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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 = 78

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 = 78
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[0.0577024767345, 0.0577024767355]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 = 0.057702476

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 = 78 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": [
    [
      "0.057702477",
      "0.057702477"
    ],
    [
      "0.256460827",
      "0.057702477"
    ],
    [
      "0.45634807",
      "0.057702477"
    ],
    [
      "0.656235312",
      "0.057702477"
    ],
    [
      "0.856122555",
      "0.057702477"
    ],
    [
      "0.356404448",
      "0.115404953"
    ],
    [
      "0.556291691",
      "0.115404953"
    ],
    [
      "0.756178934",
      "0.115404953"
    ],
    [
      "0.157081652",
      "0.116371744"
    ],
    [
      "0.942297523",
      "0.134463002"
    ],
    [
      "0.45634807",
      "0.17310743"
    ],
    [
      "0.656235312",
      "0.17310743"
    ],
    [
      "0.257025273",
      "0.17407422"
    ],
    [
      "0.057702477",
      "0.17504101"
    ],
    [
      "0.842353902",
      "0.192165478"
    ],
    [
      "0.556291691",
      "0.230809907"
    ],
    [
      "0.356968895",
      "0.231776697"
    ],
    [
      "0.157646098",
      "0.232743487"
    ],
    [
      "0.74241028",
      "0.249867955"
    ],
    [
      "0.942297523",
      "0.249867955"
    ],
    [
      "0.456912516",
      "0.289479174"
    ],
    [
      "0.057702477",
      "0.290445964"
    ],
    [
      "0.25758972",
      "0.290445964"
    ],
    [
      "0.642466659",
      "0.307570432"
    ],
    [
      "0.842353902",
      "0.307570432"
    ],
    [
      "0.157646098",
      "0.348148441"
    ],
    [
      "0.357533341",
      "0.348148441"
    ],
    [
      "0.74241028",
      "0.365272909"
    ],
    [
      "0.942297523",
      "0.365272909"
    ],
    [
      "0.540356248",
      "0.36920023"
    ],
    [
      "0.057702477",
      "0.405850917"
    ],
    [
      "0.25758972",
      "0.405850917"
    ],
    [
      "0.642466659",
      "0.422975385"
    ],
    [
      "0.842353902",
      "0.422975385"
    ],
    [
      "0.438462069",
      "0.430421427"
    ],
    [
      "0.157646098",
      "0.463553394"
    ],
    [
      "0.74241028",
      "0.480677862"
    ],
    [
      "0.942297523",
      "0.480677862"
    ],
    [
      "0.540356248",
      "0.484605184"
    ],
    [
      "0.338518447",
      "0.488123904"
    ],
    [
      "0.057702477",
      "0.521255871"
    ],
    [
      "0.642466659",
      "0.538380339"
    ],
    [
      "0.842353902",
      "0.538380339"
    ],
    [
      "0.238574826",
      "0.545826381"
    ],
    [
      "0.438462069",
      "0.545826381"
    ],
    [
      "0.74241028",
      "0.596082815"
    ],
    [
      "0.942297523",
      "0.596082815"
    ],
    [
      "0.540356248",
      "0.600010137"
    ],
    [
      "0.138631204",
      "0.603528857"
    ],
    [
      "0.338518447",
      "0.603528857"
    ],
    [
      "0.642466659",
      "0.653785292"
    ],
    [
      "0.842353902",
      "0.653785292"
    ],
    [
      "0.238574826",
      "0.661231334"
    ],
    [
      "0.438462069",
      "0.661231334"
    ],
    [
      "0.057702477",
      "0.685801844"
    ],
    [
      "0.74241028",
      "0.711487769"
    ],
    [
      "0.942297523",
      "0.711487769"
    ],
    [
      "0.540356248",
      "0.715415091"
    ],
    [
      "0.338518447",
      "0.718933811"
    ],
    [
      "0.157646098",
      "0.74350432"
    ],
    [
      "0.642466659",
      "0.769190246"
    ],
    [
      "0.842353902",
      "0.769190246"
    ],
    [
      "0.439616118",
      "0.777790337"
    ],
    [
      "0.057702477",
      "0.801206797"
    ],
    [
      "0.74241028",
      "0.826892722"
    ],
    [
      "0.942297523",
      "0.826892722"
    ],
    [
      "0.544840803",
      "0.830732878"
    ],
    [
      "0.337416721",
      "0.834333505"
    ],
    [
      "0.222115382",
      "0.839222734"
    ],
    [
      "0.840093149",
      "0.888344822"
    ],
    [
      "0.626696903",
      "0.912083244"
    ],
    [
      "0.057702477",
      "0.916611751"
    ],
    [
      "0.400467498",
      "0.930992197"
    ],
    [
      "0.170212673",
      "0.942297523"
    ],
    [
      "0.285617627",
      "0.942297523"
    ],
    [
      "0.515317369",
      "0.942297523"
    ],
    [
      "0.738076438",
      "0.942297523"
    ],
    [
      "0.94210986",
      "0.942297523"
    ]
  ],
  "radius": "0.057702476"
}
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": 78
}

