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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 = 116

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 = 116
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[0.0477759609935, 0.0477759609945]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 = 0.047775960

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 = 116 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": [
    [
      "0.047775961",
      "0.047775961"
    ],
    [
      "0.213276745",
      "0.047775961"
    ],
    [
      "0.378369621",
      "0.047775961"
    ],
    [
      "0.543870404",
      "0.047775961"
    ],
    [
      "0.709371188",
      "0.047775961"
    ],
    [
      "0.874448666",
      "0.047775961"
    ],
    [
      "0.130526353",
      "0.095551922"
    ],
    [
      "0.461120013",
      "0.095551922"
    ],
    [
      "0.626620796",
      "0.095551922"
    ],
    [
      "0.295823183",
      "0.095903452"
    ],
    [
      "0.791909927",
      "0.095916654"
    ],
    [
      "0.952224039",
      "0.10328417"
    ],
    [
      "0.047775961",
      "0.143327883"
    ],
    [
      "0.543870404",
      "0.143327883"
    ],
    [
      "0.213072791",
      "0.143679413"
    ],
    [
      "0.378573575",
      "0.143679413"
    ],
    [
      "0.709159535",
      "0.143692615"
    ],
    [
      "0.8696853",
      "0.151424863"
    ],
    [
      "0.130322399",
      "0.191455374"
    ],
    [
      "0.295823183",
      "0.191455374"
    ],
    [
      "0.461323966",
      "0.191455374"
    ],
    [
      "0.626409144",
      "0.191468576"
    ],
    [
      "0.786934908",
      "0.199200824"
    ],
    [
      "0.952224039",
      "0.199565556"
    ],
    [
      "0.213072791",
      "0.239231335"
    ],
    [
      "0.378573575",
      "0.239231335"
    ],
    [
      "0.047775961",
      "0.239582864"
    ],
    [
      "0.543862706",
      "0.239596067"
    ],
    [
      "0.704184516",
      "0.246976785"
    ],
    [
      "0.869473647",
      "0.247341517"
    ],
    [
      "0.295823183",
      "0.287007296"
    ],
    [
      "0.130526353",
      "0.287358825"
    ],
    [
      "0.461112314",
      "0.287372028"
    ],
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      "0.295104275"
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      "0.047775961",
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    ],
    [
      "0.047775961",
      "0.430686708"
    ],
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      "0.213065092",
      "0.43105144"
    ],
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      "0.373590856",
      "0.438783688"
    ],
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      "0.538676034",
      "0.43879689"
    ],
    [
      "0.704176817",
      "0.43879689"
    ],
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      "0.869677601",
      "0.43879689"
    ],
    [
      "0.1303147",
      "0.478827401"
    ],
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      "0.290840465",
      "0.486559649"
    ],
    [
      "0.786927209",
      "0.486572851"
    ],
    [
      "0.621426425",
      "0.486572851"
    ],
    [
      "0.456129596",
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    ],
    [
      "0.952224039",
      "0.486924381"
    ],
    [
      "0.047775961",
      "0.526968094"
    ],
    [
      "0.208090073",
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    ],
    [
      "0.704176817",
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      "0.373379204",
      "0.534700342"
    ],
    [
      "0.538879987",
      "0.534700342"
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    [
      "0.869473647",
      "0.534700342"
    ],
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      "0.582476303"
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      "0.582476303"
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    ],
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      "0.869677601",
      "0.726155716"
    ],
    [
      "0.373590856",
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    ],
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      "0.295815484",
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    [
      "0.130526353",
      "0.782041859"
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      "0.047775961",
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      "0.524327668",
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    [
      "0.821824093",
      "0.862883221"
    ],
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      "0.917375297",
      "0.863253623"
    ],
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      "0.23333628",
      "0.864573755"
    ],
    [
      "0.429009807",
      "0.865916653"
    ],
    [
      "0.333524114",
      "0.869473647"
    ],
    [
      "0.620047422",
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    ],
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    ],
    [
      "0.76224775",
      "0.937588168"
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  ],
  "radius": "0.04777596"
}
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": 116
}

起步布局答案

{
  "centers": [
    [
      "0.047775961",
      "0.047775961"
    ],
    [
      "0.213276745",
      "0.047775961"
    ],
    [
      "0.378369621",
      "0.047775961"
    ],
    [
      "0.543870404",
      "0.047775961"
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      "0.709371188",
      "0.047775961"
    ],
    [
      "0.874448666",
      "0.047775961"
    ],
    [
      "0.130526353",
      "0.095551922"
    ],
    [
      "0.461120013",
      "0.095551922"
    ],
    [
      "0.626620796",
      "0.095551922"
    ],
    [
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      "0.095903452"
    ],
    [
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      "0.095916654"
    ],
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      "0.952224039",
      "0.10328417"
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      "0.047775961",
      "0.143327883"
    ],
    [
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    ],
    [
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    ],
    [
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    ],
    [
      "0.130322399",
      "0.191455374"
    ],
    [
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      "0.191455374"
    ],
    [
      "0.461323966",
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    ],
    [
      "0.626409144",
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    [
      "0.786934908",
      "0.199200824"
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      "0.199565556"
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    [
      "0.213072791",
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    [
      "0.378573575",
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    [
      "0.047775961",
      "0.239582864"
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      "0.869473647"
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      "0.952224039"
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      "0.952224039"
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      "0.952224039"
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      "0.952224039"
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    [
      "0.667823383",
      "0.952224039"
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      "0.856672117",
      "0.952224039"
    ],
    [
      "0.952224039",
      "0.952224039"
    ]
  ],
  "radius": "0.04777596"
}

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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啊？？！NUE_13#310 条回复空悲切/.lird#282 条回复