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 = 176

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 = 176
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[0.0686147037244135061754508488425, 0.0686147037244135061754508488435]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 = 0.068614703

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 = 176 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": [
    [
      "1.011270316",
      "0.06893762"
    ],
    [
      "0.874283634",
      "0.077138166"
    ],
    [
      "1.151124261",
      "0.080957004"
    ],
    [
      "0.74004412",
      "0.105627868"
    ],
    [
      "1.284526852",
      "0.113138658"
    ],
    [
      "0.54278902",
      "0.188558881"
    ],
    [
      "0.945454714",
      "0.194469268"
    ],
    [
      "1.082578938",
      "0.199841187"
    ],
    [
      "1.488316452",
      "0.20688865"
    ],
    [
      "0.811952093",
      "0.226233407"
    ],
    [
      "0.674722709",
      "0.226313508"
    ],
    [
      "1.353375805",
      "0.231847259"
    ],
    [
      "1.215981529",
      "0.232022841"
    ],
    [
      "0.473419899",
      "0.306964266"
    ],
    [
      "1.014033615",
      "0.31872537"
    ],
    [
      "1.557000513",
      "0.325692733"
    ],
    [
      "0.340755991",
      "0.342066943"
    ],
    [
      "0.879481038",
      "0.345697845"
    ],
    [
      "0.742251654",
      "0.345777945"
    ],
    [
      "0.60502227",
      "0.345858046"
    ],
    [
      "1.422059866",
      "0.350651341"
    ],
    [
      "1.284830482",
      "0.350731442"
    ],
    [
      "1.147436207",
      "0.350907024"
    ],
    [
      "1.686636724",
      "0.370707875"
    ],
    [
      "0.248520704",
      "0.450118224"
    ],
    [
      "0.529159257",
      "0.462195384"
    ],
    [
      "0.941672834",
      "0.468025645"
    ],
    [
      "0.80444345",
      "0.468105746"
    ],
    [
      "0.667214066",
      "0.468185846"
    ],
    [
      "1.490743927",
      "0.469455424"
    ],
    [
      "1.353514543",
      "0.469535524"
    ],
    [
      "1.216285159",
      "0.469615625"
    ],
    [
      "1.078890884",
      "0.469791207"
    ],
    [
      "0.389334714",
      "0.470410303"
    ],
    [
      "1.771651045",
      "0.478431946"
    ],
    [
      "1.625810984",
      "0.493720637"
    ],
    [
      "0.173022017",
      "0.571530647"
    ],
    [
      "0.309831086",
      "0.582263258"
    ],
    [
      "0.873127511",
      "0.586909828"
    ],
    [
      "0.735898127",
      "0.586989929"
    ],
    [
      "0.598668743",
      "0.587070029"
    ],
    [
      "0.461439372",
      "0.587170007"
    ],
    [
      "1.422198604",
      "0.588339607"
    ],
    [
      "1.28496922",
      "0.588419707"
    ],
    [
      "1.147739836",
      "0.588499808"
    ],
    [
      "1.010345561",
      "0.58867539"
    ],
    [
      "1.710825305",
      "0.601444708"
    ],
    [
      "1.557265661",
      "0.61260482"
    ],
    [
      "1.848034995",
      "0.615662074"
    ],
    [
      "0.119039647",
      "0.697696466"
    ],
    [
      "0.804747323",
      "0.705889071"
    ],
    [
      "0.667517939",
      "0.705969172"
    ],
    [
      "0.530288555",
      "0.706049272"
    ],
    [
      "0.393059184",
      "0.70614925"
    ],
    [
      "1.353653282",
      "0.70722379"
    ],
    [
      "1.216423897",
      "0.70730389"
    ],
    [
      "1.079194513",
      "0.707383991"
    ],
    [
      "0.941965373",
      "0.707654632"
    ],
    [
      "0.255848716",
      "0.708429077"
    ],
    [
      "1.763144122",
      "0.728309406"
    ],
    [
      "1.625949722",
      "0.731408902"
    ],
    [
      "1.488720338",
      "0.731489003"
    ],
    [
      "1.898094986",
      "0.75321271"
    ],
    [
      "0.736202",
      "0.824773254"
    ],
    [
      "0.598972616",
      "0.824853355"
    ],
    [
      "0.461743245",
      "0.824953333"
    ],
    [
      "1.285273094",
      "0.826203033"
    ],
    [
      "1.14804371",
      "0.826283133"
    ],
    [
      "1.010814326",
      "0.826363234"
    ],
    [
      "0.87342005",
      "0.826538815"
    ],
    [
      "0.324532777",
      "0.82723316"
    ],
    [
      "0.08418186",
      "0.83042493"
    ],
    [
      "1.557569534",
      "0.850388145"
    ],
    [
      "1.42034015",
      "0.850468245"
    ],
    [
      "1.706608501",
      "0.854685152"
    ],
    [
      "1.863153628",
      "0.885919198"
    ],
    [
      "0.205236832",
      "0.895059318"
    ],
    [
      "0.667656678",
      "0.943657437"
    ],
    [
      "0.529096708",
      "0.944516794"
    ],
    [
      "1.216727771",
      "0.945087216"
    ],
    [
      "1.079498387",
      "0.945167316"
    ],
    [
      "0.942269003",
      "0.945247417"
    ],
    [
      "0.804874727",
      "0.945422998"
    ],
    [
      "0.39188624",
      "0.946796621"
    ],
    [
      "0.069205377",
      "0.966834663"
    ],
