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 = 135

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 = 135
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[0.0780380925643413628143289274855, 0.0780380925643413628143289274865]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 = 0.078038092

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 = 135 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": [
    [
      "0.944987045",
      "0.079680852"
    ],
    [
      "1.148830512",
      "0.090182229"
    ],
    [
      "0.788346785",
      "0.102661338"
    ],
    [
      "1.300203701",
      "0.128282444"
    ],
    [
      "0.582056926",
      "0.178210887"
    ],
    [
      "1.44294295",
      "0.191411537"
    ],
    [
      "1.040798198",
      "0.202887989"
    ],
    [
      "0.886192111",
      "0.224259313"
    ],
    [
      "1.192168638",
      "0.240924358"
    ],
    [
      "0.723332104",
      "0.244551671"
    ],
    [
      "0.449426675",
      "0.260484772"
    ],
    [
      "1.572988297",
      "0.277713236"
    ],
    [
      "1.334907887",
      "0.304053451"
    ],
    [
      "1.085377259",
      "0.354746048"
    ],
    [
      "0.62196017",
      "0.363225622"
    ],
    [
      "0.332574807",
      "0.363951755"
    ],
    [
      "0.930483209",
      "0.373919175"
    ],
    [
      "1.686612891",
      "0.384714297"
    ],
    [
      "0.775420793",
      "0.391679317"
    ],
    [
      "1.479423603",
      "0.402634905"
    ],
    [
      "0.477414308",
      "0.422100654"
    ],
    [
      "1.33432275",
      "0.460128539"
    ],
    [
      "1.179428699",
      "0.479301666"
    ],
    [
      "0.234850074",
      "0.485646669"
    ],
    [
      "1.024534649",
      "0.498474794"
    ],
    [
      "1.780560468",
      "0.509348275"
    ],
    [
      "1.593048197",
      "0.509635967"
    ],
    [
      "0.868332735",
      "0.517087237"
    ],
    [
      "0.713296631",
      "0.535101928"
    ],
    [
      "0.384771371",
      "0.548806723"
    ],
    [
      "0.558544635",
      "0.555433492"
    ],
    [
      "1.447947344",
      "0.567129601"
    ],
    [
      "1.27348014",
      "0.603857285"
    ],
    [
      "1.118586089",
      "0.623030412"
    ],
    [
      "1.686995773",
      "0.634269945"
    ],
    [
      "0.208151007",
      "0.63942227"
    ],
    [
      "0.962384175",
      "0.641642856"
    ],
    [
      "1.852138667",
      "0.648043397"
    ],
    [
      "0.807490125",
      "0.660815983"
    ],
    [
      "0.652596075",
      "0.67998911"
    ],
    [
      "1.542214579",
      "0.693244242"
    ],
    [
      "0.497702025",
      "0.699162237"
    ],
    [
      "1.387104734",
      "0.710858347"
    ],
    [
      "0.342807975",
      "0.718335364"
    ],
    [
      "1.21263753",
      "0.747586031"
    ],
    [
      "0.109684345",
      "0.760517654"
    ],
    [
      "1.056435616",
      "0.766198475"
    ],
    [
      "1.758573973",
      "0.772965067"
    ],
    [
      "0.901541565",
      "0.785371602"
    ],
    [
      "0.746647515",
      "0.804544729"
    ],
    [
      "0.591753465",
      "0.823717856"
    ],
    [
      "1.9053831",
      "0.825944834"
    ],
    [
      "1.615179139",
      "0.834590527"
    ],
    [
      "0.244341313",
      "0.839430749"
    ],
    [
      "0.436859415",
      "0.842890983"
    ],
    [
      "1.459768419",
      "0.848987867"
    ],
    [
      "1.305549472",
      "0.872993952"
    ],
    [
      "1.150487056",
      "0.890754094"
    ],
    [
      "0.995593006",
      "0.909927221"
    ],
    [
      "0.082045958",
      "0.914127208"
    ],
    [
      "0.840698956",
      "0.929100348"
    ],
    [
      "1.737028053",
      "0.936731752"
    ],
    [
      "0.685804905",
