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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 = 77

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 = 77
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[0.1020521469836900802047256289225, 0.1020521469836900802047256289235]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 = 0.102052146

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 = 77 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": [
    [
      "0.946183423",
      "0.103666288"
    ],
    [
      "1.149991001",
      "0.114667833"
    ],
    [
      "0.745156314",
      "0.138974424"
    ],
    [
      "1.346049194",
      "0.171410655"
    ],
    [
      "0.557295875",
      "0.218768021"
    ],
    [
      "1.524228523",
      "0.2709631"
    ],
    [
      "0.916821919",
      "0.305647636"
    ],
    [
      "1.15286572",
      "0.318751881"
    ],
    [
      "0.39230804",
      "0.338924491"
    ],
    [
      "0.716775535",
      "0.346144632"
    ],
    [
      "1.348923913",
      "0.375494704"
    ],
    [
      "0.5517877",
      "0.466301102"
    ],
    [
      "1.567906802",
      "0.470339053"
    ],
    [
      "1.02562055",
      "0.478336434"
    ],
    [
      "0.258717017",
      "0.493235868"
    ],
    [
      "0.825574165",
      "0.51883343"
    ],
    [
      "1.763758825",
      "0.527789449"
    ],
    [
      "1.221678742",
      "0.535079256"
    ],
    [
      "1.413753994",
      "0.604113021"
    ],
    [
      "0.418196676",
      "0.620612479"
    ],
    [
      "0.662835184",
      "0.649847638"
    ],
    [
      "1.616079311",
      "0.668677078"
    ],
    [
      "0.163424874",
      "0.673729553"
    ],
    [
      "1.061447352",
      "0.679271755"
    ],
    [
      "1.850772338",
      "0.712795585"
    ],
    [
      "0.861400967",
      "0.719768751"
    ],
    [
      "1.257505545",
      "0.736014577"
    ],
    [
      "0.520021058",
      "0.797503288"
    ],
    [
      "1.461926502",
      "0.802451046"
    ],
    [
      "0.316242334",
      "0.809026953"
    ],
    [
      "1.697557068",
      "0.855813226"
    ],
    [
      "0.714010624",
      "0.86095887"
    ],
    [
      "0.113349124",
      "0.873040974"
    ],
    [
      "1.11002641",
      "0.877510595"
    ],
    [
      "1.893663595",
      "0.912388779"
    ],
    [
      "0.909980025",
      "0.918007591"
    ],
    [
      "1.306084603",
      "0.934253418"
    ],
    [
      "0.500868304",
      "1.000706966"
    ],
    [
      "1.498149755",
      "1.003315276"
    ],
    [
      "0.297089581",
      "1.012230631"
    ],
    [
      "1.694256281",
      "1.059890828"
    ],
    [
      "0.694857871",
      "1.064162548"
    ],
    [
      "1.159035474",
      "1.075798875"
    ],
    [
      "0.10597976",
      "1.083893726"
    ],
    [
      "0.958959202",
      "1.116147953"
    ],
    [
      "1.890362807",
      "1.11646638"
    ],
    [
      "1.351100626",
      "1.144860733"
    ],
    [
      "0.542908942",
      "1.20043465"
    ],
    [
      "1.547207152",
      "1.201436286"
    ],
    [
      "0.339130219",
      "1.211958315"
    ],
    [
      "0.797273135",
      "1.240711908"
    ],
    [
      "1.743313679",
      "1.258011838"
    ],
    [
      "1.18646457",
      "1.278219371"
    ],
    [
      "0.148020398",
      "1.283621409"
    ],
    [
      "0.985991717",
      "1.318454172"
    ],
    [
      "1.378827942",
      "1.347072896"
    ],
    [
      "0.645324206",
      "1.376984009"
    ],
    [
      "0.425188952",
      "1.397032509"
    ],
    [
      "1.574934469",
      "1.403648448"
    ],
    [
      "0.836429058",
      "1.458610772"
    ],
    [
      "1.771040995",
      "1.460224001"
    ],
    [
      "0.234079131",
      "1.468695603"
    ],
    [
      "1.242010801",
      "1.498531249"
    ],
    [
      "1.038626096",
      "1.515655046"
    ],
    [
      "1.438117327",
      "1.555106802"
    ],
    [
      "0.550819498",
      "1.557891232"
    ],
    [
      "1.647191307",
      "1.622457837"
    ],
    [
      "0.359709677",
      "1.629554326"
    ],
    [
      "0.734783104",
      "1.64629912"
    ],
    [
      "1.146667358",
      "1.690550137"
    ],
    [
      "0.933959394",
      "1.69087903"
    ],
    [
      "1.489904048",
      "1.752531973"
    ],
    [
      "0.521591648",
      "1.759891963"
    ],
    [
      "1.307305569",
      "1.843726042"
    ],
    [
      "0.705555254",
      "1.848299852"
    ],
    [
      "1.108829942",
      "1.891328441"
    ],
    [
      "0.904731544",
      "1.892879761"
    ]
  ],
  "radius": "0.102052146"
}
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": 77
}

