
    [
      "1.153524547",
      "1.38602812"
    ],
    [
      "0.153167932",
      "1.388673987"
    ],
    [
      "1.017106889",
      "1.389567148"
    ],
    [
      "0.880689231",
      "1.393106176"
    ],
    [
      "0.744355302",
      "1.399052758"
    ],
    [
      "0.607937644",
      "1.402591786"
    ],
    [
      "0.469065706",
      "1.404764576"
    ],
    [
      "0.322347467",
      "1.41626106"
    ],
    [
      "1.635353582",
      "1.490974946"
    ],
    [
      "1.49763358",
      "1.495321707"
    ],
    [
      "1.787759963",
      "1.497620598"
    ],
    [
      "1.361215922",
      "1.498860735"
    ],
    [
      "1.224798264",
      "1.502399763"
    ],
    [
      "1.088380607",
      "1.505938791"
    ],
    [
      "0.219021024",
      "1.508196673"
    ],
    [
      "0.951962949",
      "1.509477819"
    ],
    [
      "0.815629019",
      "1.515424401"
    ],
    [
      "0.540339423",
      "1.52113622"
    ],
    [
      "0.678266067",
      "1.52148319"
    ],
    [
      "0.40395014",
      "1.525637961"
    ],
    [
      "1.7066273",
      "1.607346589"
    ],
    [
      "1.570209642",
      "1.610885617"
    ],
    [
      "0.30162574",
      "1.616818784"
    ],
    [
      "1.296071982",
      "1.618771406"
    ],
    [
      "1.023236666",
      "1.625849462"
    ],
    [
      "0.753162208",
      "1.636751231"
    ],
    [
      "1.160746426",
      "1.638598368"
    ],
    [
      "1.431546554",
      "1.642775122"
    ],
    [
      "0.889124916",
      "1.65380503"
    ],
    [
      "0.523827516",
      "1.656597133"
    ],
    [
      "0.399210248",
      "1.712210428"
    ],
    [
      "0.644815775",
      "1.719717204"
    ],
    [
      "1.5572939",
      "1.746736586"
    ],
    [
      "1.323780408",
      "1.752392307"
    ],
    [
      "1.057747096",
      "1.757877233"
    ],
    [
      "0.791835676",
      "1.767620116"
    ],
    [
      "1.191389436",
      "1.785482073"
    ],
    [
      "0.509681409",
      "1.7923255"
    ],
    [
      "0.925902789",
      "1.793082072"
    ],
    [
      "1.442246216",
      "1.820128224"
    ],
    [
      "0.630669667",
      "1.855445571"
    ],
    [
      "1.291524617",
      "1.884988936"
    ],
    [
      "1.025491305",
      "1.890473862"
    ],
    [
      "0.759579885",
      "1.900216745"
    ],
    [
      "1.159133645",
      "1.918078702"
    ],
    [
      "0.893646998",
      "1.925678702"
    ]
  ],
  "radius": "0.068231777"
}
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": 178
}

