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 = 104

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 = 104
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[0.0883574985178430719009729311175, 0.0883574985178430719009729311185]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 = 0.088357498

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 = 104 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": [
    [
      "1.027551065",
      "0.088773909"
    ],
    [
      "0.838211302",
      "0.10282863"
    ],
    [
      "1.250420758",
      "0.123426276"
    ],
    [
      "0.668159778",
      "0.150898052"
    ],
    [
      "1.414833212",
      "0.188208489"
    ],
    [
      "0.510577113",
      "0.230872385"
    ],
    [
      "1.118083269",
      "0.240537245"
    ],
    [
      "0.941460424",
      "0.246243444"
    ],
    [
      "1.282495723",
      "0.305319459"
    ],
    [
      "0.371384458",
      "0.339746601"
    ],
    [
      "0.649339212",
      "0.340294823"
    ],
    [
      "1.471930328",
      "0.355445177"
    ],
    [
      "1.648601752",
      "0.359369204"
    ],
    [
      "0.822262679",
      "0.376704502"
    ],
    [
      "1.004735903",
      "0.411241637"
    ],
    [
      "1.1776727",
      "0.447587952"
    ],
    [
      "0.510146557",
      "0.44916904"
    ],
    [
      "0.255811961",
      "0.473429765"
    ],
    [
      "1.350609496",
      "0.483934266"
    ],
    [
      "1.760012974",
      "0.496539644"
    ],
    [
      "0.705586993",
      "0.509425919"
    ],
    [
      "1.526921895",
      "0.523385993"
    ],
    [
      "0.885693042",
      "0.541860551"
    ],
    [
      "1.05972747",
      "0.579182454"
    ],
    [
      "0.391273355",
      "0.586912806"
    ],
    [
      "1.232664267",
      "0.615528768"
    ],
    [
      "0.565404339",
      "0.617022451"
    ],
    [
      "0.168202246",
      "0.626898745"
    ],
    [
      "1.658588559",
      "0.64125066"
    ],
    [
      "1.405601064",
      "0.651875082"
    ],
    [
      "1.847040869",
      "0.663083266"
    ],
    [
      "0.768845444",
      "0.674430642"
    ],
    [
      "0.941782241",
      "0.710776956"
    ],
    [
      "0.30366364",
      "0.740381787"
    ],
    [
      "1.114719037",
      "0.74712327"
    ],
    [
      "1.537267727",
      "0.769739749"
    ],
    [
      "0.477794624",
      "0.770491432"
    ],
    [
      "1.287655834",
      "0.783469584"
    ],
    [
      "0.111847237",
      "0.794386963"
    ],
    [
      "1.735405815",
      "0.800396189"
    ],
    [
      "0.650900214",
      "0.806025144"
    ],
    [
      "0.823837011",
      "0.842371458"
    ],
    [
      "1.901107378",
      "0.861806139"
    ],
    [
      "0.996773808",
      "0.878717772"
    ],
    [
      "1.419322497",
      "0.901334251"
    ],
    [
      "0.358655207",
      "0.908322603"
    ],
    [
      "1.169710604",
      "0.915064086"
    ],
    [
      "1.592259294",
      "0.937680566"
    ],
    [
      "0.532786191",
      "0.938432248"
