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 = 110

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 = 110
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[0.0860817696475518999332564923515, 0.0860817696475518999332564923525]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 = 0.086081769

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 = 110 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": [
    [
      "1.025324158",
      "0.086432696"
    ],
    [
      "0.853542489",
      "0.097893172"
    ],
    [
      "1.196207152",
      "0.107391864"
    ],
    [
      "0.686958139",
      "0.141366594"
    ],
    [
      "0.531482664",
      "0.215310228"
    ],
    [
      "1.479521792",
      "0.221986258"
    ],
    [
      "1.308124856",
      "0.238215072"
    ],
    [
      "1.092359094",
      "0.245009497"
    ],
    [
      "0.920577425",
      "0.256469972"
    ],
    [
      "0.689006559",
      "0.313517947"
    ],
    [
      "0.368120533",
      "0.339716068"
    ],
    [
      "0.533531084",
      "0.387461581"
    ],
    [
      "1.521647138",
      "0.388916601"
    ],
    [
      "1.178440864",
      "0.394107495"
    ],
    [
      "1.693119967",
      "0.404322871"
    ],
    [
      "1.350250202",
      "0.405145415"
    ],
    [
      "0.834495656",
      "0.405567971"
    ],
    [
      "1.006659195",
      "0.405567971"
    ],
    [
      "0.235025193",
      "0.504386528"
    ],
    [
      "0.409476981",
      "0.506838563"
    ],
    [
      "1.792686425",
      "0.553404076"
    ],
    [
      "1.437065951",
      "0.553817235"
    ],
    [
      "0.747901516",
      "0.554368977"
    ],
    [
      "0.575737977",
      "0.554371324"
    ],
    [
      "1.60922741",
      "0.554663561"
    ],
    [
      "0.920577425",
      "0.55466597"
    ],
    [
      "1.092740965",
      "0.55466597"
    ],
    [
      "1.264904504",
      "0.55466597"
    ],
    [
      "0.324586185",
      "0.656617858"
    ],
    [
      "0.152429346",
      "0.658136697"
    ],
    [
      "1.695313352",
      "0.70375915"
    ],
    [
      "0.662332116",
      "0.703761621"
    ],
    [
      "0.490168577",
      "0.703763968"
    ],
    [
      "0.834495656",
      "0.703763968"
    ],
    [
      "1.006659195",
      "0.703763968"
    ],
    [
      "1.178822734",
      "0.703763968"
    ],
    [
      "1.350986274",
      "0.703763968"
    ],
    [
      "1.523149813",
      "0.703763968"
    ],
    [
      "1.867288807",
      "0.71180448"
    ],
    [
      "0.238052692",
      "0.807498429"
    ],
    [
      "1.609234165",
      "0.852860476"
    ],
    [
      "0.576250347",
      "0.852861967"
    ],
    [
      "0.748413886",
      "0.852861967"
    ],
    [
      "0.920577425",
      "0.852861967"
    ],
    [
      "1.092740965",
      "0.852861967"
    ],
    [
      "1.264904504",
      "0.852861967"
    ],
    [
      "1.437068043",
      "0.852861967"
    ],
    [
      "0.404086887",
      "0.853027872"
    ],
    [
      "1.803051151",
      "0.871534903"
    ],
    [
      "0.09176149",
      "0.898268294"
    ],
    [
      "0.270598666",
      "0.988916997"
    ],
    [
      "1.913912436",
      "1.003254528"