    [
      "1.626253595",
      "0.969192227"
    ],
    [
      "1.489024211",
      "0.969272328"
    ],
    [
      "1.351794827",
      "0.969352428"
    ],
    [
      "1.763071617",
      "0.979810099"
    ],
    [
      "1.931371922",
      "1.004991339"
    ],
    [
      "0.191002235",
      "1.031548462"
    ],
    [
      "0.599111355",
      "1.06254162"
    ],
    [
      "1.148182448",
      "1.063971399"
    ],
    [
      "1.010953064",
      "1.064051499"
    ],
    [
      "0.87372368",
      "1.0641316"
    ],
    [
      "0.736329404",
      "1.064307181"
    ],
    [
      "0.461900886",
      "1.064821447"
    ],
    [
      "0.323844561",
      "1.065969773"
    ],
    [
      "1.557708272",
      "1.08807641"
    ],
    [
      "1.420478888",
      "1.088156511"
    ],
    [
      "1.283249504",
      "1.088236611"
    ],
    [
      "1.694526294",
      "1.098694282"
    ],
    [
      "1.831754016",
      "1.09937451"
    ],
    [
      "0.074435319",
      "1.103964375"
    ],
    [
      "0.216325066",
      "1.166421233"
    ],
    [
      "1.079637125",
      "1.182855582"
    ],
    [
      "0.942407741",
      "1.182935682"
    ],
    [
      "0.805178357",
      "1.183015783"
    ],
    [
      "0.629036137",
      "1.196468532"
    ],
    [
      "0.486393192",
      "1.199847508"
    ],
    [
      "0.349167392",
      "1.200842544"
    ],
    [
      "1.489162949",
      "1.206960593"
    ],
    [
      "1.351933565",
      "1.207040694"
    ],
    [
      "1.214704181",
      "1.207120794"
    ],
    [
      "1.906332344",
      "1.214569923"
    ],
    [
      "1.625980971",
      "1.217578465"
    ],
    [
      "1.763208693",
      "1.218258693"
    ],
    [
      "0.09975815",
      "1.238837146"
    ],
    [
      "0.725105134",
      "1.294461685"
    ],
    [
      "1.011091802",
      "1.301739765"
    ],
    [
      "0.873862418",
      "1.301819865"
    ],
    [
      "0.270216819",
      "1.313086553"
    ],
    [
      "0.560490814",
      "1.315352715"
    ],
    [
      "0.411318249",
      "1.32319115"
    ],
    [
      "1.420617626",
      "1.325844776"
    ],
    [
      "1.283388242",
      "1.325924877"
    ],
    [
      "1.146158858",
      "1.326004977"
    ],
    [
      "1.864966036",
      "1.345416164"
    ],
    [
      "1.588190275",
      "1.349501827"
    ],
    [
      "1.727844375",
      "1.350853094"
    ],
    [
      "0.144624143",
      "1.368525052"
    ],
    [
      "0.793789195",
      "1.413265768"
    ],
    [
      "0.656559811",
      "1.413345868"
    ],
    [
      "0.942546479",
      "1.420623948"
    ],
    [
      "0.491945491",
      "1.434236898"
    ],
    [
      "0.334516373",
      "1.436916178"
    ],
    [
      "1.489301687",
      "1.444648859"
    ],
    [
      "1.352072303",
      "1.444728959"
    ],
    [
      "1.214842919",
      "1.44480906"
    ],
    [
      "1.077613535",
      "1.44488916"
    ],
    [
      "1.656874336",
      "1.46830591"
    ],
    [
      "1.801837026",
      "1.473257012"
    ],
    [
      "0.208059311",
      "1.490212725"
    ],
    [
      "0.862473256",
      "1.53206985"
    ],
    [
      "0.725243872",
      "1.532149951"
    ],
    [
      "0.588014488",
      "1.532230051"
    ],
    [
      "0.415143615",
      "1.547961926"
    ],
    [
      "0.997540312",
      "1.556335063"
    ],
    [
      "1.557985749",
      "1.563452941"
    ],
    [
      "1.420756365",
      "1.563533042"
    ],
    [
      "1.28352698",
      "1.563613142"
    ],
    [
      "1.145610042",
      "1.564088095"
    ],
    [
      "1.72096666",
      "1.589648747"
    ],
    [
      "0.288686554",
      "1.601258473"
    ],
    [
      "0.511212611",
      "1.64595508"
    ],
    [
      "0.793927933",
      "1.650954033"
    ],
    [
      "0.648018901",
      "1.660178069"
    ],
    [
      "1.065083766",
      "1.675791298"
    ],
    [
      "0.927935673",
      "1.680514748"
    ],
    [
      "1.489440426",
      "1.682337124"
    ],
    [
      "1.352211042",
      "1.682417225"
    ],
    [
      "1.214981657",
      "1.682497325"
    ],
    [
      "1.626498802",
      "1.689186348"
    ],
    [
      "0.38475555",
      "1.699251626"
    ],
    [
      "0.728846608",
      "1.771769311"
    ],
    [
      "0.494292981",
      "1.782137444"
    ],
    [
      "1.134897277",
      "1.79393521"
    ],
    [
      "0.862854348",
      "1.801330025"
    ],
    [
      "1.451994023",
      "1.814358627"
    ],
    [
      "1.314764639",
      "1.814438727"
    ],
    [
      "0.999359147",
      "1.815413704"
    ],
    [
      "0.61470836",
      "1.847955731"
    ],
    [
      "0.770250148",
      "1.902603776"
    ],
    [
      "1.213148944",
      "1.906667578"
    ],
    [
      "0.920584218",
      "1.927860281"
    ],
    [
      "1.077610815",
      "1.928146072"
    ]
  ],
  "radius": "0.068614703"
}
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": 176
}

