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 = 107

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 = 107
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[0.0871683648375563932165238333485, 0.0871683648375563932165238333495]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 = 0.087168364

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 = 107 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": [
    [
      "1.001093536",
      "0.089277126"
    ],
    [
      "0.825162701",
      "0.104068355"
    ],
    [
      "1.186191338",
      "0.10635892"
    ],
    [
      "0.658024151",
      "0.153646579"
    ],
    [
      "1.352687355",
      "0.158053907"
    ],
    [
      "0.503359202",
      "0.234095625"
    ],
    [
      "1.506319067",
      "0.240458957"
    ],
    [
      "1.082944085",
      "0.246834185"
    ],
    [
      "0.908902632",
      "0.25697656"
    ],
    [
      "0.741764081",
      "0.306554785"
    ],
    [
      "1.367540845",
      "0.345976256"
    ],
    [
      "1.641482744",
      "0.350568338"
    ],
    [
      "0.351631164",
      "0.35744304"
    ],
    [
      "1.196399491",
      "0.379201727"
    ],
    [
      "0.518545902",
      "0.407769628"
    ],
    [
      "1.025038133",
      "0.411273219"
    ],
    [
      "0.853191559",
      "0.440633913"
    ],
    [
      "1.502704522",
      "0.456085637"
    ],
    [
      "0.68136189",
      "0.470093375"
    ],
    [
      "1.753248277",
      "0.484365801"
    ],
    [
      "0.241298178",
      "0.492424252"
    ],
    [
      "1.31074427",
      "0.510801753"
    ],
    [
      "1.138318959",
      "0.543790199"
    ],
    [
      "0.964541953",
      "0.574777065"
    ],
    [
      "0.545979947",
      "0.579934283"
    ],
    [
      "1.62271688",
      "0.599995164"
    ],
    [
      "0.792712283",
      "0.604236528"
    ],
    [
      "0.373596783",
      "0.605960037"
    ],
    [
      "1.445907947",
      "0.620911134"
    ],
    [
      "1.837539008",
      "0.636971069"
    ],
    [
      "0.158638913",
      "0.645919338"
    ],
    [
      "1.252836311",
      "0.67524008"
    ],
    [
      "1.077822778",
      "0.707294044"
    ],
    [
      "0.657330341",
      "0.714077435"
    ],
    [
      "0.904139761",
      "0.738315655"
    ],
    [
      "0.480635958",
      "0.743978779"
    ],
    [
      "1.707086873",
      "0.752623722"
    ],
    [
      "0.290937518",
      "0.759455123"
    ],
    [
      "1.533773593",
      "0.771486442"
    ],
    [
      "1.891280424",
      "0.802817852"
    ],
    [
      "1.363696451",
      "0.809788676"
    ],
    [
      "0.106668373",
      "0.812329549"
    ],
    [
      "1.191852864",
      "0.839166842"
    ],
    [
      "0.768757819",
      "0.848156563"
    ],
    [
      "1.017420587",
      "0.870832634"
    ],
    [
      "0.596928149",
      "0.877616026"
    ],
    [
      "0.425524666",
      "0.909461609"
    ],
    [
      "1.647203839",
      "0.916353125"
    ],
    [
      "0.252339373",
      "0.929465353"
    ],
    [
      "1.475060484",
      "0.943920506"
    ],
    [
      "1.303216896",
