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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 = 67

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 = 67
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[0.0625874295415, 0.0625874295425]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 = 0.062587429

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 = 67 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": [
    [
      "0.06258743",
      "0.06258743"
    ],
    [
      "0.286709339",
      "0.06258743"
    ],
    [
      "0.93741257",
      "0.06258743"
    ],
    [
      "0.503793788",
      "0.06258743"
    ],
    [
      "0.720603355",
      "0.06258743"
    ],
    [
      "0.174648384",
      "0.118364556"
    ],
    [
      "0.395251563",
      "0.124935892"
    ],
    [
      "0.612198571",
      "0.125174555"
    ],
    [
      "0.829007963",
      "0.125174859"
    ],
    [
      "0.070313498",
      "0.187523626"
    ],
    [
      "0.286985099",
      "0.187761985"
    ],
    [
      "0.503794139",
      "0.187762289"
    ],
    [
      "0.720603355",
      "0.187762289"
    ],
    [
      "0.93741257",
      "0.187762289"
    ],
    [
      "0.17858014",
      "0.250349414"
    ],
    [
      "0.395389531",
      "0.250349718"
    ],
    [
      "0.612198747",
      "0.250349718"
    ],
    [
      "0.829007963",
      "0.250349718"
    ],
    [
      "0.06258743",
      "0.312459823"
    ],
    [
      "0.286984748",
      "0.312936844"
    ],
    [
      "0.503794139",
      "0.312937148"
    ],
    [
      "0.93741257",
      "0.312937148"
    ],
    [
      "0.720603355",
      "0.312937148"
    ],
    [
      "0.17858014",
      "0.375524274"
    ],
    [
      "0.395389531",
      "0.375524577"
    ],
    [
      "0.612198747",
      "0.375524577"
    ],
    [
      "0.829007963",
      "0.375524577"
    ],
    [
      "0.069783471",
      "0.437427668"
    ],
    [
      "0.286984748",
      "0.438111703"
    ],
    [
      "0.503794139",
      "0.438112007"
    ],
    [
      "0.720603355",
      "0.438112007"
    ],
    [
      "0.93741257",
      "0.438112007"
    ],
    [
      "0.395389468",
      "0.500699436"
    ],
    [
      "0.829007963",
      "0.500699436"
    ],
    [
      "0.612198747",
      "0.500699436"
    ],
    [
      "0.177246832",
      "0.501617788"
    ],
    [
      "0.06258743",
      "0.562395513"
    ],
    [
      "0.286178132",
      "0.563283963"
    ],
    [
      "0.503794139",
      "0.563286866"
    ],
    [
      "0.720603355",
      "0.563286866"
    ],
    [
      "0.93741257",
      "0.563286866"
    ],
    [
      "0.395389531",
      "0.625874295"
    ],
    [
      "0.829007963",
      "0.625874295"
    ],
    [
      "0.612198747",
      "0.625874295"
    ],
    [
      "0.178050458",
      "0.626790067"
    ],
    [
      "0.068543762",
      "0.687428578"
    ],
    [
      "0.286981758",
      "0.688456243"
    ],
    [
      "0.503794139",
      "0.688461725"
    ],
    [
      "0.720603355",
      "0.688461725"
    ],
    [
      "0.93741257",
      "0.688461725"
    ],
    [
      "0.829007963",
      "0.751049155"
    ],
    [
      "0.612198747",
      "0.751049155"
    ],
    [
      "0.394160093",
      "0.75316604"
    ],
    [
      "0.173847407",
      "0.755103447"
    ],
    [
      "0.06258743",
      "0.812461644"
    ],
    [
      "0.284509579",
      "0.813606687"
    ],
    [
      "0.503794139",
      "0.813636584"
    ],
    [
      "0.720603355",
      "0.813636584"
    ],
    [
      "0.93741257",
      "0.813636584"
    ],
    [
      "0.18652158",
      "0.891498262"
    ],
    [
      "0.425371155",
      "0.911199799"
    ],
    [
      "0.795848303",
      "0.913671293"
    ],
    [
      "0.302971662",
      "0.93741257"
    ],
    [
      "0.918751099",
      "0.93741257"
    ],
    [
      "0.070071498",
      "0.93741257"
    ],
    [
      "0.672945508",
      "0.93741257"
    ],
    [
      "0.547770649",
      "0.93741257"
    ]
  ],
  "radius": "0.062587429"
}
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": 67
}

