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P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 84

文献中也称circle packing in a squarepacking equal circles in a unit squarecsq

在边长 1 的正方形内放置 n 个半径相同、互不重叠的圆，使共同半径尽可能大。

子题n = 84
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[0.0558566658875, 0.0558566658885]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器单位正方形：左下角是原点 (0, 0)，右上角是 (1, 1)
  • 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
放大来摆，然后提交 ↗

当前展示：起步布局

1y0
0x1
已验证构造r = 0.055856665

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 = 1–30 已证明最优，适合观察经典接触结构；n = 31–300 接入 Packomania 的公开前沿，除 n = 36 外最优性仍未知。大规模部分集中成一张表，供算法批量挑战。

查看来源
单位正方形内的等圆装箱 n = 84 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

容器是边长 1 的正方形，左下角是原点 (0, 0)，右上角是 (1, 1)。坐标和长度用同一个单位，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 "0.25"。

起步布局答案

{
  "centers": [
    [
      "0.055856666",
      "0.055856666"
    ],
    [
      "0.249349832",
      "0.055856666"
    ],
    [
      "0.442843012",
      "0.055856666"
    ],
    [
      "0.554556344",
      "0.055856666"
    ],
    [
      "0.748084935",
      "0.055856666"
    ],
    [
      "0.944143334",
      "0.055856666"
    ],
    [
      "0.846114134",
      "0.109430395"
    ],
    [
      "0.651320639",
      "0.111682642"
    ],
    [
      "0.346096422",
      "0.111713321"
    ],
    [
      "0.152603249",
      "0.111713332"
    ],
    [
      "0.941524945",
      "0.167539308"
    ],
    [
      "0.7480495",
      "0.167569992"
    ],
    [
      "0.442842999",
      "0.167569998"
    ],
    [
      "0.554556331",
      "0.167569998"
    ],
    [
      "0.249349832",
      "0.167569998"
    ],
    [
      "0.055856666",
      "0.167569998"
    ],
    [
      "0.844796084",
      "0.223426658"
    ],
    [
      "0.651302917",
      "0.223426658"
    ],
    [
      "0.346096416",
      "0.223426664"
    ],
    [
      "0.152603249",
      "0.223426664"
    ],
    [
      "0.944143334",
      "0.27922195"
    ],
    [
      "0.7480495",
      "0.279283324"
    ],
    [
      "0.442842999",
      "0.279283329"
    ],
    [
      "0.554556331",
      "0.279283329"
    ],
    [
      "0.249349832",
      "0.279283329"
    ],
    [
      "0.055856666",
      "0.279283329"
    ],
    [
      "0.844796084",
      "0.33513999"
    ],
    [
      "0.651302917",
      "0.33513999"
    ],
    [
      "0.346096416",
      "0.335139995"
    ],
    [
      "0.152603249",
      "0.335139995"
    ],
    [
      "0.94159483",
      "0.390906209"
    ],
    [
      "0.7480495",
      "0.390996656"
    ],
    [
      "0.442842999",
      "0.390996661"
    ],
    [
      "0.554556331",
      "0.390996661"
    ],
    [
      "0.249349832",
      "0.390996661"