      "0.973298672"
    ],
    [
      "1.912512303",
      "0.975856873"
    ],
    [
      "0.882038644",
      "0.980673542"
    ],
    [
      "0.087282188",
      "0.985585057"
    ],
    [
      "1.131387545",
      "1.002759993"
    ],
    [
      "0.708355627",
      "1.011695153"
    ],
    [
      "0.536952143",
      "1.043540737"
    ],
    [
      "1.755973922",
      "1.052596893"
    ],
    [
      "0.359130408",
      "1.074963748"
    ],
    [
      "1.583830567",
      "1.080164274"
    ],
    [
      "1.411986979",
      "1.10954244"
    ],
    [
      "0.995982375",
      "1.113312041"
    ],
    [
      "0.183614848",
      "1.130889269"
    ],
    [
      "1.240157628",
      "1.13900376"
    ],
    [
      "0.821636453",
      "1.144212133"
    ],
    [
      "1.900377165",
      "1.150274929"
    ],
    [
      "0.650232969",
      "1.176057716"
    ],
    [
      "0.478829486",
      "1.207903299"
    ],
    [
      "1.710984218",
      "1.221028546"
    ],
    [
      "1.523430143",
      "1.243703517"
    ],
    [
      "0.309918526",
      "1.251058634"
    ],
    [
      "1.107510031",
      "1.25213161"
    ],
    [
      "1.351586556",
      "1.273081683"
    ],
    [
      "0.934917279",
      "1.276729112"
    ],
    [
      "0.137535512",
      "1.299026089"
    ],
    [
      "0.763513795",
      "1.308574695"
    ],
    [
      "1.855387461",
      "1.318706582"
    ],
    [
      "0.592110312",
      "1.340420279"
    ],
    [
      "1.634794176",
      "1.377835347"
    ],
    [
      "0.423199352",
      "1.383575613"
    ],
    [
      "1.221579759",
      "1.389234721"
    ],
    [
      "1.462950588",
      "1.407213513"
    ],
    [
      "1.049387374",
      "1.416494173"
    ],
    [
      "0.25465962",
      "1.428158728"
    ],
    [
      "0.876794621",
      "1.441091674"
    ],
    [
      "0.705391138",
      "1.472937258"
    ],
    [
      "1.779197419",
      "1.475513383"
    ],
    [
      "1.335647184",
      "1.526323289"
    ],
    [
      "0.481541903",
      "1.54786025"
    ],
    [
      "1.163457102",
      "1.553597284"
    ],
    [
      "1.504544557",
      "1.576515719"
    ],
    [
      "0.991264716",
      "1.580856736"
    ],
    [
      "0.307205453",
      "1.594388181"
    ],
    [
      "0.818671964",
      "1.605454237"
    ],
    [
      "1.674586077",
      "1.614975788"
    ],
    [
      "0.634190858",
      "1.632071841"
    ],
    [
      "1.273172115",
      "1.689081272"
    ],
    [
      "0.432840517",
      "1.715256398"
    ],
    [
      "1.107649293",
      "1.743812347"
    ],
    [
      "0.933318755",
      "1.745281676"
    ],
    [
      "1.520124217",
      "1.750154912"
    ],
    [
      "0.747471684",
      "1.76458882"
    ],
    [
      "0.579162777",
      "1.810035571"
    ],
    [
      "1.368025147",
      "1.835355664"
    ],
    [
      "1.202502325",
      "1.890086739"
    ],
    [
      "0.855604682",
      "1.901338774"
    ],
    [
      "1.029593211",
      "1.912351816"
    ]
  ],
  "radius": "0.087168364"
}
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": 107
}

