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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 = 68

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 = 68
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[0.0625200779975, 0.0625200779985]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 = 0.062520077

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 = 68 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": [
    [
      "0.062520078",
      "0.062520078"
    ],
    [
      "0.279095981",
      "0.062520078"
    ],
    [
      "0.495671884",
      "0.062520078"
    ],
    [
      "0.712247788",
      "0.062520078"
    ],
    [
      "0.928823691",
      "0.062520078"
    ],
    [
      "0.17080803",
      "0.125040156"
    ],
    [
      "0.387383933",
      "0.125040156"
    ],
    [
      "0.603959836",
      "0.125040156"
    ],
    [
      "0.820535739",
      "0.125040156"
    ],
    [
      "0.937479922",
      "0.187260249"
    ],
    [
      "0.062520078",
      "0.187560234"
    ],
    [
      "0.279095981",
      "0.187560234"
    ],
    [
      "0.495671884",
      "0.187560234"
    ],
    [
      "0.712247788",
      "0.187560234"
    ],
    [
      "0.82919197",
      "0.249780327"
    ],
    [
      "0.17080803",
      "0.250080312"
    ],
    [
      "0.387383933",
      "0.250080312"
    ],
    [
      "0.603959836",
      "0.250080312"
    ],
    [
      "0.720904019",
      "0.312300405"
    ],
    [
      "0.937479922",
      "0.312300405"
    ],
    [
      "0.062520078",
      "0.31260039"
    ],
    [
      "0.279095981",
      "0.31260039"
    ],
    [
      "0.495671884",
      "0.31260039"
    ],
    [
      "0.612616067",
      "0.374820483"
    ],
    [
      "0.82919197",
      "0.374820483"
    ],
    [
      "0.17080803",
      "0.375120468"
    ],
    [
      "0.387383933",
      "0.375120468"
    ],
    [
      "0.504328116",
      "0.437340561"
    ],
    [
      "0.720904019",
      "0.437340561"
    ],
    [
      "0.937479922",
      "0.437340561"
    ],
    [
      "0.062520078",
      "0.437640546"
    ],
    [
      "0.279095981",
      "0.437640546"
    ],
    [
      "0.612616067",
      "0.499860639"
    ],
    [
      "0.82919197",
      "0.499860639"
    ],
    [
      "0.393719791",
      "0.5"
    ],
    [
      "0.17080803",
      "0.500160624"
    ],
    [
      "0.502099514",
      "0.562360855"
    ],
    [
      "0.720904019",
      "0.562380717"
    ],
    [
      "0.937479922",
      "0.562380717"
    ],
    [
      "0.285431839",
      "0.562520078"
    ],
    [
      "0.062520078",
      "0.562680702"
    ],
    [
      "0.610387571",
      "0.624880935"
    ],
    [
      "0.82919197",
      "0.624900795"
    ],
    [
      "0.393719868",
      "0.625040156"
    ],
    [
      "0.177143888",
      "0.625040156"
    ],
    [
      "0.502099619",
      "0.687401011"
    ],
    [
      "0.718675523",
      "0.687401013"
    ],
    [
      "0.937479922",
      "0.687420873"
    ],
    [
      "0.068855936",
      "0.687560234"
    ],
    [
      "0.285431839",
      "0.687560234"
    ],
    [
      "0.827561444",
      "0.74993032"
    ],
    [
      "0.610399755",
      "0.749942285"
    ],
    [
      "0.177143888",
      "0.750080312"
    ],
    [
      "0.393719791",
      "0.750080312"
    ],
    [
      "0.062520078",
      "0.812439766"
    ],
    [
      "0.719267356",
      "0.812439768"
    ],
    [
      "0.502099514",
      "0.812441167"
    ],
    [
      "0.935849395",
      "0.812450398"
    ],
    [
      "0.285431839",
      "0.81260039"
    ],
    [
      "0.17080803",
      "0.874959844"
    ],
    [
      "0.610991682",
      "0.874981106"
    ],
    [
      "0.828385914",
      "0.876377237"
    ],
    [
      "0.391049386",
      "0.880509074"
    ],
    [
      "0.062520078",
      "0.937479922"
    ],
    [
      "0.279095981",
      "0.937479922"
    ],
    [
      "0.502691457",
      "0.937479922"
    ],
    [
      "0.719291906",
      "0.937479922"
    ],
    [
      "0.937479922",
      "0.937479922"
    ]
  ],
  "radius": "0.062520077"
}
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": 68
}