起步布局答案

{
  "centers": [
    [
      "0.057702477",
      "0.057702477"
    ],
    [
      "0.256460827",
      "0.057702477"
    ],
    [
      "0.45634807",
      "0.057702477"
    ],
    [
      "0.656235312",
      "0.057702477"
    ],
    [
      "0.856122555",
      "0.057702477"
    ],
    [
      "0.356404448",
      "0.115404953"
    ],
    [
      "0.556291691",
      "0.115404953"
    ],
    [
      "0.756178934",
      "0.115404953"
    ],
    [
      "0.157081652",
      "0.116371744"
    ],
    [
      "0.942297523",
      "0.134463002"
    ],
    [
      "0.45634807",
      "0.17310743"
    ],
    [
      "0.656235312",
      "0.17310743"
    ],
    [
      "0.257025273",
      "0.17407422"
    ],
    [
      "0.057702477",
      "0.17504101"
    ],
    [
      "0.842353902",
      "0.192165478"
    ],
    [
      "0.556291691",
      "0.230809907"
    ],
    [
      "0.356968895",
      "0.231776697"
    ],
    [
      "0.157646098",
      "0.232743487"
    ],
    [
      "0.74241028",
      "0.249867955"
    ],
    [
      "0.942297523",
      "0.249867955"
    ],
    [
      "0.456912516",
      "0.289479174"
    ],
    [
      "0.057702477",
      "0.290445964"
    ],
    [
      "0.25758972",
      "0.290445964"
    ],
    [
      "0.642466659",
      "0.307570432"
    ],
    [
      "0.842353902",
      "0.307570432"
    ],
    [
      "0.157646098",
      "0.348148441"
    ],
    [
      "0.357533341",
      "0.348148441"
    ],
    [
      "0.74241028",
      "0.365272909"
    ],
    [
      "0.942297523",
      "0.365272909"
    ],
    [
      "0.540356248",
      "0.36920023"
    ],
    [
      "0.057702477",
      "0.405850917"
    ],
    [
      "0.25758972",
      "0.405850917"
    ],
    [
      "0.642466659",
      "0.422975385"
    ],
    [
      "0.842353902",
      "0.422975385"
    ],
    [
      "0.438462069",
      "0.430421427"
    ],
    [
      "0.157646098",
      "0.463553394"
    ],
    [
      "0.74241028",
      "0.480677862"
    ],
    [
      "0.942297523",
      "0.480677862"
    ],
    [
      "0.540356248",
      "0.484605184"
    ],
    [
      "0.338518447",
      "0.488123904"
    ],
    [
      "0.057702477",
      "0.521255871"
    ],
    [
      "0.642466659",
      "0.538380339"
    ],
    [
      "0.842353902",
      "0.538380339"
    ],
    [
      "0.238574826",
      "0.545826381"
    ],
    [
      "0.438462069",
      "0.545826381"
    ],
    [
      "0.74241028",
      "0.596082815"
    ],
    [
      "0.942297523",
      "0.596082815"
    ],
    [
      "0.540356248",
      "0.600010137"
    ],
    [
      "0.138631204",
      "0.603528857"
    ],
    [
      "0.338518447",
      "0.603528857"
    ],
    [
      "0.642466659",
      "0.653785292"
    ],
    [
      "0.842353902",
      "0.653785292"
    ],
    [
      "0.238574826",
      "0.661231334"
    ],
    [
      "0.438462069",
      "0.661231334"
    ],
    [
      "0.057702477",
      "0.685801844"
    ],
    [
      "0.74241028",
      "0.711487769"
    ],
    [
      "0.942297523",
      "0.711487769"
    ],
    [
      "0.540356248",
      "0.715415091"
    ],
    [
      "0.338518447",
      "0.718933811"
    ],
    [
      "0.157646098",
      "0.74350432"
    ],
    [
      "0.642466659",
      "0.769190246"
    ],
    [
      "0.842353902",
      "0.769190246"
    ],
    [
      "0.439616118",
      "0.777790337"
    ],
    [
      "0.057702477",
      "0.801206797"
    ],
    [
      "0.74241028",
      "0.826892722"
    ],
    [
      "0.942297523",
      "0.826892722"
    ],
    [
      "0.544840803",
      "0.830732878"
    ],
    [
      "0.337416721",
      "0.834333505"
    ],
    [
      "0.222115382",
      "0.839222734"
    ],
    [
      "0.840093149",
      "0.888344822"
    ],
    [
      "0.626696903",
      "0.912083244"
    ],
    [
      "0.057702477",
      "0.916611751"
    ],
    [
      "0.400467498",
      "0.930992197"
    ],
    [
      "0.170212673",
      "0.942297523"
    ],
    [
      "0.285617627",
      "0.942297523"
    ],
    [
      "0.515317369",
      "0.942297523"
    ],
    [
      "0.738076438",
      "0.942297523"
    ],
    [
      "0.94210986",
      "0.942297523"
    ]
  ],
  "radius": "0.057702476"
}

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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