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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 = 77

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 = 77
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[0.0578525779155, 0.0578525779165]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 = 0.057852577

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 = 77 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": [
    [
      "0.057852578",
      "0.057852578"
    ],
    [
      "0.257276452",
      "0.057852578"
    ],
    [
      "0.452703754",
      "0.057852578"
    ],
    [
      "0.653110962",
      "0.057852578"
    ],
    [
      "0.853518171",
      "0.057852578"
    ],
    [
      "0.552907358",
      "0.115705156"
    ],
    [
      "0.753314567",
      "0.115705156"
    ],
    [
      "0.157564515",
      "0.116548512"
    ],
    [
      "0.354990103",
      "0.119818097"
    ],
    [
      "0.942147422",
      "0.132233617"
    ],
    [
      "0.653110962",
      "0.173557734"
    ],
    [
      "0.057852578",
      "0.175244446"
    ],
    [
      "0.455193707",
      "0.177670675"
    ],
    [
      "0.255278166",
      "0.178514031"
    ],
    [
      "0.841943818",
      "0.190086195"
    ],
    [
      "0.555397311",
      "0.235523253"
    ],
    [
      "0.35548177",
      "0.236366609"
    ],
    [
      "0.155566229",
      "0.237209965"
    ],
    [
      "0.741740213",
      "0.247938772"
    ],
    [
      "0.942147422",
      "0.247938772"
    ],
    [
      "0.455685374",
      "0.294219187"
    ],
    [
      "0.255769833",
      "0.295062543"
    ],
    [
      "0.057852578",
      "0.299175484"
    ],
    [
      "0.841943818",
      "0.30579135"
    ],
    [
      "0.63465589",
      "0.319818928"
    ],
    [
      "0.355973437",
      "0.352915121"
    ],
    [
      "0.158056182",
      "0.357028062"
    ],
    [
      "0.741740213",
      "0.363643928"
    ],
    [
      "0.942147422",
      "0.363643928"
    ],
    [
      "0.532959626",
      "0.380337554"
    ],
    [
      "0.057852578",
      "0.41488064"
    ],
    [
      "0.258259787",
      "0.41488064"
    ],
    [
      "0.841943818",
      "0.421496506"
    ],
    [
      "0.63465589",
      "0.435524084"
    ],
    [
      "0.428596154",
      "0.442990778"
    ],
    [
      "0.158056182",
      "0.472733218"
    ],
    [
      "0.741740213",
      "0.479349084"
    ],
    [
      "0.942147422",
      "0.479349084"
    ],
    [
      "0.531427337",
      "0.496032563"
    ],
    [
      "0.330882504",
      "0.504956297"
    ],
    [
      "0.057852578",
      "0.530585796"
    ],
    [
      "0.841943818",
      "0.537201662"
    ],
    [
      "0.63465589",
      "0.55122924"
    ],
    [
      "0.230678899",
      "0.562808875"
    ],
    [
      "0.431086108",
      "0.562808875"
    ],
    [
      "0.942147422",
      "0.59505424"
    ],
    [
      "0.741740213",
      "0.59505424"
    ],
    [
      "0.53597846",
      "0.611648178"
    ],
    [
      "0.130475295",
      "0.620661453"
    ],
    [
      "0.330882504",
      "0.620661453"
    ],
    [
      "0.841943818",
      "0.652906818"
    ],
    [
      "0.63077031",
      "0.677997166"
    ],
    [
      "0.431086108",
      "0.678514031"
    ],
    [
      "0.230678899",
      "0.678514031"
    ],
    [
      "0.057852578",
      "0.71073711"
    ],
    [
      "0.942147422",
      "0.710759396"
    ],
    [
      "0.741740213",
      "0.710759396"
    ],
    [
      "0.330882504",
      "0.736366609"
    ],
    [
      "0.531079563",
      "0.736729082"
    ],
    [
      "0.158056182",
      "0.768589688"
    ],
    [
      "0.841943818",
      "0.768611974"
    ],
    [
      "0.632525346",
      "0.793689011"
    ],
    [
      "0.430875958",
      "0.794581659"
    ],
    [
      "0.057852578",
      "0.826442266"
    ],
    [
      "0.258259787",
      "0.826442266"
    ],
    [
      "0.942147422",
      "0.826464552"
    ],
    [
      "0.742894917",
      "0.828419987"
    ],
    [
      "0.532834598",
      "0.852420926"
    ],
    [
      "0.158056182",
      "0.884294844"
    ],
    [
      "0.359020356",
      "0.885270443"
    ],
    [
      "0.83073502",
      "0.903731337"
    ],
    [
      "0.057852578",
      "0.942147422"
    ],
    [
      "0.258259787",
      "0.942147422"
    ],
    [
      "0.459780925",
      "0.942147422"
    ],
    [
      "0.60588827",
      "0.942147422"
    ],
    [
      "0.721593426",
      "0.942147422"
    ],
    [
      "0.939876614",
      "0.942147422"
    ]
  ],
  "radius": "0.057852577"
}
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": 77
}

