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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 = 103

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 = 103
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[0.0505344976075, 0.0505344976085]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 = 0.050534497

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 = 103 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": [
    [
      "0.050534498",
      "0.050534498"
    ],
    [
      "0.225591132",
      "0.050534498"
    ],
    [
      "0.400647767",
      "0.050534498"
    ],
    [
      "0.599352233",
      "0.050534498"
    ],
    [
      "0.774408868",
      "0.050534498"
    ],
    [
      "0.949465502",
      "0.050534498"
    ],
    [
      "0.5",
      "0.069083773"
    ],
    [
      "0.138062815",
      "0.101068995"
    ],
    [
      "0.31311945",
      "0.101068995"
    ],
    [
      "0.68688055",
      "0.101068995"
    ],
    [
      "0.861937185",
      "0.101068995"
    ],
    [
      "0.050534498",
      "0.151603493"
    ],
    [
      "0.225591132",
      "0.151603493"
    ],
    [
      "0.400647767",
      "0.151603493"
    ],
    [
      "0.599352233",
      "0.151603493"
    ],
    [
      "0.774408868",
      "0.151603493"
    ],
    [
      "0.949465502",
      "0.151603493"
    ],
    [
      "0.5",
      "0.170152768"
    ],
    [
      "0.138062815",
      "0.20213799"
    ],
    [
      "0.31311945",
      "0.20213799"
    ],
    [
      "0.68688055",
      "0.20213799"
    ],
    [
      "0.861937185",
      "0.20213799"
    ],
    [
      "0.050534498",
      "0.252672488"
    ],
    [
      "0.225591132",
      "0.252672488"
    ],
    [
      "0.400647767",
      "0.252672488"
    ],
    [
      "0.599352233",
      "0.252672488"
    ],
    [
      "0.774408868",
      "0.252672488"
    ],
    [
      "0.949465502",
      "0.252672488"
    ],
    [
      "0.5",
      "0.271221764"
    ],
    [
      "0.138062815",
      "0.303206986"
    ],
    [
      "0.31311945",
      "0.303206986"
    ],
    [
      "0.68688055",
      "0.303206986"
    ],
    [
      "0.861937185",
      "0.303206986"
    ],
    [
      "0.050534498",
      "0.353741483"
    ],
    [
      "0.225591132",
      "0.353741483"
    ],
    [
      "0.400647767",
      "0.353741483"
    ],
    [
      "0.599352233",
      "0.353741483"
    ],
    [
      "0.774408868",
      "0.353741483"
    ],
    [
      "0.949465502",
      "0.353741483"
    ],
    [
      "0.5",
      "0.372290759"
    ],
    [
      "0.138062815",
      "0.404275981"
    ],
    [
      "0.31311945",
      "0.404275981"
    ],
    [
      "0.68688055",
      "0.404275981"
    ],
    [
      "0.861937185",
      "0.404275981"
    ],
    [
      "0.050534498",
      "0.454810478"
    ],
    [
      "0.225591132",
      "0.454810478"
    ],
    [
      "0.400647767",
      "0.454810478"
    ],
    [
      "0.599352233",
      "0.454810478"
    ],
    [
