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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 = 68

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 = 68
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[0.1083450177044750438137458049165, 0.1083450177044750438137458049175]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 = 0.108345017

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 = 68 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": [
    [
      "0.99198595",
      "0.108381033"
    ],
    [
      "1.207298319",
      "0.132776837"
    ],
    [
      "0.777146881",
      "0.136643124"
    ],
    [
      "1.535237553",
      "0.286859502"
    ],
    [
      "0.453636343",
      "0.295347346"
    ],
    [
      "1.320712833",
      "0.317416268"
    ],
    [
      "1.101893999",
      "0.322103277"
    ],
    [
      "0.66841334",
      "0.324077349"
    ],
    [
      "0.885216788",
      "0.324460766"
    ],
    [
      "1.691455589",
      "0.437027731"
    ],
    [
      "0.299794286",
      "0.447948765"
    ],
    [
      "1.428937176",
      "0.505144963"
    ],
    [
      "1.21227425",
      "0.508572501"
    ],
    [
      "0.995597039",
      "0.51092999"
    ],
    [
      "0.562230029",
      "0.512967994"
    ],
    [
      "0.778919828",
      "0.51328748"
    ],
    [
      "1.602645631",
      "0.634682386"
    ],
    [
      "0.390286382",
      "0.644838968"
    ],
    [
      "1.818963819",
      "0.647371483"
    ],
    [
      "0.17437996",
      "0.663250603"
    ],
    [
      "1.323576551",
      "0.694497644"
    ],
    [
      "1.10597729",
      "0.697399215"
    ],
    [
      "0.88930008",
      "0.699756704"
    ],
    [
      "0.670272581",
      "0.700803572"
    ],
    [
      "1.646926555",
      "0.846799739"
    ],
    [
      "1.879841142",
      "0.855334275"
    ],
    [
      "0.352132297",
      "0.858143533"
    ],
    [
      "0.117529541",
      "0.872350104"
    ],
    [
      "1.433034752",
      "0.88151095"
    ],
    [
      "1.216357542",
      "0.883868439"
    ],
    [
      "0.999680331",
      "0.886225928"
    ],
    [
      "0.78300312",
      "0.888583418"
    ],
    [
      "0.566325909",
      "0.890940907"
    ],
    [
      "1.756694306",
      "1.033630176"
    ],
    [
      "0.245835337",
      "1.046970246"
    ],
    [
      "1.326737793",
      "1.070337663"
    ],
    [
      "1.543421735",
      "1.071962692"
    ],
    [
      "1.110060582",
      "1.072695153"
    ],
    [
      "0.893383371",
      "1.075052642"
    ],
    [
      "0.67670616",
      "1.077410132"
    ],
    [
      "0.460028949",
      "1.079767621"
    ],
    [
      "1.863538654",
      "1.2221477"
    ],
    [
      "0.140188119",
      "1.236161253"
    ],
    [
      "1.433672447",
      "1.258803976"
    ],
    [
      "1.650351814",
      "1.260954179"
    ],
    [
      "1.216995237",
      "1.261161466"
    ],
    [
      "0.787086411",
      "1.263879356"
    ],
    [
      "0.570409201",
      "1.266236845"
    ],
    [
      "0.35380966",
      "1.272498681"
    ],
    [
      "1.002213647",
      "1.289857114"
    ],
    [
      "1.327375488",
      "1.44763069"
    ],
    [
      "1.770027466",
      "1.448801167"
    ],
    [
      "1.544054855",
      "1.449780893"
    ],
    [
      "0.680789452",
      "1.45270607"
    ],
    [
      "0.236618643",
      "1.457937079"
    ],
    [
      "0.464189911",
      "1.458967905"
    ],
    [
      "1.112593898",
      "1.476326339"
    ],
    [
      "0.895916687",
      "1.478683828"
    ],
    [
      "1.244835854",
      "1.647984827"
    ],
    [
      "0.765855671",
      "1.6520006"
    ],
    [
      "1.600713428",
      "1.658932459"
    ],
    [
      "0.410368966",
      "1.668867589"
    ],
    [
      "1.000739425",
      "1.671340744"
    ],
    [
      "1.4240275",
      "1.78437828"
    ],
    [
      "0.589124183",
      "1.791346745"
    ],
    [
      "1.222299041",
      "1.863499707"
    ],
    [
      "0.79214521",
      "1.867089957"
    ],
    [
      "1.007441879",
      "1.891623926"
    ]
  ],
  "radius": "0.108345017"
}
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": 68
}