    ],
    [
      "0.055856666",
      "0.390996661"
    ],
    [
      "0.844796084",
      "0.446853321"
    ],
    [
      "0.651302917",
      "0.446853321"
    ],
    [
      "0.346096416",
      "0.446853327"
    ],
    [
      "0.152603249",
      "0.446853327"
    ],
    [
      "0.944143334",
      "0.502590467"
    ],
    [
      "0.7480495",
      "0.502709987"
    ],
    [
      "0.442842999",
      "0.502709993"
    ],
    [
      "0.554556331",
      "0.502709993"
    ],
    [
      "0.249349832",
      "0.502709993"
    ],
    [
      "0.055856666",
      "0.502709993"
    ],
    [
      "0.847344588",
      "0.55853758"
    ],
    [
      "0.651302917",
      "0.558566653"
    ],
    [
      "0.346096416",
      "0.558566659"
    ],
    [
      "0.152603249",
      "0.558566659"
    ],
    [
      "0.944143334",
      "0.614303799"
    ],
    [
      "0.750598004",
      "0.614394246"
    ],
    [
      "0.442842999",
      "0.614423325"
    ],
    [
      "0.554556331",
      "0.614423325"
    ],
    [
      "0.249349832",
      "0.614423325"
    ],
    [
      "0.055856666",
      "0.614423325"
    ],
    [
      "0.847344588",
      "0.670250912"
    ],
    [
      "0.653851421",
      "0.670250912"
    ],
    [
      "0.152603249",
      "0.670279991"
    ],
    [
      "0.346160913",
      "0.670391704"
    ],
    [
      "0.944143334",
      "0.726017131"
    ],
    [
      "0.750598004",
      "0.726107578"
    ],
    [
      "0.557104841",
      "0.726107583"
    ],
    [
      "0.443152339",
      "0.726136228"
    ],
    [
      "0.249349832",
      "0.726136657"
    ],
    [
      "0.055856666",
      "0.726136657"
    ],
    [
      "0.847344588",
      "0.781964244"
    ],
    [
      "0.653851424",
      "0.781964249"
    ],
    [
      "0.153248226",
      "0.783110456"
    ],
    [
      "0.353073533",
      "0.7922095"
    ],
    [
      "0.500152745",
      "0.822213396"
    ],
    [
      "0.944143334",
      "0.837730463"
    ],
    [
      "0.750598008",
      "0.837820915"
    ],
    [
      "0.251102609",
      "0.837836237"
    ],
    [
      "0.055856666",
      "0.837849988"
    ],
    [
      "0.596899328",
      "0.878070062"
    ],
    [
      "0.41007394",
      "0.888286668"
    ],
    [
      "0.201779166",
      "0.938071293"
    ],
    [
      "0.798559489",
      "0.938714744"
    ],
    [
      "0.090230975",
      "0.944143334"
    ],
    [
      "0.313327356",
      "0.944143334"
    ],
    [
      "0.506820523",
      "0.944143334"
    ],
    [
      "0.686978134",
      "0.944143334"
    ],
    [
      "0.910140845",
      "0.944143334"
    ]
  ],
  "radius": "0.055856665"
}
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": 84
}