起步布局答案

{
  "centers": [
    [
      "0.057852578",
      "0.057852578"
    ],
    [
      "0.257276452",
      "0.057852578"
    ],
    [
      "0.452703754",
      "0.057852578"
    ],
    [
      "0.653110962",
      "0.057852578"
    ],
    [
      "0.853518171",
      "0.057852578"
    ],
    [
      "0.552907358",
      "0.115705156"
    ],
    [
      "0.753314567",
      "0.115705156"
    ],
    [
      "0.157564515",
      "0.116548512"
    ],
    [
      "0.354990103",
      "0.119818097"
    ],
    [
      "0.942147422",
      "0.132233617"
    ],
    [
      "0.653110962",
      "0.173557734"
    ],
    [
      "0.057852578",
      "0.175244446"
    ],
    [
      "0.455193707",
      "0.177670675"
    ],
    [
      "0.255278166",
      "0.178514031"
    ],
    [
      "0.841943818",
      "0.190086195"
    ],
    [
      "0.555397311",
      "0.235523253"
    ],
    [
      "0.35548177",
      "0.236366609"
    ],
    [
      "0.155566229",
      "0.237209965"
    ],
    [
      "0.741740213",
      "0.247938772"
    ],
    [
      "0.942147422",
      "0.247938772"
    ],
    [
      "0.455685374",
      "0.294219187"
    ],
    [
      "0.255769833",
      "0.295062543"
    ],
    [
      "0.057852578",
      "0.299175484"
    ],
    [
      "0.841943818",
      "0.30579135"
    ],
    [
      "0.63465589",
      "0.319818928"
    ],
    [
      "0.355973437",
      "0.352915121"
    ],
    [
      "0.158056182",
      "0.357028062"
    ],
    [
      "0.741740213",
      "0.363643928"
    ],
    [
      "0.942147422",
      "0.363643928"
    ],
    [
      "0.532959626",
      "0.380337554"
    ],
    [
      "0.057852578",
      "0.41488064"
    ],
    [
      "0.258259787",
      "0.41488064"
    ],
    [
      "0.841943818",
      "0.421496506"
    ],
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      "0.63465589",
      "0.435524084"
    ],
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    ],
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      "0.158056182",
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    ],
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      "0.479349084"
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      "0.479349084"
    ],
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      "0.531427337",
      "0.496032563"
    ],
    [
      "0.330882504",
      "0.504956297"
    ],
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      "0.057852578",
      "0.530585796"
    ],
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      "0.537201662"
    ],
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      "0.55122924"
    ],
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      "0.562808875"
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      "0.431086108",
      "0.562808875"
    ],
    [
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      "0.59505424"
    ],
    [
      "0.741740213",
      "0.59505424"
    ],
    [
      "0.53597846",
      "0.611648178"
    ],
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      "0.130475295",
      "0.620661453"
    ],
    [
      "0.330882504",
      "0.620661453"
    ],
    [
      "0.841943818",
      "0.652906818"
    ],
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      "0.63077031",
      "0.677997166"
    ],
    [
      "0.431086108",
      "0.678514031"
    ],
    [
      "0.230678899",
      "0.678514031"
    ],
    [
      "0.057852578",
      "0.71073711"
    ],
    [
      "0.942147422",
      "0.710759396"
    ],
    [
      "0.741740213",
      "0.710759396"
    ],
    [
      "0.330882504",
      "0.736366609"
    ],
    [
      "0.531079563",
      "0.736729082"
    ],
    [
      "0.158056182",
      "0.768589688"
    ],
    [
      "0.841943818",
      "0.768611974"
    ],
    [
      "0.632525346",
      "0.793689011"
    ],
    [
      "0.430875958",
      "0.794581659"
    ],
    [
      "0.057852578",
      "0.826442266"
    ],
    [
      "0.258259787",
      "0.826442266"
    ],
    [
      "0.942147422",
      "0.826464552"
    ],
    [
      "0.742894917",
      "0.828419987"
    ],
    [
      "0.532834598",
      "0.852420926"
    ],
    [
      "0.158056182",
      "0.884294844"
    ],
    [
      "0.359020356",
      "0.885270443"
    ],
    [
      "0.83073502",
      "0.903731337"
    ],
    [
      "0.057852578",
      "0.942147422"
    ],
    [
      "0.258259787",
      "0.942147422"
    ],
    [
      "0.459780925",
      "0.942147422"
    ],
    [
      "0.60588827",
      "0.942147422"
    ],
    [
      "0.721593426",
      "0.942147422"
    ],
    [
      "0.939876614",
      "0.942147422"
    ]
  ],
  "radius": "0.057852577"
}

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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