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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 = 85

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 = 85
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[0.0556801817675, 0.0556801817685]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 = 0.055680181

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 = 85 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": [
    [
      "0.055680182",
      "0.055680182"
    ],
    [
      "0.167040545",
      "0.055680182"
    ],
    [
      "0.278400909",
      "0.055680182"
    ],
    [
      "0.389761272",
      "0.055680182"
    ],
    [
      "0.501121636",
      "0.055680182"
    ],
    [
      "0.722659527",
      "0.055680182"
    ],
    [
      "0.944319818",
      "0.055680182"
    ],
    [
      "0.833489673",
      "0.066534185"
    ],
    [
      "0.611890581",
      "0.067141913"
    ],
    [
      "0.111360364",
      "0.152121086"
    ],
    [
      "0.222720727",
      "0.152121086"
    ],
    [
      "0.334081091",
      "0.152121086"
    ],
    [
      "0.445441454",
      "0.152121086"
    ],
    [
      "0.898304588",
      "0.157088905"
    ],
    [
      "0.5562104",
      "0.163582817"
    ],
    [
      "0.667570763",
      "0.163582817"
    ],
    [
      "0.778931122",
      "0.163614024"
    ],
    [
      "0.055680182",
      "0.248561989"
    ],
    [
      "0.167040545",
      "0.248561989"
    ],
    [
      "0.278400909",
      "0.248561989"
    ],
    [
      "0.389761272",
      "0.248561989"
    ],
    [
      "0.944319818",
      "0.258497628"
    ],
    [
      "0.610238728",
      "0.26095874"
    ],
    [
      "0.721599091",
      "0.26095874"
    ],
    [
      "0.832987007",
      "0.260974652"
    ],
    [
      "0.499186222",
      "0.269233492"
    ],
    [
      "0.111360364",
      "0.345002893"
    ],
    [
      "0.222720727",
      "0.345002893"
    ],
    [
      "0.334081091",
      "0.345002893"
    ],
    [
      "0.890798578",
      "0.356153183"
    ],
    [
      "0.665918909",
      "0.357399643"
    ],
    [
      "0.777279273",
      "0.357399643"
    ],
    [
      "0.44350604",
      "0.365674395"
    ],
    [
      "0.554866404",
      "0.365674395"
    ],
    [
      "0.055680182",
      "0.441443797"
    ],
    [
      "0.167040545",
      "0.441443797"
    ],
    [
      "0.278400909",
      "0.441443797"
    ],
    [
      "0.944319818",
      "0.453808738"
    ],
    [
      "0.832959459",
      "0.453840545"
    ],
    [
      "0.721599091",
      "0.453840547"
    ],
    [
      "0.387825858",
      "0.462115299"
    ],
    [
      "0.499186222",
      "0.462115299"
    ],
    [
      "0.610546585",
      "0.462115299"
    ],
    [
      "0.111360364",
      "0.537884701"
    ],
    [
      "0.222720727",
      "0.537884701"
    ],
    [
      "0.888667184",
      "0.550265541"
    ],
    [
      "0.777279277",
      "0.550281448"
    ],
    [
      "0.666226768",
      "0.558556203"
    ],
    [
      "0.332145677",
      "0.558556203"
    ],
    [
      "0.44350604",
      "0.558556203"
    ],
    [
      "0.554866404",
      "0.558556203"
    ],
    [
      "0.055680182",
      "0.634325605"
    ],
    [
      "0.167040545",
      "0.634325605"
    ],
    [
      "0.944319818",
      "0.646722344"
    ],
    [
      "0.832959455",
      "0.646722355"
    ],
    [
      "0.276465495",
      "0.654997107"
    ],
    [
      "0.387825858",
      "0.654997107"
    ],
    [
      "0.499186222",
      "0.654997107"
    ],
    [
      "0.610546585",
      "0.654997107"
    ],
    [
      "0.721906949",
      "0.654997107"
    ],
    [
      "0.111360364",
      "0.730766508"
    ],
    [
      "0.888639646",
      "0.743163253"
    ],
    [
      "0.777587136",
      "0.751438007"
    ],
    [
      "0.220785313",
      "0.751438011"
    ],
    [
      "0.332145677",
      "0.751438011"
    ],
    [
      "0.44350604",
      "0.751438011"
    ],
    [
      "0.554866404",
      "0.751438011"
    ],
    [
      "0.666226767",
      "0.751438011"
    ],
    [
      "0.055680182",
      "0.827207412"
    ],
    [
      "0.944319818",
      "0.839604162"
    ],
    [
      "0.276465495",
      "0.847878914"
    ],
    [
      "0.387825858",
      "0.847878914"
    ],
    [
      "0.499186222",
      "0.847878914"
    ],
    [
      "0.610546585",
      "0.847878914"
    ],
    [
      "0.721906949",
      "0.847878914"
    ],
    [
      "0.833267312",
      "0.847878914"
    ],
    [
      "0.165105131",
      "0.847878914"
    ],
    [
      "0.083520273",
      "0.941535809"
    ],
    [
      "0.896450926",
      "0.941535809"
    ],
    [
      "0.220785313",
      "0.944319818"
    ],
    [
      "0.332145677",
      "0.944319818"
    ],
    [
      "0.44350604",
      "0.944319818"
    ],
    [
      "0.554866404",
      "0.944319818"
    ],
    [
      "0.666226767",
      "0.944319818"
    ],
    [
      "0.777587131",
      "0.944319818"
    ]
  ],
  "radius": "0.055680181"
}
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": 85
}