起步布局答案

{
  "centers": [
    [
      "1.058461791",
      "0.070067616"
    ],
    [
      "0.922005771",
      "0.071501793"
    ],
    [
      "1.211001729",
      "0.09243717"
    ],
    [
      "0.787222691",
      "0.092851811"
    ],
    [
      "0.657003581",
      "0.133659723"
    ],
    [
      "1.447430051",
      "0.182688328"
    ],
    [
      "1.310991574",
      "0.185304457"
    ],
    [
      "1.118393166",
      "0.192666688"
    ],
    [
      "0.53414158",
      "0.193050217"
    ],
    [
      "0.981845462",
      "0.194208703"
    ],
    [
      "0.846666007",
      "0.215688264"
    ],
    [
      "0.661839108",
      "0.270037579"
    ],
    [
      "0.413373922",
      "0.276078827"
    ],
    [
      "1.624638111",
      "0.308610638"
    ],
    [
      "1.206326859",
      "0.312701763"
    ],
    [
      "1.488242176",
      "0.312906117"
    ],
    [
      "1.350741502",
      "0.315850409"
    ],
    [
      "1.067571069",
      "0.319313506"
    ],
    [
      "0.931153411",
      "0.322852534"
    ],
    [
      "0.538977107",
      "0.329428074"
    ],
    [
      "0.781522067",
      "0.335598936"
    ],
    [
      "0.313927001",
      "0.369527273"
    ],
    [
      "1.718925657",
      "0.407262352"
    ],
    [
      "0.439530186",
      "0.422876519"
    ],
    [
      "1.559515894",
      "0.42927776"
    ],
    [
      "1.277305583",
      "0.430869833"
    ],
    [
      "1.422015219",
      "0.432222052"
    ],
    [
      "1.140887925",
      "0.434408861"
    ],
    [
      "1.002427129",
      "0.439224177"
    ],
    [
      "0.866009471",
      "0.442763205"
    ],
    [
      "0.715854431",
      "0.455223731"
    ],
    [
      "0.579461563",
      "0.459748106"
    ],
    [
      "0.229196017",
      "0.476499056"
    ],
    [
      "1.797800818",
      "0.519330428"
    ],
    [
      "1.654299557",
      "0.527452906"
    ],
    [
      "1.3485793",
      "0.547241476"
    ],
    [
      "1.493288936",
      "0.548593695"
    ],
    [
      "1.212161642",
      "0.550780504"
    ],
    [
      "0.343569802",
      "0.550936542"
    ],
    [
      "0.480014642",
      "0.553196552"
    ],
    [
      "1.075743984",
      "0.554319532"
    ],
    [
      "0.937283188",
      "0.559134849"
    ],
    [
      "0.800340781",
      "0.562389685"
    ],
    [
      "0.649242526",
      "0.577020948"
    ],
    [
      "0.160998407",
      "0.594699683"
    ],
    [
      "1.860699959",
      "0.643086005"
    ],
    [
      "1.724245112",
      "0.644627654"
    ],
    [
      "1.587802681",
      "0.64702872"
    ],
    [
      "1.419853018",
      "0.663613119"
    ],
    [
      "1.28343536",
      "0.667152147"
    ],
    [
      "0.409834996",
      "0.670231245"
    ],
    [
      "0.549795605",
      "0.670469393"
    ],
    [
      "1.147017702",
      "0.670691175"
    ],
    [
      "0.27336751",
      "0.672222812"
    ],
    [
      "1.010600044",
      "0.674230204"
    ],
    [
      "0.872139247",
      "0.67904552"
    ],
    [
      "0.735721589",
      "0.682584548"
    ],
    [
      "0.149989393",
      "0.730718445"
    ],
    [
      "1.903601209",
      "0.772630534"
    ],
    [
      "1.766592211",
      "0.774354387"
    ],
    [
      "1.628859473",
      "0.777169575"
    ],
    [
      "1.492384888",
      "0.779204779"
    ],
    [
      "1.354709077",
      "0.78352379"
    ],
    [
      "1.218291419",
      "0.787062819"
    ],
    [
      "1.081873761",
      "0.790601847"
    ],
    [
      "0.945456103",
      "0.794140875"
    ],
    [
      "0.806995307",
      "0.798956191"
    ],
    [
      "0.670577649",
      "0.802495219"
    ],
    [
      "0.534159991",
      "0.806034248"
    ],
    [
      "0.397696487",
      "0.806153865"
    ],
    [
      "0.261278829",
      "0.809692893"
    ],
    [
      "0.080669618",
      "0.848264482"
    ],
    [
      "1.836581524",
      "0.891503003"
    ],
    [
      "1.70013319",
      "0.893541218"
    ],
    [
      "1.562400452",
      "0.896356405"
    ],
    [
      "1.425982794",
      "0.899895434"
    ],
    [
      "1.289565136",
      "0.903434462"
    ],
    [
      "1.153147478",
      "0.90697349"
    ],
    [
      "1.01672982",
      "0.910512518"
    ],
    [
      "0.880312163",
      "0.914051546"
    ],
    [
      "0.741851366",
      "0.918866863"
    ],
    [
      "0.605433708",
      "0.922405891"
    ],
    [
      "0.46901605",
      "0.925944919"
    ],
    [
      "0.332552547",
      "0.926064536"
    ],
    [
      "0.192542429",
      "0.927581016"
    ],
    [
      "0.068366219",
      "0.984172276"
    ],
    [
      "1.931707318",
      "0.989346681"
    ],
    [
      "1.770122503",
      "1.010689834"
    ],
    [
      "1.633674169",
      "1.012728049"
    ],
    [
      "1.497256511",
      "1.016267077"
    ],
    [