      "0.948273475"
    ],
    [
      "0.530910855",
      "0.967446602"
    ],
    [
      "1.554496241",
      "0.978386776"
    ],
    [
      "1.921769361",
      "0.981158448"
    ],
    [
      "0.376016805",
      "0.986619729"
    ],
    [
      "0.216702926",
      "0.993040303"
    ],
    [
      "1.399600912",
      "0.99754957"
    ],
    [
      "1.244706862",
      "1.016722697"
    ],
    [
      "1.089644446",
      "1.03448284"
    ],
    [
      "0.934750396",
      "1.053655967"
    ],
    [
      "0.080714386",
      "1.070197713"
    ],
    [
      "0.779856346",
      "1.072829094"
    ],
    [
      "1.803317316",
      "1.082789588"
    ],
    [
      "0.624962295",
      "1.092002221"
    ],
    [
      "1.648547681",
      "1.102942395"
    ],
    [
      "0.470068245",
      "1.111175348"
    ],
    [
      "1.493653631",
      "1.122115522"
    ],
    [
      "0.301763118",
      "1.123901066"
    ],
    [
      "1.338758302",
      "1.141278316"
    ],
    [
      "1.183864252",
      "1.160451443"
    ],
    [
      "1.028801836",
      "1.178211585"
    ],
    [
      "0.873907786",
      "1.197384712"
    ],
    [
      "1.898551601",
      "1.206443163"
    ],
    [
      "0.719013736",
      "1.21655784"
    ],
    [
      "0.105727789",
      "1.224256485"
    ],
    [
      "1.743627916",
      "1.227928224"
    ],
    [
      "0.564119686",
      "1.235730967"
    ],
    [
      "1.587705071",
      "1.246671141"
    ],
    [
      "0.409225635",
      "1.254904094"
    ],
    [
      "1.432811021",
      "1.265844268"
    ],
    [
      "0.2541487",
      "1.272537006"
    ],
    [
      "1.277915692",
      "1.285007062"
    ],
    [
      "1.123021642",
      "1.304180189"
    ],
    [
      "0.967959226",
      "1.321940331"
    ],
    [
      "0.813065176",
      "1.341113458"
    ],
    [
      "1.850853482",
      "1.355052265"
    ],
    [
      "0.658171126",
      "1.360286585"
    ],
    [
      "1.681706111",
      "1.371264801"
    ],
    [
      "0.503277076",
      "1.379459712"
    ],
    [
      "1.527195857",
      "1.3933183"
    ],
    [
      "0.171821123",
      "1.405133936"
    ],
    [
      "1.371968411",
      "1.409573014"
    ],
    [
      "0.32781722",
      "1.410133244"
    ],
    [
      "1.217073082",
      "1.428735808"
    ],
    [
      "1.062179032",
      "1.447908935"
    ],
    [
      "0.907116616",
      "1.465669077"
    ],
    [
      "0.752222566",
      "1.484842204"
    ],
    [
      "1.778771523",
      "1.493486244"
    ],
    [
      "0.597328516",
      "1.504015331"
    ],
    [
      "0.437619148",
      "1.521053541"
    ],
    [
      "1.544549426",
      "1.548426744"
    ],
    [
      "0.261928694",
      "1.552340797"
    ],
    [
      "1.311125801",
      "1.55330176"
    ],
    [
      "1.156230472",
      "1.572464554"
    ],
    [
      "1.001735732",
      "1.594766261"
    ],
    [
      "0.846274006",
      "1.609397823"
    ],
    [
      "1.684371455",
      "1.617777848"
    ],
    [
      "0.690220589",
      "1.641954751"
    ],
    [
      "0.531670589",
      "1.645609159"
    ],
    [
      "1.430000734",
      "1.654437935"
    ],
    [
      "0.369519578",
      "1.670186955"
    ],
    [
      "1.273319878",
      "1.716447907"
    ],
    [
      "1.098967541",
      "1.717656607"
    ],
    [
      "1.570358605",
      "1.724365115"
    ],
    [
      "0.797090458",
      "1.75752194"
    ],
    [
      "0.952526341",
      "1.771645038"
    ],
    [
      "0.54384574",
      "1.801209742"
    ],
    [
      "1.440000363",
      "1.810193458"
    ],
    [
      "0.685529656",
      "1.866673042"
    ],
    [
      "1.141031601",
      "1.867957606"
    ],
    [
      "1.297032549",
      "1.872803199"
