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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 = 85

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 = 85
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[0.0983950636926061735556696115065, 0.0983950636926061735556696115075]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 = 0.098395063

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 = 85 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": [
    [
      "0.829443571",
      "0.114674091"
    ],
    [
      "1.170556429",
      "0.114674091"
    ],
    [
      "0.641423464",
      "0.172767065"
    ],
    [
      "1.36271286",
      "0.175192352"
    ],
    [
      "1",
      "0.212841594"
    ],
    [
      "0.46714792",
      "0.273295707"
    ],
    [
      "1.537116469",
      "0.275847143"
    ],
    [
      "1.179631649",
      "0.311254849"
    ],
    [
      "0.828655582",
      "0.31146264"
    ],
    [
      "0.640635475",
      "0.369555615"
    ],
    [
      "1.366298591",
      "0.373558999"
    ],
    [
      "0.999212011",
      "0.409630144"
    ],
    [
      "0.318563487",
      "0.409630845"
    ],
    [
      "1.681436513",
      "0.409630845"
    ],
    [
      "0.493345027",
      "0.500061853"
    ],
    [
      "1.510618636",
      "0.507342701"
    ],
    [
      "1.16976844",
      "0.507797647"
    ],
    [
      "0.828918245",
      "0.508252593"
    ],
    [
      "1.34032487",
      "0.60596515"
    ],
    [
      "0.999474674",
      "0.606420096"
    ],
    [
      "0.318300824",
      "0.606420797"
    ],
    [
      "1.681699176",
      "0.606420797"
    ],
    [
      "0.658624479",
      "0.606875041"
    ],
    [
      "1.510881299",
      "0.704132653"
    ],
    [
      "1.170031103",
      "0.704587599"
    ],
    [
      "0.829180908",
      "0.705042545"
    ],
    [
      "0.148007057",
      "0.705043246"
    ],
    [
      "1.851992943",
      "0.705043246"
    ],
    [
      "0.488330712",
      "0.70549749"
    ],
    [
      "1.340587532",
      "0.802755102"
    ],
    [
      "0.999737337",
      "0.803210048"
    ],
    [
      "0.658887142",
      "0.803664993"
    ],
    [
      "0.318036946",
      "0.804119939"
    ],
    [
      "1.686286622",
      "0.811192996"
    ],
    [
      "0.104306995",
      "0.896919922"
    ],
    [
      "1.511143962",
      "0.900922606"
    ],
    [
      "1.170293766",
      "0.901377551"
    ],
    [
      "1.895744523",
      "0.901652567"
    ],
    [
      "0.829443571",
      "0.901832497"
    ],
    [
      "0.488593375",
      "0.902287442"
    ],
    [
      "0.274336884",
      "0.995996615"
    ],
    [
      "1.340850195",
      "0.999545054"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.659149805",
      "1.000454946"
    ],
    [
      "1.725663116",
      "1.004003385"
    ],
    [
      "1.511406625",
      "1.097712558"
    ],
    [
      "1.170556429",
      "1.098167503"
    ],
    [
      "0.104255787",
      "1.098350257"
    ],
    [
      "0.829706234",
      "1.098622449"
    ],
    [
      "0.488856038",
      "1.099077394"
    ],
    [
      "1.895693005",
      "1.103080078"
    ],
    [
      "0.313713378",
      "1.188807004"
    ],
    [
      "1.681963054",
      "1.195880061"
    ],
    [
      "1.341112858",
      "1.196335007"
    ],
    [
      "1.000262663",
      "1.196789952"
    ],
    [
      "0.659412468",
      "1.197244898"
    ],
    [
      "1.511669288",
      "1.29450251"
    ],
    [
      "0.148007057",
      "1.294956754"
    ],
    [
      "1.851992943",
      "1.294956754"
    ],
    [
      "1.170819092",
      "1.294957455"
    ],
    [
      "0.829968897",
      "1.295412401"
    ],
    [
      "0.489118701",
      "1.295867347"
    ],
    [
      "1.341375521",
      "1.393124959"
    ],
    [
      "0.318300824",
      "1.393579203"
    ],
    [
      "1.681699176",
      "1.393579203"
    ],
    [
      "1.000525326",
      "1.393579904"
    ],
    [
      "0.65967513",
      "1.39403485"
    ],
    [
      "1.171081755",
      "1.491747407"
    ],
    [
      "0.83023156",
      "1.492202353"
    ],
    [
      "0.489381364",
      "1.492657299"
    ],
    [
      "1.506654973",
      "1.499938147"
    ],
    [
      "0.318563487",
      "1.590369155"
    ],
    [
      "1.681436513",
      "1.590369155"
    ],
    [
      "1.000787989",
      "1.590369856"
    ],
    [
      "0.633701409",
      "1.626441001"
    ],
    [
      "1.359364525",
      "1.630444385"
    ],
    [
      "1.171344418",
      "1.68853736"
    ],
    [
      "0.820368351",
      "1.688745151"
    ],
    [
      "0.462883531",
      "1.724152857"
    ],
    [
      "1.532843473",
      "1.726710604"
    ],
    [
      "1",
      "1.787158406"
    ],
    [
      "0.637304678",
      "1.824777285"
    ],
    [
      "1.358576536",
      "1.827232935"
    ],
    [
      "0.829443571",
      "1.885325909"
    ],
    [
      "1.170556429",
      "1.885325909"
    ]
  ],
  "radius": "0.098395063"
}
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": 85
}