起步布局答案

{
  "centers": [
    [
      "0.06258743",
      "0.06258743"
    ],
    [
      "0.286709339",
      "0.06258743"
    ],
    [
      "0.93741257",
      "0.06258743"
    ],
    [
      "0.503793788",
      "0.06258743"
    ],
    [
      "0.720603355",
      "0.06258743"
    ],
    [
      "0.174648384",
      "0.118364556"
    ],
    [
      "0.395251563",
      "0.124935892"
    ],
    [
      "0.612198571",
      "0.125174555"
    ],
    [
      "0.829007963",
      "0.125174859"
    ],
    [
      "0.070313498",
      "0.187523626"
    ],
    [
      "0.286985099",
      "0.187761985"
    ],
    [
      "0.503794139",
      "0.187762289"
    ],
    [
      "0.720603355",
      "0.187762289"
    ],
    [
      "0.93741257",
      "0.187762289"
    ],
    [
      "0.17858014",
      "0.250349414"
    ],
    [
      "0.395389531",
      "0.250349718"
    ],
    [
      "0.612198747",
      "0.250349718"
    ],
    [
      "0.829007963",
      "0.250349718"
    ],
    [
      "0.06258743",
      "0.312459823"
    ],
    [
      "0.286984748",
      "0.312936844"
    ],
    [
      "0.503794139",
      "0.312937148"
    ],
    [
      "0.93741257",
      "0.312937148"
    ],
    [
      "0.720603355",
      "0.312937148"
    ],
    [
      "0.17858014",
      "0.375524274"
    ],
    [
      "0.395389531",
      "0.375524577"
    ],
    [
      "0.612198747",
      "0.375524577"
    ],
    [
      "0.829007963",
      "0.375524577"
    ],
    [
      "0.069783471",
      "0.437427668"
    ],
    [
      "0.286984748",
      "0.438111703"
    ],
    [
      "0.503794139",
      "0.438112007"
    ],
    [
      "0.720603355",
      "0.438112007"
    ],
    [
      "0.93741257",
      "0.438112007"
    ],
    [
      "0.395389468",
      "0.500699436"
    ],
    [
      "0.829007963",
      "0.500699436"
    ],
    [
      "0.612198747",
      "0.500699436"
    ],
    [
      "0.177246832",
      "0.501617788"
    ],
    [
      "0.06258743",
      "0.562395513"
    ],
    [
      "0.286178132",
      "0.563283963"
    ],
    [
      "0.503794139",
      "0.563286866"
    ],
    [
      "0.720603355",
      "0.563286866"
    ],
    [
      "0.93741257",
      "0.563286866"
    ],
    [
      "0.395389531",
      "0.625874295"
    ],
    [
      "0.829007963",
      "0.625874295"
    ],
    [
      "0.612198747",
      "0.625874295"
    ],
    [
      "0.178050458",
      "0.626790067"
    ],
    [
      "0.068543762",
      "0.687428578"
    ],
    [
      "0.286981758",
      "0.688456243"
    ],
    [
      "0.503794139",
      "0.688461725"
    ],
    [
      "0.720603355",
      "0.688461725"
    ],
    [
      "0.93741257",
      "0.688461725"
    ],
    [
      "0.829007963",
      "0.751049155"
    ],
    [
      "0.612198747",
      "0.751049155"
    ],
    [
      "0.394160093",
      "0.75316604"
    ],
    [
      "0.173847407",
      "0.755103447"
    ],
    [
      "0.06258743",
      "0.812461644"
    ],
    [
      "0.284509579",
      "0.813606687"
    ],
    [
      "0.503794139",
      "0.813636584"
    ],
    [
      "0.720603355",
      "0.813636584"
    ],
    [
      "0.93741257",
      "0.813636584"
    ],
    [
      "0.18652158",
      "0.891498262"
    ],
    [
      "0.425371155",
      "0.911199799"
    ],
    [
      "0.795848303",
      "0.913671293"
    ],
    [
      "0.302971662",
      "0.93741257"
    ],
    [
      "0.918751099",
      "0.93741257"
    ],
    [
      "0.070071498",
      "0.93741257"
    ],
    [
      "0.672945508",
      "0.93741257"
    ],
    [
      "0.547770649",
      "0.93741257"
    ]
  ],
  "radius": "0.062587429"
}

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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