起步布局答案

{
  "centers": [
    [
      "0.062520078",
      "0.062520078"
    ],
    [
      "0.279095981",
      "0.062520078"
    ],
    [
      "0.495671884",
      "0.062520078"
    ],
    [
      "0.712247788",
      "0.062520078"
    ],
    [
      "0.928823691",
      "0.062520078"
    ],
    [
      "0.17080803",
      "0.125040156"
    ],
    [
      "0.387383933",
      "0.125040156"
    ],
    [
      "0.603959836",
      "0.125040156"
    ],
    [
      "0.820535739",
      "0.125040156"
    ],
    [
      "0.937479922",
      "0.187260249"
    ],
    [
      "0.062520078",
      "0.187560234"
    ],
    [
      "0.279095981",
      "0.187560234"
    ],
    [
      "0.495671884",
      "0.187560234"
    ],
    [
      "0.712247788",
      "0.187560234"
    ],
    [
      "0.82919197",
      "0.249780327"
    ],
    [
      "0.17080803",
      "0.250080312"
    ],
    [
      "0.387383933",
      "0.250080312"
    ],
    [
      "0.603959836",
      "0.250080312"
    ],
    [
      "0.720904019",
      "0.312300405"
    ],
    [
      "0.937479922",
      "0.312300405"
    ],
    [
      "0.062520078",
      "0.31260039"
    ],
    [
      "0.279095981",
      "0.31260039"
    ],
    [
      "0.495671884",
      "0.31260039"
    ],
    [
      "0.612616067",
      "0.374820483"
    ],
    [
      "0.82919197",
      "0.374820483"
    ],
    [
      "0.17080803",
      "0.375120468"
    ],
    [
      "0.387383933",
      "0.375120468"
    ],
    [
      "0.504328116",
      "0.437340561"
    ],
    [
      "0.720904019",
      "0.437340561"
    ],
    [
      "0.937479922",
      "0.437340561"
    ],
    [
      "0.062520078",
      "0.437640546"
    ],
    [
      "0.279095981",
      "0.437640546"
    ],
    [
      "0.612616067",
      "0.499860639"
    ],
    [
      "0.82919197",
      "0.499860639"
    ],
    [
      "0.393719791",
      "0.5"
    ],
    [
      "0.17080803",
      "0.500160624"
    ],
    [
      "0.502099514",
      "0.562360855"
    ],
    [
      "0.720904019",
      "0.562380717"
    ],
    [
      "0.937479922",
      "0.562380717"
    ],
    [
      "0.285431839",
      "0.562520078"
    ],
    [
      "0.062520078",
      "0.562680702"
    ],
    [
      "0.610387571",
      "0.624880935"
    ],
    [
      "0.82919197",
      "0.624900795"
    ],
    [
      "0.393719868",
      "0.625040156"
    ],
    [
      "0.177143888",
      "0.625040156"
    ],
    [
      "0.502099619",
      "0.687401011"
    ],
    [
      "0.718675523",
      "0.687401013"
    ],
    [
      "0.937479922",
      "0.687420873"
    ],
    [
      "0.068855936",
      "0.687560234"
    ],
    [
      "0.285431839",
      "0.687560234"
    ],
    [
      "0.827561444",
      "0.74993032"
    ],
    [
      "0.610399755",
      "0.749942285"
    ],
    [
      "0.177143888",
      "0.750080312"
    ],
    [
      "0.393719791",
      "0.750080312"
    ],
    [
      "0.062520078",
      "0.812439766"
    ],
    [
      "0.719267356",
      "0.812439768"
    ],
    [
      "0.502099514",
      "0.812441167"
    ],
    [
      "0.935849395",
      "0.812450398"
    ],
    [
      "0.285431839",
      "0.81260039"
    ],
    [
      "0.17080803",
      "0.874959844"
    ],
    [
      "0.610991682",
      "0.874981106"
    ],
    [
      "0.828385914",
      "0.876377237"
    ],
    [
      "0.391049386",
      "0.880509074"
    ],
    [
      "0.062520078",
      "0.937479922"
    ],
    [
      "0.279095981",
      "0.937479922"
    ],
    [
      "0.502691457",
      "0.937479922"
    ],
    [
      "0.719291906",
      "0.937479922"
    ],
    [
      "0.937479922",
      "0.937479922"
    ]
  ],
  "radius": "0.062520077"
}

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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