起步布局答案

{
  "centers": [
    [
      "0.055856666",
      "0.055856666"
    ],
    [
      "0.249349832",
      "0.055856666"
    ],
    [
      "0.442843012",
      "0.055856666"
    ],
    [
      "0.554556344",
      "0.055856666"
    ],
    [
      "0.748084935",
      "0.055856666"
    ],
    [
      "0.944143334",
      "0.055856666"
    ],
    [
      "0.846114134",
      "0.109430395"
    ],
    [
      "0.651320639",
      "0.111682642"
    ],
    [
      "0.346096422",
      "0.111713321"
    ],
    [
      "0.152603249",
      "0.111713332"
    ],
    [
      "0.941524945",
      "0.167539308"
    ],
    [
      "0.7480495",
      "0.167569992"
    ],
    [
      "0.442842999",
      "0.167569998"
    ],
    [
      "0.554556331",
      "0.167569998"
    ],
    [
      "0.249349832",
      "0.167569998"
    ],
    [
      "0.055856666",
      "0.167569998"
    ],
    [
      "0.844796084",
      "0.223426658"
    ],
    [
      "0.651302917",
      "0.223426658"
    ],
    [
      "0.346096416",
      "0.223426664"
    ],
    [
      "0.152603249",
      "0.223426664"
    ],
    [
      "0.944143334",
      "0.27922195"
    ],
    [
      "0.7480495",
      "0.279283324"
    ],
    [
      "0.442842999",
      "0.279283329"
    ],
    [
      "0.554556331",
      "0.279283329"
    ],
    [
      "0.249349832",
      "0.279283329"
    ],
    [
      "0.055856666",
      "0.279283329"
    ],
    [
      "0.844796084",
      "0.33513999"
    ],
    [
      "0.651302917",
      "0.33513999"
    ],
    [
      "0.346096416",
      "0.335139995"
    ],
    [
      "0.152603249",
      "0.335139995"
    ],
    [
      "0.94159483",
      "0.390906209"
    ],
    [
      "0.7480495",
      "0.390996656"
    ],
    [
      "0.442842999",
      "0.390996661"
    ],
    [
      "0.554556331",
      "0.390996661"
    ],
    [
      "0.249349832",
      "0.390996661"
    ],
    [
      "0.055856666",
      "0.390996661"
    ],
    [
      "0.844796084",
      "0.446853321"
    ],
    [
      "0.651302917",
      "0.446853321"
    ],
    [
      "0.346096416",
      "0.446853327"
    ],
    [
      "0.152603249",
      "0.446853327"
    ],
    [
      "0.944143334",
      "0.502590467"
    ],
    [
      "0.7480495",
      "0.502709987"
    ],
    [
      "0.442842999",
      "0.502709993"
    ],
    [
      "0.554556331",
      "0.502709993"
    ],
    [
      "0.249349832",
      "0.502709993"
    ],
    [
      "0.055856666",
      "0.502709993"
    ],
    [
      "0.847344588",
      "0.55853758"
    ],
    [
      "0.651302917",
      "0.558566653"
    ],
    [
      "0.346096416",
      "0.558566659"
    ],
    [
      "0.152603249",
      "0.558566659"
    ],
    [
      "0.944143334",
      "0.614303799"
    ],
    [
      "0.750598004",
      "0.614394246"
    ],
    [
      "0.442842999",
      "0.614423325"
    ],
    [
      "0.554556331",
      "0.614423325"
    ],
    [
      "0.249349832",
      "0.614423325"
    ],
    [
      "0.055856666",
      "0.614423325"
    ],
    [
      "0.847344588",
      "0.670250912"
    ],
    [
      "0.653851421",
      "0.670250912"
    ],
    [
      "0.152603249",
      "0.670279991"
    ],
    [
      "0.346160913",
      "0.670391704"
    ],
    [
      "0.944143334",
      "0.726017131"
    ],
    [
      "0.750598004",
      "0.726107578"
    ],
    [
      "0.557104841",
      "0.726107583"
    ],
    [
      "0.443152339",
      "0.726136228"
    ],
    [
      "0.249349832",
      "0.726136657"
    ],
    [
      "0.055856666",
      "0.726136657"
    ],
    [
      "0.847344588",
      "0.781964244"
    ],
    [
      "0.653851424",
      "0.781964249"
    ],
    [
      "0.153248226",
      "0.783110456"
    ],
    [
      "0.353073533",
      "0.7922095"
    ],
    [
      "0.500152745",
      "0.822213396"
    ],
    [
      "0.944143334",
      "0.837730463"
    ],
    [
      "0.750598008",
      "0.837820915"
    ],
    [
      "0.251102609",
      "0.837836237"
    ],
    [
      "0.055856666",
      "0.837849988"
    ],
    [
      "0.596899328",
      "0.878070062"
    ],
    [
      "0.41007394",
      "0.888286668"
    ],
    [
      "0.201779166",
      "0.938071293"
    ],
    [
      "0.798559489",
      "0.938714744"
    ],
    [
      "0.090230975",
      "0.944143334"
    ],
    [
      "0.313327356",
      "0.944143334"
    ],
    [
      "0.506820523",
      "0.944143334"
    ],
    [
      "0.686978134",
      "0.944143334"
    ],
    [
      "0.910140845",
      "0.944143334"
    ]
  ],
  "radius": "0.055856665"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.25"。 · 验证器 v1.0.0

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