      "1.360838853",
      "1.019806105"
    ],
    [
      "1.224421196",
      "1.023345133"
    ],
    [
      "1.088003538",
      "1.026884161"
    ],
    [
      "0.95158588",
      "1.030423189"
    ],
    [
      "0.815168222",
      "1.033962218"
    ],
    [
      "0.676707425",
      "1.038777534"
    ],
    [
      "0.540289767",
      "1.042316562"
    ],
    [
      "0.263816147",
      "1.043952659"
    ],
    [
      "0.40088789",
      "1.044185588"
    ],
    [
      "1.865248297",
      "1.108533512"
    ],
    [
      "0.076045917",
      "1.120419567"
    ],
    [
      "1.704978562",
      "1.130600505"
    ],
    [
      "1.568530229",
      "1.13263872"
    ],
    [
      "1.432112571",
      "1.136177748"
    ],
    [
      "1.295694913",
      "1.139716776"
    ],
    [
      "1.159277255",
      "1.143255804"
    ],
    [
      "1.022859597",
      "1.146794833"
    ],
    [
      "0.886441939",
      "1.150333861"
    ],
    [
      "0.750024281",
      "1.153872889"
    ],
    [
      "0.613690352",
      "1.159819471"
    ],
    [
      "0.477272694",
      "1.163358499"
    ],
    [
      "0.340850866",
      "1.166732958"
    ],
    [
      "0.204387325",
      "1.166796125"
    ],
    [
      "1.900210824",
      "1.240442283"
    ],
    [
      "1.764369795",
      "1.253462149"
    ],
    [
      "1.627921461",
      "1.255500364"
    ],
    [
      "1.491503804",
      "1.259039392"
    ],
    [
      "0.105478982",
      "1.260814431"
    ],
    [
      "1.355086146",
      "1.26257842"
    ],
    [
      "1.218668488",
      "1.266117448"
    ],
    [
      "1.08225083",
      "1.269656477"
    ],
    [
      "0.945833172",
      "1.273195505"
    ],
    [
      "0.809415514",
      "1.276734533"
    ],
    [
      "0.673081585",
      "1.282681115"
    ],
    [
      "0.536663927",
      "1.286220143"
    ],
    [
      "0.388539817",
      "1.294592514"
    ],
    [
      "0.252076275",
      "1.294655682"
    ],
    [
      "1.853737027",
      "1.371271762"
    ],
    [
      "1.562777521",
      "1.375411035"
    ],
    [
      "1.426359863",
      "1.378950063"
    ],
    [
      "1.716486246",
      "1.381248955"
    ],
    [
      "1.289942205",
      "1.382489092"
    ],
    [
      "1.153524547",
      "1.38602812"
    ],
    [
      "0.153167932",
      "1.388673987"
    ],
    [
      "1.017106889",
      "1.389567148"
    ],
    [
      "0.880689231",
      "1.393106176"
    ],
    [
      "0.744355302",
      "1.399052758"
    ],
    [
      "0.607937644",
      "1.402591786"
    ],
    [
      "0.469065706",
      "1.404764576"
    ],
    [
      "0.322347467",
      "1.41626106"
    ],
    [
      "1.635353582",
      "1.490974946"
    ],
    [
      "1.49763358",
      "1.495321707"
    ],
    [
      "1.787759963",
      "1.497620598"
    ],
    [
      "1.361215922",
      "1.498860735"
    ],
    [
      "1.224798264",
      "1.502399763"
    ],
    [
      "1.088380607",
      "1.505938791"
    ],
    [
      "0.219021024",
      "1.508196673"
    ],
    [
      "0.951962949",
      "1.509477819"
    ],
    [
      "0.815629019",
      "1.515424401"
    ],
    [
      "0.540339423",
      "1.52113622"
    ],
    [
      "0.678266067",
      "1.52148319"
    ],
    [
      "0.40395014",
      "1.525637961"
    ],
    [
      "1.7066273",
      "1.607346589"
    ],
    [
      "1.570209642",
      "1.610885617"
    ],
    [
      "0.30162574",
      "1.616818784"
    ],
    [
      "1.296071982",
      "1.618771406"
    ],
    [
      "1.023236666",
      "1.625849462"
    ],
    [
      "0.753162208",
      "1.636751231"
    ],
    [
      "1.160746426",
      "1.638598368"
    ],
    [
      "1.431546554",
      "1.642775122"
    ],
    [
      "0.889124916",
      "1.65380503"
    ],
    [
      "0.523827516",
      "1.656597133"
    ],
    [
      "0.399210248",
      "1.712210428"
    ],
    [
      "0.644815775",
      "1.719717204"
    ],
    [
      "1.5572939",
      "1.746736586"
    ],
    [
      "1.323780408",
      "1.752392307"
    ],
    [
      "1.057747096",
      "1.757877233"
    ],
    [
      "0.791835676",
      "1.767620116"
    ],
    [
      "1.191389436",
      "1.785482073"
    ],
    [
      "0.509681409",
      "1.7923255"
    ],
    [
      "0.925902789",
      "1.793082072"
    ],
    [
      "1.442246216",
      "1.820128224"
    ],
    [
      "0.630669667",
      "1.855445571"
    ],
    [
      "1.291524617",
      "1.884988936"
    ],
    [
      "1.025491305",
      "1.890473862"
    ],
    [
      "0.759579885",
      "1.900216745"
    ],
    [
      "1.159133645",
      "1.918078702"
    ],
    [
      "0.893646998",
      "1.925678702"
    ]
  ],
  "radius": "0.068231777"
}

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

讨论区

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?