    ],
    [
      "0.659543447",
      "1.0035356"
    ],
    [
      "0.831706986",
      "1.0035356"
    ],
    [
      "1.003870526",
      "1.0035356"
    ],
    [
      "1.176034065",
      "1.0035356"
    ],
    [
      "1.348197604",
      "1.0035356"
    ],
    [
      "1.520361144",
      "1.0035356"
    ],
    [
      "1.692524683",
      "1.0035356"
    ],
    [
      "0.487379988",
      "1.003701505"
    ],
    [
      "0.088797782",
      "1.070406322"
    ],
    [
      "1.803385967",
      "1.135255226"
    ],
    [
      "0.745625217",
      "1.152633599"
    ],
    [
      "0.917788756",
      "1.152633599"
    ],
    [
      "1.089952295",
      "1.152633599"
    ],
    [
      "1.262115835",
      "1.152633599"
    ],
    [
      "1.434279374",
      "1.152633599"
    ],
    [
      "1.606442913",
      "1.152633599"
    ],
    [
      "0.401298218",
      "1.152799504"
    ],
    [
      "0.573461757",
      "1.152799504"
    ],
    [
      "0.229902884",
      "1.169045232"
    ],
    [
      "1.864598344",
      "1.296169269"
    ],
    [
      "0.831706986",
      "1.301731597"
    ],
    [
      "1.003870526",
      "1.301731597"
    ],
    [
      "1.176034065",
      "1.301731597"
    ],
    [
      "1.348197604",
      "1.301731597"
    ],
    [
      "1.520361144",
      "1.301731597"
    ],
    [
      "1.692524683",
      "1.301731597"
    ],
    [
      "0.659543527",
      "1.301897502"
    ],
    [
      "0.486949579",
      "1.302145999"
    ],
    [
      "0.143085088",
      "1.317715856"
    ],
    [
      "0.315246523",
      "1.318566961"
    ],
    [
      "0.917788756",
      "1.450829596"
    ],
    [
      "1.089952295",
      "1.450829596"
    ],
    [
      "1.262115835",
      "1.450829596"
    ],
    [
      "1.434279374",
      "1.450829596"
    ],
    [
      "1.606442913",
      "1.450829596"
    ],
    [
      "0.573461757",
      "1.450995501"
    ],
    [
      "0.745625297",
      "1.450995501"
    ],
    [
      "1.787214629",
      "1.455994502"
    ],
    [
      "0.402065986",
      "1.467236612"
    ],
    [
      "0.226990813",
      "1.480302272"
    ],
    [
      "1.003870526",
      "1.599927594"
    ],
    [
      "1.176034065",
      "1.599927594"
    ],
    [
      "1.348197604",
      "1.599927594"
    ],
    [
      "0.659543527",
      "1.6000935"
    ],
    [
      "0.831707066",
      "1.6000935"
    ],
    [
      "1.68657715",
      "1.603206721"
    ],
    [
      "0.333243487",
      "1.625045826"
    ],
    [
      "1.517079049",
      "1.633384161"
    ],
    [
      "0.500475939",
      "1.665955376"
    ],
    [
      "1.089952295",
      "1.749025593"
    ],
    [
      "0.745625297",
      "1.749191498"
    ],
    [
      "0.917788836",
      "1.749191498"
    ],
    [
      "1.284706919",
      "1.759956395"
    ],
    [
      "1.453588363",
      "1.793412962"
    ],
    [
      "0.586557709",
      "1.815053375"
    ],
    [
      "1.179379328",
      "1.8961415"
    ],
    [
      "0.831707066",
      "1.898289497"
    ],
    [
      "1.005543197",
      "1.912196595"
    ]
  ],
  "radius": "0.086081769"
}
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": 110
}