    ],
    [
      "0.839154519",
      "1.90782294"
    ],
    [
      "0.994590402",
      "1.921946037"
    ]
  ],
  "radius": "0.078038092"
}
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": 135
}

起步布局答案

{
  "centers": [
    [
      "0.944987045",
      "0.079680852"
    ],
    [
      "1.148830512",
      "0.090182229"
    ],
    [
      "0.788346785",
      "0.102661338"
    ],
    [
      "1.300203701",
      "0.128282444"
    ],
    [
      "0.582056926",
      "0.178210887"
    ],
    [
      "1.44294295",
      "0.191411537"
    ],
    [
      "1.040798198",
      "0.202887989"
    ],
    [
      "0.886192111",
      "0.224259313"
    ],
    [
      "1.192168638",
      "0.240924358"
    ],
    [
      "0.723332104",
      "0.244551671"
    ],
    [
      "0.449426675",
      "0.260484772"
    ],
    [
      "1.572988297",
      "0.277713236"
    ],
    [
      "1.334907887",
      "0.304053451"
    ],
    [
      "1.085377259",
      "0.354746048"
    ],
    [
      "0.62196017",
      "0.363225622"
    ],
    [
      "0.332574807",
      "0.363951755"
    ],
    [
      "0.930483209",
      "0.373919175"
    ],
    [
      "1.686612891",
      "0.384714297"
    ],
    [
      "0.775420793",
      "0.391679317"
    ],
    [
      "1.479423603",
      "0.402634905"
    ],
    [
      "0.477414308",
      "0.422100654"
    ],
    [
      "1.33432275",
      "0.460128539"
    ],
    [
      "1.179428699",
      "0.479301666"
    ],
    [
      "0.234850074",
      "0.485646669"
    ],
    [
      "1.024534649",
      "0.498474794"
    ],
    [
      "1.780560468",
      "0.509348275"
    ],
    [
      "1.593048197",
      "0.509635967"
    ],
    [
      "0.868332735",
      "0.517087237"
    ],
    [
      "0.713296631",
      "0.535101928"
    ],
    [
      "0.384771371",
      "0.548806723"
    ],
    [
      "0.558544635",
      "0.555433492"
    ],
    [
      "1.447947344",
      "0.567129601"
    ],
    [
      "1.27348014",
      "0.603857285"
    ],
    [
      "1.118586089",
      "0.623030412"
    ],
    [
      "1.686995773",
      "0.634269945"
    ],
    [
      "0.208151007",
      "0.63942227"
    ],
    [
      "0.962384175",
      "0.641642856"
    ],
    [
      "1.852138667",
      "0.648043397"
    ],
    [
      "0.807490125",
      "0.660815983"
    ],
    [
      "0.652596075",
      "0.67998911"
    ],
    [
      "1.542214579",
      "0.693244242"
    ],
    [
      "0.497702025",
      "0.699162237"
    ],
    [
      "1.387104734",
      "0.710858347"
    ],
    [
      "0.342807975",
      "0.718335364"
    ],
    [
      "1.21263753",
      "0.747586031"
    ],
    [
      "0.109684345",
      "0.760517654"
    ],
    [
      "1.056435616",
      "0.766198475"
    ],
    [
      "1.758573973",
      "0.772965067"
    ],
    [
      "0.901541565",
      "0.785371602"
    ],
    [
      "0.746647515",
      "0.804544729"
    ],
    [
      "0.591753465",
      "0.823717856"
    ],
    [
      "1.9053831",
      "0.825944834"
    ],
    [
      "1.615179139",
      "0.834590527"
    ],
    [
      "0.244341313",
      "0.839430749"
    ],
    [
      "0.436859415",
      "0.842890983"
    ],
    [
      "1.459768419",
      "0.848987867"
    ],
    [
      "1.305549472",
      "0.872993952"
    ],
    [
      "1.150487056",
      "0.890754094"
    ],
    [
      "0.995593006",
      "0.909927221"
    ],
    [
      "0.082045958",
      "0.914127208"
    ],
    [
      "0.840698956",
      "0.929100348"
    ],
    [
      "1.737028053",
      "0.936731752"
    ],
    [
      "0.685804905",
      "0.948273475"
    ],
    [
      "0.530910855",
      "0.967446602"
    ],
    [
      "1.554496241",
      "0.978386776"
    ],
    [
      "1.921769361",
      "0.981158448"
    ],
    [