    ],
    [
      "0.088864469",
      "0.969601068"
    ],
    [
      "0.705891781",
      "0.97396596"
    ],
    [
      "1.76507402",
      "0.974602928"
    ],
    [
      "0.878828578",
      "1.010312274"
    ],
    [
      "1.301377268",
      "1.032928754"
    ],
    [
      "0.248116556",
      "1.046197172"
    ],
    [
      "1.051765375",
      "1.046658588"
    ],
    [
      "1.474681946",
      "1.069604881"
    ],
    [
      "0.422247539",
      "1.076306817"
    ],
    [
      "1.908288371",
      "1.078129928"
    ],
    [
      "1.649909894",
      "1.108638052"
    ],
    [
      "0.596024125",
      "1.112375816"
    ],
    [
      "0.768960922",
      "1.14872213"
    ],
    [
      "0.106547595",
      "1.157414879"
    ],
    [
      "1.183432038",
      "1.164523256"
    ],
    [
      "0.950720578",
      "1.191695812"
    ],
    [
      "1.356368835",
      "1.20086957"
    ],
    [
      "0.277051449",
      "1.22052721"
    ],
    [
      "1.529202425",
      "1.237703527"
    ],
    [
      "0.478078896",
      "1.243970318"
    ],
    [
      "1.876150361",
      "1.251897987"
    ],
    [
      "1.70133014",
      "1.277706525"
    ],
    [
      "0.651015692",
      "1.280316632"
    ],
    [
      "1.083191501",
      "1.310056842"
    ],
    [
      "0.824780549",
      "1.316389537"
    ],
    [
      "1.256128297",
      "1.346403156"
    ],
    [
      "0.159106219",
      "1.352121712"
    ],
    [
      "0.360784476",
      "1.376145237"
    ],
    [
      "1.431424563",
      "1.384902984"
    ],
    [
      "0.533721272",
      "1.412491551"
    ],
    [
      "1.811091099",
      "1.416201008"
    ],
    [
      "1.605794222",
      "1.426370839"
    ],
    [
      "0.95693854",
      "1.433703031"
    ],
    [
      "0.70683532",
      "1.447984039"
    ],
    [
      "1.138183068",
      "1.477997658"
    ],
    [
      "0.242839246",
      "1.507739739"
    ],
    [
      "1.311119864",
      "1.514343972"
    ],
    [
      "0.415776043",
      "1.544086053"
    ],
    [
      "1.483371398",
      "1.553810403"
    ],
    [
      "1.715555181",
      "1.564865323"
    ],
    [
      "0.83899331",
      "1.565297533"
    ],
    [
      "0.588478096",
      "1.584501833"
    ],
    [
      "1.011930107",
      "1.601643847"
    ],
    [
      "1.304678164",
      "1.690941523"
    ],
    [
      "1.593132358",
      "1.692304887"
    ],
    [
      "0.723311904",
      "1.698886464"
    ],
    [
      "0.439742109",
      "1.719168371"
    ],
    [
      "1.132521126",
      "1.730818134"
    ],
    [
      "0.896248701",
      "1.735232778"
    ],
    [
      "1.448422653",
      "1.793731173"
    ],
    [
      "0.589016954",
      "1.813747495"
    ],
    [
      "1.283219819",
      "1.866437458"
    ],
    [
      "0.754874123",
      "1.878027183"
    ],
    [
      "1.105730455",
      "1.905490542"
    ],
    [
      "0.929047914",
      "1.908877248"
    ]
  ],
  "radius": "0.088357498"
}
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": 104
}