      "0.774408868",
      "0.454810478"
    ],
    [
      "0.949465502",
      "0.454810478"
    ],
    [
      "0.5",
      "0.473359754"
    ],
    [
      "0.138062815",
      "0.505344976"
    ],
    [
      "0.31311945",
      "0.505344976"
    ],
    [
      "0.68688055",
      "0.505344976"
    ],
    [
      "0.861937185",
      "0.505344976"
    ],
    [
      "0.050534498",
      "0.555879474"
    ],
    [
      "0.225591132",
      "0.555879474"
    ],
    [
      "0.400647767",
      "0.555879474"
    ],
    [
      "0.599352233",
      "0.555879474"
    ],
    [
      "0.774408868",
      "0.555879474"
    ],
    [
      "0.949465502",
      "0.555879474"
    ],
    [
      "0.5",
      "0.574428749"
    ],
    [
      "0.138062815",
      "0.606413971"
    ],
    [
      "0.31311945",
      "0.606413971"
    ],
    [
      "0.68688055",
      "0.606413971"
    ],
    [
      "0.861937185",
      "0.606413971"
    ],
    [
      "0.050534498",
      "0.656948469"
    ],
    [
      "0.225591132",
      "0.656948469"
    ],
    [
      "0.400647767",
      "0.656948469"
    ],
    [
      "0.599352233",
      "0.656948469"
    ],
    [
      "0.774408868",
      "0.656948469"
    ],
    [
      "0.949465502",
      "0.656948469"
    ],
    [
      "0.5",
      "0.675497745"
    ],
    [
      "0.138062815",
      "0.707482967"
    ],
    [
      "0.31311945",
      "0.707482967"
    ],
    [
      "0.68688055",
      "0.707482967"
    ],
    [
      "0.861937185",
      "0.707482967"
    ],
    [
      "0.050534498",
      "0.758017464"
    ],
    [
      "0.225591132",
      "0.758017464"
    ],
    [
      "0.400647767",
      "0.758017464"
    ],
    [
      "0.599352233",
      "0.758017464"
    ],
    [
      "0.774408868",
      "0.758017464"
    ],
    [
      "0.949465502",
      "0.758017464"
    ],
    [
      "0.5",
      "0.77656674"
    ],
    [
      "0.14242562",
      "0.816054577"
    ],
    [
      "0.85757438",
      "0.816054577"
    ],
    [
      "0.339056426",
      "0.838151405"
    ],
    [
      "0.660943574",
      "0.838151405"
    ],
    [
      "0.438408659",
      "0.85670068"
    ],
    [
      "0.561591341",
      "0.85670068"
    ],
    [
      "0.239967384",
      "0.858058781"
    ],
    [
      "0.760032616",
      "0.858058781"
    ],
    [
      "0.050534498",
      "0.859086459"
    ],
    [
      "0.949465502",
      "0.859086459"
    ],
    [
      "0.5",
      "0.939069437"
    ],
    [
      "0.297565499",
      "0.941109362"
    ],
    [
      "0.702434501",
      "0.941109362"
    ],
    [
      "0.095773534",
      "0.949465502"
    ],
    [
      "0.196842529",
      "0.949465502"
    ],
    [
      "0.398288469",
      "0.949465502"
    ],
    [
      "0.601711531",
      "0.949465502"
    ],
    [
      "0.803157471",
      "0.949465502"
    ],
    [
      "0.904226466",
      "0.949465502"
    ]
  ],
  "radius": "0.050534497"
}
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": 103
}