起步布局答案

{
  "centers": [
    [
      "1.025324158",
      "0.086432696"
    ],
    [
      "0.853542489",
      "0.097893172"
    ],
    [
      "1.196207152",
      "0.107391864"
    ],
    [
      "0.686958139",
      "0.141366594"
    ],
    [
      "0.531482664",
      "0.215310228"
    ],
    [
      "1.479521792",
      "0.221986258"
    ],
    [
      "1.308124856",
      "0.238215072"
    ],
    [
      "1.092359094",
      "0.245009497"
    ],
    [
      "0.920577425",
      "0.256469972"
    ],
    [
      "0.689006559",
      "0.313517947"
    ],
    [
      "0.368120533",
      "0.339716068"
    ],
    [
      "0.533531084",
      "0.387461581"
    ],
    [
      "1.521647138",
      "0.388916601"
    ],
    [
      "1.178440864",
      "0.394107495"
    ],
    [
      "1.693119967",
      "0.404322871"
    ],
    [
      "1.350250202",
      "0.405145415"
    ],
    [
      "0.834495656",
      "0.405567971"
    ],
    [
      "1.006659195",
      "0.405567971"
    ],
    [
      "0.235025193",
      "0.504386528"
    ],
    [
      "0.409476981",
      "0.506838563"
    ],
    [
      "1.792686425",
      "0.553404076"
    ],
    [
      "1.437065951",
      "0.553817235"
    ],
    [
      "0.747901516",
      "0.554368977"
    ],
    [
      "0.575737977",
      "0.554371324"
    ],
    [
      "1.60922741",
      "0.554663561"
    ],
    [
      "0.920577425",
      "0.55466597"
    ],
    [
      "1.092740965",
      "0.55466597"
    ],
    [
      "1.264904504",
      "0.55466597"
    ],
    [
      "0.324586185",
      "0.656617858"
    ],
    [
      "0.152429346",
      "0.658136697"
    ],
    [
      "1.695313352",
      "0.70375915"
    ],
    [
      "0.662332116",
      "0.703761621"
    ],
    [
      "0.490168577",
      "0.703763968"
    ],
    [
      "0.834495656",
      "0.703763968"
    ],
    [
      "1.006659195",
      "0.703763968"
    ],
    [
      "1.178822734",
      "0.703763968"
    ],
    [
      "1.350986274",
      "0.703763968"
    ],
    [
      "1.523149813",
      "0.703763968"
    ],
    [
      "1.867288807",
      "0.71180448"
    ],
    [
      "0.238052692",
      "0.807498429"
    ],
    [
      "1.609234165",
      "0.852860476"
    ],
    [
      "0.576250347",
      "0.852861967"
    ],
    [
      "0.748413886",
      "0.852861967"
    ],
    [
      "0.920577425",
      "0.852861967"
    ],
    [
      "1.092740965",
      "0.852861967"
    ],
    [
      "1.264904504",
      "0.852861967"
    ],
    [
      "1.437068043",
      "0.852861967"
    ],
    [
      "0.404086887",
      "0.853027872"
    ],
    [
      "1.803051151",
      "0.871534903"
    ],
    [
      "0.09176149",
      "0.898268294"
    ],
    [
      "0.270598666",
      "0.988916997"
    ],
    [
      "1.913912436",
      "1.003254528"
    ],
    [
      "0.659543447",
      "1.0035356"
    ],
    [
      "0.831706986",
      "1.0035356"
    ],
    [
      "1.003870526",
      "1.0035356"
    ],
    [
      "1.176034065",
      "1.0035356"
    ],
    [
      "1.348197604",
      "1.0035356"
    ],
    [
      "1.520361144",
      "1.0035356"
    ],
    [
      "1.692524683",
      "1.0035356"
    ],
    [
      "0.487379988",
      "1.003701505"
    ],
    [
      "0.088797782",
      "1.070406322"
    ],
    [
      "1.803385967",
      "1.135255226"
    ],
    [
      "0.745625217",
      "1.152633599"
    ],
    [
      "0.917788756",
      "1.152633599"
    ],
    [
      "1.089952295",
      "1.152633599"
    ],
    [
      "1.262115835",
      "1.152633599"
    ],
    [
      "1.434279374",
      "1.152633599"
    ],
    [
      "1.606442913",
      "1.152633599"
    ],
    [
      "0.401298218",
      "1.152799504"
    ],
    [
      "0.573461757",
      "1.152799504"
    ],
    [
      "0.229902884",
      "1.169045232"
    ],
    [
      "1.864598344",
      "1.296169269"
    ],
    [
      "0.831706986",
      "1.301731597"
    ],
    [
      "1.003870526",
      "1.301731597"
    ],
    [
      "1.176034065",
      "1.301731597"
    ],
    [
      "1.348197604",
      "1.301731597"
    ],
    [
      "1.520361144",
      "1.301731597"
    ],
    [
      "1.692524683",
      "1.301731597"
    ],
    [
      "0.659543527",
      "1.301897502"
    ],
    [
      "0.486949579",
      "1.302145999"
    ],
    [
      "0.143085088",
      "1.317715856"
    ],
    [
      "0.315246523",
      "1.318566961"
    ],
    [
      "0.917788756",
      "1.450829596"
    ],
    [
      "1.089952295",
      "1.450829596"
    ],
    [
      "1.262115835",
      "1.450829596"
    ],
    [
      "1.434279374",
      "1.450829596"
    ],
    [
      "1.606442913",
      "1.450829596"
    ],
    [
      "0.573461757",
      "1.450995501"
    ],
    [
      "0.745625297",
      "1.450995501"
    ],
    [
      "1.787214629",
      "1.455994502"
    ],
    [
      "0.402065986",
      "1.467236612"
    ],
    [
      "0.226990813",
      "1.480302272"
    ],
    [
      "1.003870526",
      "1.599927594"
    ],
    [
      "1.176034065",
      "1.599927594"
    ],
    [
      "1.348197604",
      "1.599927594"
    ],
    [
      "0.659543527",
      "1.6000935"
    ],
    [
      "0.831707066",
      "1.6000935"
    ],
    [
      "1.68657715",
      "1.603206721"
    ],
    [
      "0.333243487",
      "1.625045826"
    ],
    [
      "1.517079049",
      "1.633384161"
    ],
    [
      "0.500475939",
      "1.665955376"
    ],
    [
      "1.089952295",
      "1.749025593"
    ],
    [
      "0.745625297",
      "1.749191498"
    ],
    [
      "0.917788836",
      "1.749191498"
    ],
    [
      "1.284706919",
      "1.759956395"
    ],
    [
      "1.453588363",
      "1.793412962"
    ],
    [
      "0.586557709",
      "1.815053375"
    ],
    [
      "1.179379328",
      "1.8961415"
    ],
    [
      "0.831707066",
      "1.898289497"
    ],
    [
      "1.005543197",
      "1.912196595"
    ]
  ],
  "radius": "0.086081769"
}

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。

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