起步布局答案

{
  "centers": [
    [
      "1.027551065",
      "0.088773909"
    ],
    [
      "0.838211302",
      "0.10282863"
    ],
    [
      "1.250420758",
      "0.123426276"
    ],
    [
      "0.668159778",
      "0.150898052"
    ],
    [
      "1.414833212",
      "0.188208489"
    ],
    [
      "0.510577113",
      "0.230872385"
    ],
    [
      "1.118083269",
      "0.240537245"
    ],
    [
      "0.941460424",
      "0.246243444"
    ],
    [
      "1.282495723",
      "0.305319459"
    ],
    [
      "0.371384458",
      "0.339746601"
    ],
    [
      "0.649339212",
      "0.340294823"
    ],
    [
      "1.471930328",
      "0.355445177"
    ],
    [
      "1.648601752",
      "0.359369204"
    ],
    [
      "0.822262679",
      "0.376704502"
    ],
    [
      "1.004735903",
      "0.411241637"
    ],
    [
      "1.1776727",
      "0.447587952"
    ],
    [
      "0.510146557",
      "0.44916904"
    ],
    [
      "0.255811961",
      "0.473429765"
    ],
    [
      "1.350609496",
      "0.483934266"
    ],
    [
      "1.760012974",
      "0.496539644"
    ],
    [
      "0.705586993",
      "0.509425919"
    ],
    [
      "1.526921895",
      "0.523385993"
    ],
    [
      "0.885693042",
      "0.541860551"
    ],
    [
      "1.05972747",
      "0.579182454"
    ],
    [
      "0.391273355",
      "0.586912806"
    ],
    [
      "1.232664267",
      "0.615528768"
    ],
    [
      "0.565404339",
      "0.617022451"
    ],
    [
      "0.168202246",
      "0.626898745"
    ],
    [
      "1.658588559",
      "0.64125066"
    ],
    [
      "1.405601064",
      "0.651875082"
    ],
    [
      "1.847040869",
      "0.663083266"
    ],
    [
      "0.768845444",
      "0.674430642"
    ],
    [
      "0.941782241",
      "0.710776956"
    ],
    [
      "0.30366364",
      "0.740381787"
    ],
    [
      "1.114719037",
      "0.74712327"
    ],
    [
      "1.537267727",
      "0.769739749"
    ],
    [
      "0.477794624",
      "0.770491432"
    ],
    [
      "1.287655834",
      "0.783469584"
    ],
    [
      "0.111847237",
      "0.794386963"
    ],
    [
      "1.735405815",
      "0.800396189"
    ],
    [
      "0.650900214",
      "0.806025144"
    ],
    [
      "0.823837011",
      "0.842371458"
    ],
    [
      "1.901107378",
      "0.861806139"
    ],
    [
      "0.996773808",
      "0.878717772"
    ],
    [
      "1.419322497",
      "0.901334251"
    ],
    [
      "0.358655207",
      "0.908322603"
    ],
    [
      "1.169710604",
      "0.915064086"
    ],
    [
      "1.592259294",
      "0.937680566"
    ],
    [
      "0.532786191",
      "0.938432248"
    ],
    [
      "0.088864469",
      "0.969601068"
    ],
    [
      "0.705891781",
      "0.97396596"
    ],
    [
      "1.76507402",
      "0.974602928"
    ],
    [
      "0.878828578",
      "1.010312274"
    ],
    [
      "1.301377268",
      "1.032928754"
    ],
    [
      "0.248116556",
      "1.046197172"
    ],
    [
      "1.051765375",
      "1.046658588"
    ],
    [
      "1.474681946",
      "1.069604881"
    ],
    [
      "0.422247539",
      "1.076306817"
    ],
    [
      "1.908288371",
      "1.078129928"
    ],
    [
      "1.649909894",
      "1.108638052"
    ],
    [
      "0.596024125",
      "1.112375816"
    ],
    [
      "0.768960922",
      "1.14872213"
    ],
    [
      "0.106547595",
      "1.157414879"
    ],
    [
      "1.183432038",
      "1.164523256"
    ],
    [
      "0.950720578",
      "1.191695812"
    ],
    [
      "1.356368835",
      "1.20086957"
    ],
    [
      "0.277051449",
      "1.22052721"
    ],
    [
      "1.529202425",
      "1.237703527"
    ],
    [
      "0.478078896",
      "1.243970318"
    ],
    [
      "1.876150361",
      "1.251897987"
    ],
    [
      "1.70133014",
      "1.277706525"
    ],
    [
      "0.651015692",
      "1.280316632"
    ],
    [
      "1.083191501",
      "1.310056842"
    ],
    [
      "0.824780549",
      "1.316389537"
    ],
    [
      "1.256128297",
      "1.346403156"
    ],
    [
      "0.159106219",
      "1.352121712"
    ],
    [
      "0.360784476",
      "1.376145237"
    ],
    [
      "1.431424563",
      "1.384902984"
    ],
    [
      "0.533721272",
      "1.412491551"
    ],
    [
      "1.811091099",
      "1.416201008"
    ],
    [
      "1.605794222",
      "1.426370839"
    ],
    [
      "0.95693854",
      "1.433703031"
    ],
    [
      "0.70683532",
      "1.447984039"
    ],
    [
      "1.138183068",
      "1.477997658"
    ],
    [
      "0.242839246",
      "1.507739739"
    ],
    [
      "1.311119864",
      "1.514343972"
    ],
    [
      "0.415776043",
      "1.544086053"
    ],
    [
      "1.483371398",
      "1.553810403"
    ],
    [
      "1.715555181",
      "1.564865323"
    ],
    [
      "0.83899331",
      "1.565297533"
    ],
    [
      "0.588478096",
      "1.584501833"
    ],
    [
      "1.011930107",
      "1.601643847"
    ],
    [
      "1.304678164",
      "1.690941523"
    ],
    [
      "1.593132358",
      "1.692304887"
    ],
    [
      "0.723311904",
      "1.698886464"
    ],
    [
      "0.439742109",
      "1.719168371"
    ],
    [
      "1.132521126",
      "1.730818134"
    ],
    [
      "0.896248701",
      "1.735232778"
    ],
    [
      "1.448422653",
      "1.793731173"
    ],
    [
      "0.589016954",
      "1.813747495"
    ],
    [
      "1.283219819",
      "1.866437458"
    ],
    [
      "0.754874123",
      "1.878027183"
    ],
    [
      "1.105730455",
      "1.905490542"
    ],
    [
      "0.929047914",
      "1.908877248"
    ]
  ],
  "radius": "0.088357498"
}

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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