起步布局答案

{
  "centers": [
    [
      "0.99198595",
      "0.108381033"
    ],
    [
      "1.207298319",
      "0.132776837"
    ],
    [
      "0.777146881",
      "0.136643124"
    ],
    [
      "1.535237553",
      "0.286859502"
    ],
    [
      "0.453636343",
      "0.295347346"
    ],
    [
      "1.320712833",
      "0.317416268"
    ],
    [
      "1.101893999",
      "0.322103277"
    ],
    [
      "0.66841334",
      "0.324077349"
    ],
    [
      "0.885216788",
      "0.324460766"
    ],
    [
      "1.691455589",
      "0.437027731"
    ],
    [
      "0.299794286",
      "0.447948765"
    ],
    [
      "1.428937176",
      "0.505144963"
    ],
    [
      "1.21227425",
      "0.508572501"
    ],
    [
      "0.995597039",
      "0.51092999"
    ],
    [
      "0.562230029",
      "0.512967994"
    ],
    [
      "0.778919828",
      "0.51328748"
    ],
    [
      "1.602645631",
      "0.634682386"
    ],
    [
      "0.390286382",
      "0.644838968"
    ],
    [
      "1.818963819",
      "0.647371483"
    ],
    [
      "0.17437996",
      "0.663250603"
    ],
    [
      "1.323576551",
      "0.694497644"
    ],
    [
      "1.10597729",
      "0.697399215"
    ],
    [
      "0.88930008",
      "0.699756704"
    ],
    [
      "0.670272581",
      "0.700803572"
    ],
    [
      "1.646926555",
      "0.846799739"
    ],
    [
      "1.879841142",
      "0.855334275"
    ],
    [
      "0.352132297",
      "0.858143533"
    ],
    [
      "0.117529541",
      "0.872350104"
    ],
    [
      "1.433034752",
      "0.88151095"
    ],
    [
      "1.216357542",
      "0.883868439"
    ],
    [
      "0.999680331",
      "0.886225928"
    ],
    [
      "0.78300312",
      "0.888583418"
    ],
    [
      "0.566325909",
      "0.890940907"
    ],
    [
      "1.756694306",
      "1.033630176"
    ],
    [
      "0.245835337",
      "1.046970246"
    ],
    [
      "1.326737793",
      "1.070337663"
    ],
    [
      "1.543421735",
      "1.071962692"
    ],
    [
      "1.110060582",
      "1.072695153"
    ],
    [
      "0.893383371",
      "1.075052642"
    ],
    [
      "0.67670616",
      "1.077410132"
    ],
    [
      "0.460028949",
      "1.079767621"
    ],
    [
      "1.863538654",
      "1.2221477"
    ],
    [
      "0.140188119",
      "1.236161253"
    ],
    [
      "1.433672447",
      "1.258803976"
    ],
    [
      "1.650351814",
      "1.260954179"
    ],
    [
      "1.216995237",
      "1.261161466"
    ],
    [
      "0.787086411",
      "1.263879356"
    ],
    [
      "0.570409201",
      "1.266236845"
    ],
    [
      "0.35380966",
      "1.272498681"
    ],
    [
      "1.002213647",
      "1.289857114"
    ],
    [
      "1.327375488",
      "1.44763069"
    ],
    [
      "1.770027466",
      "1.448801167"
    ],
    [
      "1.544054855",
      "1.449780893"
    ],
    [
      "0.680789452",
      "1.45270607"
    ],
    [
      "0.236618643",
      "1.457937079"
    ],
    [
      "0.464189911",
      "1.458967905"
    ],
    [
      "1.112593898",
      "1.476326339"
    ],
    [
      "0.895916687",
      "1.478683828"
    ],
    [
      "1.244835854",
      "1.647984827"
    ],
    [
      "0.765855671",
      "1.6520006"
    ],
    [
      "1.600713428",
      "1.658932459"
    ],
    [
      "0.410368966",
      "1.668867589"
    ],
    [
      "1.000739425",
      "1.671340744"
    ],
    [
      "1.4240275",
      "1.78437828"
    ],
    [
      "0.589124183",
      "1.791346745"
    ],
    [
      "1.222299041",
      "1.863499707"
    ],
    [
      "0.79214521",
      "1.867089957"
    ],
    [
      "1.007441879",
      "1.891623926"
    ]
  ],
  "radius": "0.108345017"
}

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。

还没有帖子。第一个聊聊这道题的思路？