起步布局答案

{
  "centers": [
    [
      "0.946183423",
      "0.103666288"
    ],
    [
      "1.149991001",
      "0.114667833"
    ],
    [
      "0.745156314",
      "0.138974424"
    ],
    [
      "1.346049194",
      "0.171410655"
    ],
    [
      "0.557295875",
      "0.218768021"
    ],
    [
      "1.524228523",
      "0.2709631"
    ],
    [
      "0.916821919",
      "0.305647636"
    ],
    [
      "1.15286572",
      "0.318751881"
    ],
    [
      "0.39230804",
      "0.338924491"
    ],
    [
      "0.716775535",
      "0.346144632"
    ],
    [
      "1.348923913",
      "0.375494704"
    ],
    [
      "0.5517877",
      "0.466301102"
    ],
    [
      "1.567906802",
      "0.470339053"
    ],
    [
      "1.02562055",
      "0.478336434"
    ],
    [
      "0.258717017",
      "0.493235868"
    ],
    [
      "0.825574165",
      "0.51883343"
    ],
    [
      "1.763758825",
      "0.527789449"
    ],
    [
      "1.221678742",
      "0.535079256"
    ],
    [
      "1.413753994",
      "0.604113021"
    ],
    [
      "0.418196676",
      "0.620612479"
    ],
    [
      "0.662835184",
      "0.649847638"
    ],
    [
      "1.616079311",
      "0.668677078"
    ],
    [
      "0.163424874",
      "0.673729553"
    ],
    [
      "1.061447352",
      "0.679271755"
    ],
    [
      "1.850772338",
      "0.712795585"
    ],
    [
      "0.861400967",
      "0.719768751"
    ],
    [
      "1.257505545",
      "0.736014577"
    ],
    [
      "0.520021058",
      "0.797503288"
    ],
    [
      "1.461926502",
      "0.802451046"
    ],
    [
      "0.316242334",
      "0.809026953"
    ],
    [
      "1.697557068",
      "0.855813226"
    ],
    [
      "0.714010624",
      "0.86095887"
    ],
    [
      "0.113349124",
      "0.873040974"
    ],
    [
      "1.11002641",
      "0.877510595"
    ],
    [
      "1.893663595",
      "0.912388779"
    ],
    [
      "0.909980025",
      "0.918007591"
    ],
    [
      "1.306084603",
      "0.934253418"
    ],
    [
      "0.500868304",
      "1.000706966"
    ],
    [
      "1.498149755",
      "1.003315276"
    ],
    [
      "0.297089581",
      "1.012230631"
    ],
    [
      "1.694256281",
      "1.059890828"
    ],
    [
      "0.694857871",
      "1.064162548"
    ],
    [
      "1.159035474",
      "1.075798875"
    ],
    [
      "0.10597976",
      "1.083893726"
    ],
    [
      "0.958959202",
      "1.116147953"
    ],
    [
      "1.890362807",
      "1.11646638"
    ],
    [
      "1.351100626",
      "1.144860733"
    ],
    [
      "0.542908942",
      "1.20043465"
    ],
    [
      "1.547207152",
      "1.201436286"
    ],
    [
      "0.339130219",
      "1.211958315"
    ],
    [
      "0.797273135",
      "1.240711908"
    ],
    [
      "1.743313679",
      "1.258011838"
    ],
    [
      "1.18646457",
      "1.278219371"
    ],
    [
      "0.148020398",
      "1.283621409"
    ],
    [
      "0.985991717",
      "1.318454172"
    ],
    [
      "1.378827942",
      "1.347072896"
    ],
    [
      "0.645324206",
      "1.376984009"
    ],
    [
      "0.425188952",
      "1.397032509"
    ],
    [
      "1.574934469",
      "1.403648448"
    ],
    [
      "0.836429058",
      "1.458610772"
    ],
    [
      "1.771040995",
      "1.460224001"
    ],
    [
      "0.234079131",
      "1.468695603"
    ],
    [
      "1.242010801",
      "1.498531249"
    ],
    [
      "1.038626096",
      "1.515655046"
    ],
    [
      "1.438117327",
      "1.555106802"
    ],
    [
      "0.550819498",
      "1.557891232"
    ],
    [
      "1.647191307",
      "1.622457837"
    ],
    [
      "0.359709677",
      "1.629554326"
    ],
    [
      "0.734783104",
      "1.64629912"
    ],
    [
      "1.146667358",
      "1.690550137"
    ],
    [
      "0.933959394",
      "1.69087903"
    ],
    [
      "1.489904048",
      "1.752531973"
    ],
    [
      "0.521591648",
      "1.759891963"
    ],
    [
      "1.307305569",
      "1.843726042"
    ],
    [
      "0.705555254",
      "1.848299852"
    ],
    [
      "1.108829942",
      "1.891328441"
    ],
    [
      "0.904731544",
      "1.892879761"
    ]
  ],
  "radius": "0.102052146"
}

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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