起步布局答案

{
  "centers": [
    [
      "0.055680182",
      "0.055680182"
    ],
    [
      "0.167040545",
      "0.055680182"
    ],
    [
      "0.278400909",
      "0.055680182"
    ],
    [
      "0.389761272",
      "0.055680182"
    ],
    [
      "0.501121636",
      "0.055680182"
    ],
    [
      "0.722659527",
      "0.055680182"
    ],
    [
      "0.944319818",
      "0.055680182"
    ],
    [
      "0.833489673",
      "0.066534185"
    ],
    [
      "0.611890581",
      "0.067141913"
    ],
    [
      "0.111360364",
      "0.152121086"
    ],
    [
      "0.222720727",
      "0.152121086"
    ],
    [
      "0.334081091",
      "0.152121086"
    ],
    [
      "0.445441454",
      "0.152121086"
    ],
    [
      "0.898304588",
      "0.157088905"
    ],
    [
      "0.5562104",
      "0.163582817"
    ],
    [
      "0.667570763",
      "0.163582817"
    ],
    [
      "0.778931122",
      "0.163614024"
    ],
    [
      "0.055680182",
      "0.248561989"
    ],
    [
      "0.167040545",
      "0.248561989"
    ],
    [
      "0.278400909",
      "0.248561989"
    ],
    [
      "0.389761272",
      "0.248561989"
    ],
    [
      "0.944319818",
      "0.258497628"
    ],
    [
      "0.610238728",
      "0.26095874"
    ],
    [
      "0.721599091",
      "0.26095874"
    ],
    [
      "0.832987007",
      "0.260974652"
    ],
    [
      "0.499186222",
      "0.269233492"
    ],
    [
      "0.111360364",
      "0.345002893"
    ],
    [
      "0.222720727",
      "0.345002893"
    ],
    [
      "0.334081091",
      "0.345002893"
    ],
    [
      "0.890798578",
      "0.356153183"
    ],
    [
      "0.665918909",
      "0.357399643"
    ],
    [
      "0.777279273",
      "0.357399643"
    ],
    [
      "0.44350604",
      "0.365674395"
    ],
    [
      "0.554866404",
      "0.365674395"
    ],
    [
      "0.055680182",
      "0.441443797"
    ],
    [
      "0.167040545",
      "0.441443797"
    ],
    [
      "0.278400909",
      "0.441443797"
    ],
    [
      "0.944319818",
      "0.453808738"
    ],
    [
      "0.832959459",
      "0.453840545"
    ],
    [
      "0.721599091",
      "0.453840547"
    ],
    [
      "0.387825858",
      "0.462115299"
    ],
    [
      "0.499186222",
      "0.462115299"
    ],
    [
      "0.610546585",
      "0.462115299"
    ],
    [
      "0.111360364",
      "0.537884701"
    ],
    [
      "0.222720727",
      "0.537884701"
    ],
    [
      "0.888667184",
      "0.550265541"
    ],
    [
      "0.777279277",
      "0.550281448"
    ],
    [
      "0.666226768",
      "0.558556203"
    ],
    [
      "0.332145677",
      "0.558556203"
    ],
    [
      "0.44350604",
      "0.558556203"
    ],
    [
      "0.554866404",
      "0.558556203"
    ],
    [
      "0.055680182",
      "0.634325605"
    ],
    [
      "0.167040545",
      "0.634325605"
    ],
    [
      "0.944319818",
      "0.646722344"
    ],
    [
      "0.832959455",
      "0.646722355"
    ],
    [
      "0.276465495",
      "0.654997107"
    ],
    [
      "0.387825858",
      "0.654997107"
    ],
    [
      "0.499186222",
      "0.654997107"
    ],
    [
      "0.610546585",
      "0.654997107"
    ],
    [
      "0.721906949",
      "0.654997107"
    ],
    [
      "0.111360364",
      "0.730766508"
    ],
    [
      "0.888639646",
      "0.743163253"
    ],
    [
      "0.777587136",
      "0.751438007"
    ],
    [
      "0.220785313",
      "0.751438011"
    ],
    [
      "0.332145677",
      "0.751438011"
    ],
    [
      "0.44350604",
      "0.751438011"
    ],
    [
      "0.554866404",
      "0.751438011"
    ],
    [
      "0.666226767",
      "0.751438011"
    ],
    [
      "0.055680182",
      "0.827207412"
    ],
    [
      "0.944319818",
      "0.839604162"
    ],
    [
      "0.276465495",
      "0.847878914"
    ],
    [
      "0.387825858",
      "0.847878914"
    ],
    [
      "0.499186222",
      "0.847878914"
    ],
    [
      "0.610546585",
      "0.847878914"
    ],
    [
      "0.721906949",
      "0.847878914"
    ],
    [
      "0.833267312",
      "0.847878914"
    ],
    [
      "0.165105131",
      "0.847878914"
    ],
    [
      "0.083520273",
      "0.941535809"
    ],
    [
      "0.896450926",
      "0.941535809"
    ],
    [
      "0.220785313",
      "0.944319818"
    ],
    [
      "0.332145677",
      "0.944319818"
    ],
    [
      "0.44350604",
      "0.944319818"
    ],
    [
      "0.554866404",
      "0.944319818"
    ],
    [
      "0.666226767",
      "0.944319818"
    ],
    [
      "0.777587131",
      "0.944319818"
    ]
  ],
  "radius": "0.055680181"
}

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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