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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 = 76

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 = 76
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[0.0581985249355, 0.0581985249365]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 = 0.058198524

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 = 76 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": [
    [
      "0.058198525",
      "0.058198525"
    ],
    [
      "0.259804129",
      "0.058198525"
    ],
    [
      "0.461409733",
      "0.058198525"
    ],
    [
      "0.663015338",
      "0.058198525"
    ],
    [
      "0.864620942",
      "0.058198525"
    ],
    [
      "0.159001327",
      "0.11639705"
    ],
    [
      "0.360606931",
      "0.11639705"
    ],
    [
      "0.562212536",
      "0.11639705"
    ],
    [
      "0.76381814",
      "0.11639705"
    ],
    [
      "0.941801475",
      "0.145327386"
    ],
    [
      "0.058198525",
      "0.174595575"
    ],
    [
      "0.259804129",
      "0.174595575"
    ],
    [
      "0.461409733",
      "0.174595575"
    ],
    [
      "0.663015338",
      "0.174595575"
    ],
    [
      "0.840998673",
      "0.203525911"
    ],
    [
      "0.159001327",
      "0.2327941"
    ],
    [
      "0.360606931",
      "0.2327941"
    ],
    [
      "0.562212536",
      "0.2327941"
    ],
    [
      "0.740195871",
      "0.261724436"
    ],
    [
      "0.941801475",
      "0.261724436"
    ],
    [
      "0.058198525",
      "0.290992625"
    ],
    [
      "0.259804129",
      "0.290992625"
    ],
    [
      "0.461409733",
      "0.290992625"
    ],
    [
      "0.639393069",
      "0.319922961"
    ],
    [
      "0.840998673",
      "0.319922961"
    ],
    [
      "0.159001327",
      "0.34919115"
    ],
    [
      "0.360606931",
      "0.34919115"
    ],
    [
      "0.538590267",
      "0.378121486"
    ],
    [
      "0.740195871",
      "0.378121486"
    ],
    [
      "0.941801475",
      "0.378121486"
    ],
    [
      "0.058198525",
      "0.407389675"
    ],
    [
      "0.259804129",
      "0.407389675"
    ],
    [
      "0.437787464",
      "0.436320011"
    ],
    [
      "0.639393069",
      "0.436320011"
    ],
    [
      "0.840998673",
      "0.436320011"
    ],
    [
      "0.159672656",
      "0.466765378"
    ],
    [
      "0.336984662",
      "0.494518536"
    ],
    [
      "0.538590267",
      "0.494518536"
    ],
    [
      "0.740195871",
      "0.494518536"
    ],
    [
      "0.941801475",
      "0.494518536"
    ],
    [
      "0.058198525",
      "0.523786724"
    ],
    [
      "0.437787464",
      "0.552717061"
    ],
    [
      "0.639393069",
      "0.552717061"
    ],
    [
      "0.840998673",
      "0.552717061"
    ],
    [
      "0.158311579",
      "0.583163906"
    ],
    [
      "0.274324675",
      "0.592610325"
    ],
    [
      "0.538590267",
      "0.610915586"
    ],
    [
      "0.740195871",
      "0.610915586"
    ],
    [
      "0.941801475",
      "0.610915586"
    ],
    [
      "0.058198525",
      "0.642541087"
    ],
    [
      "0.375127477",
      "0.65080885"
    ],
    [
      "0.639393069",
      "0.669114111"
    ],
    [
      "0.840998673",
      "0.669114111"
    ],
    [
      "0.158125002",
      "0.702231725"
    ],
    [
      "0.475930279",
      "0.709007375"
    ],
    [
      "0.274324675",
      "0.709007375"
    ],
    [
      "0.941801475",
      "0.727312636"
    ],
    [
      "0.740195871",
      "0.727312636"
    ],
    [
      "0.058198525",
      "0.761922364"
    ],
    [
      "0.375127477",
      "0.7672059"
    ],
    [
      "0.576733081",
      "0.7672059"
    ],
    [
      "0.840998673",
      "0.785511161"
    ],
    [
      "0.157981847",
      "0.821852005"
    ],
    [
      "0.274324675",
      "0.825404425"
    ],
    [
      "0.475930279",
      "0.825404425"
    ],
    [
      "0.677535883",
      "0.825404425"
    ],
    [
      "0.941801475",
      "0.843709686"
    ],
    [
      "0.058198525",
      "0.881781646"
    ],
    [
      "0.375127477",
      "0.88360295"
    ],
    [
      "0.576733081",
      "0.88360295"
    ],
    [
      "0.778338685",
      "0.88360295"
    ],
    [
      "0.157927625",
      "0.941801475"
    ],
    [
      "0.274324675",
      "0.941801475"
    ],
    [
      "0.475930279",
      "0.941801475"
    ],
    [
      "0.677535883",
      "0.941801475"
    ],
    [
      "0.879141487",
      "0.941801475"
    ]
  ],
  "radius": "0.058198524"
}
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": 76
}