起步布局答案

{
  "centers": [
    [
      "0.050534498",
      "0.050534498"
    ],
    [
      "0.225591132",
      "0.050534498"
    ],
    [
      "0.400647767",
      "0.050534498"
    ],
    [
      "0.599352233",
      "0.050534498"
    ],
    [
      "0.774408868",
      "0.050534498"
    ],
    [
      "0.949465502",
      "0.050534498"
    ],
    [
      "0.5",
      "0.069083773"
    ],
    [
      "0.138062815",
      "0.101068995"
    ],
    [
      "0.31311945",
      "0.101068995"
    ],
    [
      "0.68688055",
      "0.101068995"
    ],
    [
      "0.861937185",
      "0.101068995"
    ],
    [
      "0.050534498",
      "0.151603493"
    ],
    [
      "0.225591132",
      "0.151603493"
    ],
    [
      "0.400647767",
      "0.151603493"
    ],
    [
      "0.599352233",
      "0.151603493"
    ],
    [
      "0.774408868",
      "0.151603493"
    ],
    [
      "0.949465502",
      "0.151603493"
    ],
    [
      "0.5",
      "0.170152768"
    ],
    [
      "0.138062815",
      "0.20213799"
    ],
    [
      "0.31311945",
      "0.20213799"
    ],
    [
      "0.68688055",
      "0.20213799"
    ],
    [
      "0.861937185",
      "0.20213799"
    ],
    [
      "0.050534498",
      "0.252672488"
    ],
    [
      "0.225591132",
      "0.252672488"
    ],
    [
      "0.400647767",
      "0.252672488"
    ],
    [
      "0.599352233",
      "0.252672488"
    ],
    [
      "0.774408868",
      "0.252672488"
    ],
    [
      "0.949465502",
      "0.252672488"
    ],
    [
      "0.5",
      "0.271221764"
    ],
    [
      "0.138062815",
      "0.303206986"
    ],
    [
      "0.31311945",
      "0.303206986"
    ],
    [
      "0.68688055",
      "0.303206986"
    ],
    [
      "0.861937185",
      "0.303206986"
    ],
    [
      "0.050534498",
      "0.353741483"
    ],
    [
      "0.225591132",
      "0.353741483"
    ],
    [
      "0.400647767",
      "0.353741483"
    ],
    [
      "0.599352233",
      "0.353741483"
    ],
    [
      "0.774408868",
      "0.353741483"
    ],
    [
      "0.949465502",
      "0.353741483"
    ],
    [
      "0.5",
      "0.372290759"
    ],
    [
      "0.138062815",
      "0.404275981"
    ],
    [
      "0.31311945",
      "0.404275981"
    ],
    [
      "0.68688055",
      "0.404275981"
    ],
    [
      "0.861937185",
      "0.404275981"
    ],
    [
      "0.050534498",
      "0.454810478"
    ],
    [
      "0.225591132",
      "0.454810478"
    ],
    [
      "0.400647767",
      "0.454810478"
    ],
    [
      "0.599352233",
      "0.454810478"
    ],
    [
      "0.774408868",
      "0.454810478"
    ],
    [
      "0.949465502",
      "0.454810478"
    ],
    [
      "0.5",
      "0.473359754"
    ],
    [
      "0.138062815",
      "0.505344976"
    ],
    [
      "0.31311945",
      "0.505344976"
    ],
    [
      "0.68688055",
      "0.505344976"
    ],
    [
      "0.861937185",
      "0.505344976"
    ],
    [
      "0.050534498",
      "0.555879474"
    ],
    [
      "0.225591132",
      "0.555879474"
    ],
    [
      "0.400647767",
      "0.555879474"
    ],
    [
      "0.599352233",
      "0.555879474"
    ],
    [
      "0.774408868",
      "0.555879474"
    ],
    [
      "0.949465502",
      "0.555879474"
    ],
    [
      "0.5",
      "0.574428749"
    ],
    [
      "0.138062815",
      "0.606413971"
    ],
    [
      "0.31311945",
      "0.606413971"
    ],
    [
      "0.68688055",
      "0.606413971"
    ],
    [
      "0.861937185",
      "0.606413971"
    ],
    [
      "0.050534498",
      "0.656948469"
    ],
    [
      "0.225591132",
      "0.656948469"
    ],
    [
      "0.400647767",
      "0.656948469"
    ],
    [
      "0.599352233",
      "0.656948469"
    ],
    [
      "0.774408868",
      "0.656948469"
    ],
    [
      "0.949465502",
      "0.656948469"
    ],
    [
      "0.5",
      "0.675497745"
    ],
    [
      "0.138062815",
      "0.707482967"
    ],
    [
      "0.31311945",
      "0.707482967"
    ],
    [
      "0.68688055",
      "0.707482967"
    ],
    [
      "0.861937185",
      "0.707482967"
    ],
    [
      "0.050534498",
      "0.758017464"
    ],
    [
      "0.225591132",
      "0.758017464"
    ],
    [
      "0.400647767",
      "0.758017464"
    ],
    [
      "0.599352233",
      "0.758017464"
    ],
    [
      "0.774408868",
      "0.758017464"
    ],
    [
      "0.949465502",
      "0.758017464"
    ],
    [
      "0.5",
      "0.77656674"
    ],
    [
      "0.14242562",
      "0.816054577"
    ],
    [
      "0.85757438",
      "0.816054577"
    ],
    [
      "0.339056426",
      "0.838151405"
    ],
    [
      "0.660943574",
      "0.838151405"
    ],
    [
      "0.438408659",
      "0.85670068"
    ],
    [
      "0.561591341",
      "0.85670068"
    ],
    [
      "0.239967384",
      "0.858058781"
    ],
    [
      "0.760032616",
      "0.858058781"
    ],
    [
      "0.050534498",
      "0.859086459"
    ],
    [
      "0.949465502",
      "0.859086459"
    ],
    [
      "0.5",
      "0.939069437"
    ],
    [
      "0.297565499",
      "0.941109362"
    ],
    [
      "0.702434501",
      "0.941109362"
    ],
    [
      "0.095773534",
      "0.949465502"
    ],
    [
      "0.196842529",
      "0.949465502"
    ],
    [
      "0.398288469",
      "0.949465502"
    ],
    [
      "0.601711531",
      "0.949465502"
    ],
    [
      "0.803157471",
      "0.949465502"
    ],
    [
      "0.904226466",
      "0.949465502"
    ]
  ],
  "radius": "0.050534497"
}

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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