起步布局答案

{
  "centers": [
    [
      "0.829443571",
      "0.114674091"
    ],
    [
      "1.170556429",
      "0.114674091"
    ],
    [
      "0.641423464",
      "0.172767065"
    ],
    [
      "1.36271286",
      "0.175192352"
    ],
    [
      "1",
      "0.212841594"
    ],
    [
      "0.46714792",
      "0.273295707"
    ],
    [
      "1.537116469",
      "0.275847143"
    ],
    [
      "1.179631649",
      "0.311254849"
    ],
    [
      "0.828655582",
      "0.31146264"
    ],
    [
      "0.640635475",
      "0.369555615"
    ],
    [
      "1.366298591",
      "0.373558999"
    ],
    [
      "0.999212011",
      "0.409630144"
    ],
    [
      "0.318563487",
      "0.409630845"
    ],
    [
      "1.681436513",
      "0.409630845"
    ],
    [
      "0.493345027",
      "0.500061853"
    ],
    [
      "1.510618636",
      "0.507342701"
    ],
    [
      "1.16976844",
      "0.507797647"
    ],
    [
      "0.828918245",
      "0.508252593"
    ],
    [
      "1.34032487",
      "0.60596515"
    ],
    [
      "0.999474674",
      "0.606420096"
    ],
    [
      "0.318300824",
      "0.606420797"
    ],
    [
      "1.681699176",
      "0.606420797"
    ],
    [
      "0.658624479",
      "0.606875041"
    ],
    [
      "1.510881299",
      "0.704132653"
    ],
    [
      "1.170031103",
      "0.704587599"
    ],
    [
      "0.829180908",
      "0.705042545"
    ],
    [
      "0.148007057",
      "0.705043246"
    ],
    [
      "1.851992943",
      "0.705043246"
    ],
    [
      "0.488330712",
      "0.70549749"
    ],
    [
      "1.340587532",
      "0.802755102"
    ],
    [
      "0.999737337",
      "0.803210048"
    ],
    [
      "0.658887142",
      "0.803664993"
    ],
    [
      "0.318036946",
      "0.804119939"
    ],
    [
      "1.686286622",
      "0.811192996"
    ],
    [
      "0.104306995",
      "0.896919922"
    ],
    [
      "1.511143962",
      "0.900922606"
    ],
    [
      "1.170293766",
      "0.901377551"
    ],
    [
      "1.895744523",
      "0.901652567"
    ],
    [
      "0.829443571",
      "0.901832497"
    ],
    [
      "0.488593375",
      "0.902287442"
    ],
    [
      "0.274336884",
      "0.995996615"
    ],
    [
      "1.340850195",
      "0.999545054"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.659149805",
      "1.000454946"
    ],
    [
      "1.725663116",
      "1.004003385"
    ],
    [
      "1.511406625",
      "1.097712558"
    ],
    [
      "1.170556429",
      "1.098167503"
    ],
    [
      "0.104255787",
      "1.098350257"
    ],
    [
      "0.829706234",
      "1.098622449"
    ],
    [
      "0.488856038",
      "1.099077394"
    ],
    [
      "1.895693005",
      "1.103080078"
    ],
    [
      "0.313713378",
      "1.188807004"
    ],
    [
      "1.681963054",
      "1.195880061"
    ],
    [
      "1.341112858",
      "1.196335007"
    ],
    [
      "1.000262663",
      "1.196789952"
    ],
    [
      "0.659412468",
      "1.197244898"
    ],
    [
      "1.511669288",
      "1.29450251"
    ],
    [
      "0.148007057",
      "1.294956754"
    ],
    [
      "1.851992943",
      "1.294956754"
    ],
    [
      "1.170819092",
      "1.294957455"
    ],
    [
      "0.829968897",
      "1.295412401"
    ],
    [
      "0.489118701",
      "1.295867347"
    ],
    [
      "1.341375521",
      "1.393124959"
    ],
    [
      "0.318300824",
      "1.393579203"
    ],
    [
      "1.681699176",
      "1.393579203"
    ],
    [
      "1.000525326",
      "1.393579904"
    ],
    [
      "0.65967513",
      "1.39403485"
    ],
    [
      "1.171081755",
      "1.491747407"
    ],
    [
      "0.83023156",
      "1.492202353"
    ],
    [
      "0.489381364",
      "1.492657299"
    ],
    [
      "1.506654973",
      "1.499938147"
    ],
    [
      "0.318563487",
      "1.590369155"
    ],
    [
      "1.681436513",
      "1.590369155"
    ],
    [
      "1.000787989",
      "1.590369856"
    ],
    [
      "0.633701409",
      "1.626441001"
    ],
    [
      "1.359364525",
      "1.630444385"
    ],
    [
      "1.171344418",
      "1.68853736"
    ],
    [
      "0.820368351",
      "1.688745151"
    ],
    [
      "0.462883531",
      "1.724152857"
    ],
    [
      "1.532843473",
      "1.726710604"
    ],
    [
      "1",
      "1.787158406"
    ],
    [
      "0.637304678",
      "1.824777285"
    ],
    [
      "1.358576536",
      "1.827232935"
    ],
    [
      "0.829443571",
      "1.885325909"
    ],
    [
      "1.170556429",
      "1.885325909"
    ]
  ],
  "radius": "0.098395063"
}

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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