      "0.376016805",
      "0.986619729"
    ],
    [
      "0.216702926",
      "0.993040303"
    ],
    [
      "1.399600912",
      "0.99754957"
    ],
    [
      "1.244706862",
      "1.016722697"
    ],
    [
      "1.089644446",
      "1.03448284"
    ],
    [
      "0.934750396",
      "1.053655967"
    ],
    [
      "0.080714386",
      "1.070197713"
    ],
    [
      "0.779856346",
      "1.072829094"
    ],
    [
      "1.803317316",
      "1.082789588"
    ],
    [
      "0.624962295",
      "1.092002221"
    ],
    [
      "1.648547681",
      "1.102942395"
    ],
    [
      "0.470068245",
      "1.111175348"
    ],
    [
      "1.493653631",
      "1.122115522"
    ],
    [
      "0.301763118",
      "1.123901066"
    ],
    [
      "1.338758302",
      "1.141278316"
    ],
    [
      "1.183864252",
      "1.160451443"
    ],
    [
      "1.028801836",
      "1.178211585"
    ],
    [
      "0.873907786",
      "1.197384712"
    ],
    [
      "1.898551601",
      "1.206443163"
    ],
    [
      "0.719013736",
      "1.21655784"
    ],
    [
      "0.105727789",
      "1.224256485"
    ],
    [
      "1.743627916",
      "1.227928224"
    ],
    [
      "0.564119686",
      "1.235730967"
    ],
    [
      "1.587705071",
      "1.246671141"
    ],
    [
      "0.409225635",
      "1.254904094"
    ],
    [
      "1.432811021",
      "1.265844268"
    ],
    [
      "0.2541487",
      "1.272537006"
    ],
    [
      "1.277915692",
      "1.285007062"
    ],
    [
      "1.123021642",
      "1.304180189"
    ],
    [
      "0.967959226",
      "1.321940331"
    ],
    [
      "0.813065176",
      "1.341113458"
    ],
    [
      "1.850853482",
      "1.355052265"
    ],
    [
      "0.658171126",
      "1.360286585"
    ],
    [
      "1.681706111",
      "1.371264801"
    ],
    [
      "0.503277076",
      "1.379459712"
    ],
    [
      "1.527195857",
      "1.3933183"
    ],
    [
      "0.171821123",
      "1.405133936"
    ],
    [
      "1.371968411",
      "1.409573014"
    ],
    [
      "0.32781722",
      "1.410133244"
    ],
    [
      "1.217073082",
      "1.428735808"
    ],
    [
      "1.062179032",
      "1.447908935"
    ],
    [
      "0.907116616",
      "1.465669077"
    ],
    [
      "0.752222566",
      "1.484842204"
    ],
    [
      "1.778771523",
      "1.493486244"
    ],
    [
      "0.597328516",
      "1.504015331"
    ],
    [
      "0.437619148",
      "1.521053541"
    ],
    [
      "1.544549426",
      "1.548426744"
    ],
    [
      "0.261928694",
      "1.552340797"
    ],
    [
      "1.311125801",
      "1.55330176"
    ],
    [
      "1.156230472",
      "1.572464554"
    ],
    [
      "1.001735732",
      "1.594766261"
    ],
    [
      "0.846274006",
      "1.609397823"
    ],
    [
      "1.684371455",
      "1.617777848"
    ],
    [
      "0.690220589",
      "1.641954751"
    ],
    [
      "0.531670589",
      "1.645609159"
    ],
    [
      "1.430000734",
      "1.654437935"
    ],
    [
      "0.369519578",
      "1.670186955"
    ],
    [
      "1.273319878",
      "1.716447907"
    ],
    [
      "1.098967541",
      "1.717656607"
    ],
    [
      "1.570358605",
      "1.724365115"
    ],
    [
      "0.797090458",
      "1.75752194"
    ],
    [
      "0.952526341",
      "1.771645038"
    ],
    [
      "0.54384574",
      "1.801209742"
    ],
    [
      "1.440000363",
      "1.810193458"
    ],
    [
      "0.685529656",
      "1.866673042"
    ],
    [
      "1.141031601",
      "1.867957606"
    ],
    [
      "1.297032549",
      "1.872803199"
    ],
    [
      "0.839154519",
      "1.90782294"
    ],
    [
      "0.994590402",
      "1.921946037"
    ]
  ],
  "radius": "0.078038092"
}

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?