起步布局答案

{
  "centers": [
    [
      "0.058198525",
      "0.058198525"
    ],
    [
      "0.259804129",
      "0.058198525"
    ],
    [
      "0.461409733",
      "0.058198525"
    ],
    [
      "0.663015338",
      "0.058198525"
    ],
    [
      "0.864620942",
      "0.058198525"
    ],
    [
      "0.159001327",
      "0.11639705"
    ],
    [
      "0.360606931",
      "0.11639705"
    ],
    [
      "0.562212536",
      "0.11639705"
    ],
    [
      "0.76381814",
      "0.11639705"
    ],
    [
      "0.941801475",
      "0.145327386"
    ],
    [
      "0.058198525",
      "0.174595575"
    ],
    [
      "0.259804129",
      "0.174595575"
    ],
    [
      "0.461409733",
      "0.174595575"
    ],
    [
      "0.663015338",
      "0.174595575"
    ],
    [
      "0.840998673",
      "0.203525911"
    ],
    [
      "0.159001327",
      "0.2327941"
    ],
    [
      "0.360606931",
      "0.2327941"
    ],
    [
      "0.562212536",
      "0.2327941"
    ],
    [
      "0.740195871",
      "0.261724436"
    ],
    [
      "0.941801475",
      "0.261724436"
    ],
    [
      "0.058198525",
      "0.290992625"
    ],
    [
      "0.259804129",
      "0.290992625"
    ],
    [
      "0.461409733",
      "0.290992625"
    ],
    [
      "0.639393069",
      "0.319922961"
    ],
    [
      "0.840998673",
      "0.319922961"
    ],
    [
      "0.159001327",
      "0.34919115"
    ],
    [
      "0.360606931",
      "0.34919115"
    ],
    [
      "0.538590267",
      "0.378121486"
    ],
    [
      "0.740195871",
      "0.378121486"
    ],
    [
      "0.941801475",
      "0.378121486"
    ],
    [
      "0.058198525",
      "0.407389675"
    ],
    [
      "0.259804129",
      "0.407389675"
    ],
    [
      "0.437787464",
      "0.436320011"
    ],
    [
      "0.639393069",
      "0.436320011"
    ],
    [
      "0.840998673",
      "0.436320011"
    ],
    [
      "0.159672656",
      "0.466765378"
    ],
    [
      "0.336984662",
      "0.494518536"
    ],
    [
      "0.538590267",
      "0.494518536"
    ],
    [
      "0.740195871",
      "0.494518536"
    ],
    [
      "0.941801475",
      "0.494518536"
    ],
    [
      "0.058198525",
      "0.523786724"
    ],
    [
      "0.437787464",
      "0.552717061"
    ],
    [
      "0.639393069",
      "0.552717061"
    ],
    [
      "0.840998673",
      "0.552717061"
    ],
    [
      "0.158311579",
      "0.583163906"
    ],
    [
      "0.274324675",
      "0.592610325"
    ],
    [
      "0.538590267",
      "0.610915586"
    ],
    [
      "0.740195871",
      "0.610915586"
    ],
    [
      "0.941801475",
      "0.610915586"
    ],
    [
      "0.058198525",
      "0.642541087"
    ],
    [
      "0.375127477",
      "0.65080885"
    ],
    [
      "0.639393069",
      "0.669114111"
    ],
    [
      "0.840998673",
      "0.669114111"
    ],
    [
      "0.158125002",
      "0.702231725"
    ],
    [
      "0.475930279",
      "0.709007375"
    ],
    [
      "0.274324675",
      "0.709007375"
    ],
    [
      "0.941801475",
      "0.727312636"
    ],
    [
      "0.740195871",
      "0.727312636"
    ],
    [
      "0.058198525",
      "0.761922364"
    ],
    [
      "0.375127477",
      "0.7672059"
    ],
    [
      "0.576733081",
      "0.7672059"
    ],
    [
      "0.840998673",
      "0.785511161"
    ],
    [
      "0.157981847",
      "0.821852005"
    ],
    [
      "0.274324675",
      "0.825404425"
    ],
    [
      "0.475930279",
      "0.825404425"
    ],
    [
      "0.677535883",
      "0.825404425"
    ],
    [
      "0.941801475",
      "0.843709686"
    ],
    [
      "0.058198525",
      "0.881781646"
    ],
    [
      "0.375127477",
      "0.88360295"
    ],
    [
      "0.576733081",
      "0.88360295"
    ],
    [
      "0.778338685",
      "0.88360295"
    ],
    [
      "0.157927625",
      "0.941801475"
    ],
    [
      "0.274324675",
      "0.941801475"
    ],
    [
      "0.475930279",
      "0.941801475"
    ],
    [
      "0.677535883",
      "0.941801475"
    ],
    [
      "0.879141487",
      "0.941801475"
    ]
  ],
  "radius": "0.058198524"
}

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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