起步布局答案

{
  "centers": [
    [
      "1.011270316",
      "0.06893762"
    ],
    [
      "0.874283634",
      "0.077138166"
    ],
    [
      "1.151124261",
      "0.080957004"
    ],
    [
      "0.74004412",
      "0.105627868"
    ],
    [
      "1.284526852",
      "0.113138658"
    ],
    [
      "0.54278902",
      "0.188558881"
    ],
    [
      "0.945454714",
      "0.194469268"
    ],
    [
      "1.082578938",
      "0.199841187"
    ],
    [
      "1.488316452",
      "0.20688865"
    ],
    [
      "0.811952093",
      "0.226233407"
    ],
    [
      "0.674722709",
      "0.226313508"
    ],
    [
      "1.353375805",
      "0.231847259"
    ],
    [
      "1.215981529",
      "0.232022841"
    ],
    [
      "0.473419899",
      "0.306964266"
    ],
    [
      "1.014033615",
      "0.31872537"
    ],
    [
      "1.557000513",
      "0.325692733"
    ],
    [
      "0.340755991",
      "0.342066943"
    ],
    [
      "0.879481038",
      "0.345697845"
    ],
    [
      "0.742251654",
      "0.345777945"
    ],
    [
      "0.60502227",
      "0.345858046"
    ],
    [
      "1.422059866",
      "0.350651341"
    ],
    [
      "1.284830482",
      "0.350731442"
    ],
    [
      "1.147436207",
      "0.350907024"
    ],
    [
      "1.686636724",
      "0.370707875"
    ],
    [
      "0.248520704",
      "0.450118224"
    ],
    [
      "0.529159257",
      "0.462195384"
    ],
    [
      "0.941672834",
      "0.468025645"
    ],
    [
      "0.80444345",
      "0.468105746"
    ],
    [
      "0.667214066",
      "0.468185846"
    ],
    [
      "1.490743927",
      "0.469455424"
    ],
    [
      "1.353514543",
      "0.469535524"
    ],
    [
      "1.216285159",
      "0.469615625"
    ],
    [
      "1.078890884",
      "0.469791207"
    ],
    [
      "0.389334714",
      "0.470410303"
    ],
    [
      "1.771651045",
      "0.478431946"
    ],
    [
      "1.625810984",
      "0.493720637"
    ],
    [
      "0.173022017",
      "0.571530647"
    ],
    [
      "0.309831086",
      "0.582263258"
    ],
    [
      "0.873127511",
      "0.586909828"
    ],
    [
      "0.735898127",
      "0.586989929"
    ],
    [
      "0.598668743",
      "0.587070029"
    ],
    [
      "0.461439372",
      "0.587170007"
    ],
    [
      "1.422198604",
      "0.588339607"
    ],
    [
      "1.28496922",
      "0.588419707"
    ],
    [
      "1.147739836",
      "0.588499808"
    ],
    [
      "1.010345561",
      "0.58867539"
    ],
    [
      "1.710825305",
      "0.601444708"
    ],
    [
      "1.557265661",
      "0.61260482"
    ],
    [
      "1.848034995",
      "0.615662074"
    ],
    [
      "0.119039647",
      "0.697696466"
    ],
    [
      "0.804747323",
      "0.705889071"
    ],
    [
      "0.667517939",
      "0.705969172"
    ],
    [
      "0.530288555",
      "0.706049272"
    ],
    [
      "0.393059184",
      "0.70614925"
    ],
    [
      "1.353653282",
      "0.70722379"
    ],
    [
      "1.216423897",
      "0.70730389"
    ],
    [
      "1.079194513",
      "0.707383991"
    ],
    [
      "0.941965373",
      "0.707654632"
    ],
    [
      "0.255848716",
      "0.708429077"
    ],
    [
      "1.763144122",
      "0.728309406"
    ],
    [
      "1.625949722",
      "0.731408902"
    ],
    [
      "1.488720338",
      "0.731489003"
    ],
    [
      "1.898094986",
      "0.75321271"
    ],
    [
      "0.736202",
      "0.824773254"
    ],
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  "radius": "0.068614703"
}

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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