起步布局答案

{
  "centers": [
    [
      "1.001093536",
      "0.089277126"
    ],
    [
      "0.825162701",
      "0.104068355"
    ],
    [
      "1.186191338",
      "0.10635892"
    ],
    [
      "0.658024151",
      "0.153646579"
    ],
    [
      "1.352687355",
      "0.158053907"
    ],
    [
      "0.503359202",
      "0.234095625"
    ],
    [
      "1.506319067",
      "0.240458957"
    ],
    [
      "1.082944085",
      "0.246834185"
    ],
    [
      "0.908902632",
      "0.25697656"
    ],
    [
      "0.741764081",
      "0.306554785"
    ],
    [
      "1.367540845",
      "0.345976256"
    ],
    [
      "1.641482744",
      "0.350568338"
    ],
    [
      "0.351631164",
      "0.35744304"
    ],
    [
      "1.196399491",
      "0.379201727"
    ],
    [
      "0.518545902",
      "0.407769628"
    ],
    [
      "1.025038133",
      "0.411273219"
    ],
    [
      "0.853191559",
      "0.440633913"
    ],
    [
      "1.502704522",
      "0.456085637"
    ],
    [
      "0.68136189",
      "0.470093375"
    ],
    [
      "1.753248277",
      "0.484365801"
    ],
    [
      "0.241298178",
      "0.492424252"
    ],
    [
      "1.31074427",
      "0.510801753"
    ],
    [
      "1.138318959",
      "0.543790199"
    ],
    [
      "0.964541953",
      "0.574777065"
    ],
    [
      "0.545979947",
      "0.579934283"
    ],
    [
      "1.62271688",
      "0.599995164"
    ],
    [
      "0.792712283",
      "0.604236528"
    ],
    [
      "0.373596783",
      "0.605960037"
    ],
    [
      "1.445907947",
      "0.620911134"
    ],
    [
      "1.837539008",
      "0.636971069"
    ],
    [
      "0.158638913",
      "0.645919338"
    ],
    [
      "1.252836311",
      "0.67524008"
    ],
    [
      "1.077822778",
      "0.707294044"
    ],
    [
      "0.657330341",
      "0.714077435"
    ],
    [
      "0.904139761",
      "0.738315655"
    ],
    [
      "0.480635958",
      "0.743978779"
    ],
    [
      "1.707086873",
      "0.752623722"
    ],
    [
      "0.290937518",
      "0.759455123"
    ],
    [
      "1.533773593",
      "0.771486442"
    ],
    [
      "1.891280424",
      "0.802817852"
    ],
    [
      "1.363696451",
      "0.809788676"
    ],
    [
      "0.106668373",
      "0.812329549"
    ],
    [
      "1.191852864",
      "0.839166842"
    ],
    [
      "0.768757819",
      "0.848156563"
    ],
    [
      "1.017420587",
      "0.870832634"
    ],
    [
      "0.596928149",
      "0.877616026"
    ],
    [
      "0.425524666",
      "0.909461609"
    ],
    [
      "1.647203839",
      "0.916353125"
    ],
    [
      "0.252339373",
      "0.929465353"
    ],
    [
      "1.475060484",
      "0.943920506"
    ],
    [
      "1.303216896",
      "0.973298672"
    ],
    [
      "1.912512303",
      "0.975856873"
    ],
    [
      "0.882038644",
      "0.980673542"
    ],
    [
      "0.087282188",
      "0.985585057"
    ],
    [
      "1.131387545",
      "1.002759993"
    ],
    [
      "0.708355627",
      "1.011695153"
    ],
    [
      "0.536952143",
      "1.043540737"
    ],
    [
      "1.755973922",
      "1.052596893"
    ],
    [
      "0.359130408",
      "1.074963748"
    ],
    [
      "1.583830567",
      "1.080164274"
    ],
    [
      "1.411986979",
      "1.10954244"
    ],
    [
      "0.995982375",
      "1.113312041"
    ],
    [
      "0.183614848",
      "1.130889269"
    ],
    [
      "1.240157628",
      "1.13900376"
    ],
    [
      "0.821636453",
      "1.144212133"
    ],
    [
      "1.900377165",
      "1.150274929"
    ],
    [
      "0.650232969",
      "1.176057716"
    ],
    [
      "0.478829486",
      "1.207903299"
    ],
    [
      "1.710984218",
      "1.221028546"
    ],
    [
      "1.523430143",
      "1.243703517"
    ],
    [
      "0.309918526",
      "1.251058634"
    ],
    [
      "1.107510031",
      "1.25213161"
    ],
    [
      "1.351586556",
      "1.273081683"
    ],
    [
      "0.934917279",
      "1.276729112"
    ],
    [
      "0.137535512",
      "1.299026089"
    ],
    [
      "0.763513795",
      "1.308574695"
    ],
    [
      "1.855387461",
      "1.318706582"
    ],
    [
      "0.592110312",
      "1.340420279"
    ],
    [
      "1.634794176",
      "1.377835347"
    ],
    [
      "0.423199352",
      "1.383575613"
    ],
    [
      "1.221579759",
      "1.389234721"
    ],
    [
      "1.462950588",
      "1.407213513"
    ],
    [
      "1.049387374",
      "1.416494173"
    ],
    [
      "0.25465962",
      "1.428158728"
    ],
    [
      "0.876794621",
      "1.441091674"
    ],
    [
      "0.705391138",
      "1.472937258"
    ],
    [
      "1.779197419",
      "1.475513383"
    ],
    [
      "1.335647184",
      "1.526323289"
    ],
    [
      "0.481541903",
      "1.54786025"
    ],
    [
      "1.163457102",
      "1.553597284"
    ],
    [
      "1.504544557",
      "1.576515719"
    ],
    [
      "0.991264716",
      "1.580856736"
    ],
    [
      "0.307205453",
      "1.594388181"
    ],
    [
      "0.818671964",
      "1.605454237"
    ],
    [
      "1.674586077",
      "1.614975788"
    ],
    [
      "0.634190858",
      "1.632071841"
    ],
    [
      "1.273172115",
      "1.689081272"
    ],
    [
      "0.432840517",
      "1.715256398"
    ],
    [
      "1.107649293",
      "1.743812347"
    ],
    [
      "0.933318755",
      "1.745281676"
    ],
    [
      "1.520124217",
      "1.750154912"
    ],
    [
      "0.747471684",
      "1.76458882"
    ],
    [
      "0.579162777",
      "1.810035571"
    ],
    [
      "1.368025147",
      "1.835355664"
    ],
    [
      "1.202502325",
      "1.890086739"
    ],
    [
      "0.855604682",
      "1.901338774"
    ],
    [
      "1.029593211",
      "1.912351816"
    ]
  ],
  "radius": "0.087168364"
}

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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