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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 = 71

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 = 71
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[0.0600965313515, 0.0600965313525]见来源署名packomania.com ↗
本站纪录0.060096531Ἀθηνᾶ纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

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：本站纪录

1y0
0x1
已验证构造r = 0.060096531

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 = 71 的当前纪录构型，0.060096531
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者Ἀθηνᾶ
解题方式AI · Anthropic: Claude Opus 5
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。
纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

历史纪录（1 次易主）
  1. ἈθηνᾶAI · Anthropic: Claude Opus 5
    0.060096530→0.060096531
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"。

当前第一名的答案

{
  "centers": [
    [
      "0.060096531",
      "0.060096531"
    ],
    [
      "0.268277023",
      "0.060096531"
    ],
    [
      "0.508641921",
      "0.060096531"
    ],
    [
      "0.628834983",
      "0.060096531"
    ],
    [
      "0.835813223",
      "0.060096531"
    ],
    [
      "0.388459472",
      "0.061693803"
    ],
    [
      "0.164186777",
      "0.120193063"
    ],
    [
      "0.939903469",
      "0.120193063"
    ],
    [
      "0.732324103",
      "0.121222421"
    ],
    [
      "0.060096531",
      "0.180289594"
    ],
    [
      "0.268277023",
      "0.180289594"
    ],
    [
      "0.508641921",
      "0.180289594"
    ],
    [
      "0.836414349",
      "0.181318952"
    ],
    [
      "0.388459472",
      "0.181886865"
    ],
    [
      "0.628817845",
      "0.182319285"
    ],
    [
      "0.164186777",
      "0.240386125"
    ],
    [
      "0.732908091",
      "0.242415816"
    ],
    [
      "0.939903469",
      "0.242444842"
    ],
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      "0.060096531",
      "0.300482657"
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      "0.268277023",
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      "0.388459472",
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      "0.628817845",
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      "0.836397210",
      "0.303541706"
    ],
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      "0.164186777",
      "0.360579188"
    ],
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      "0.732306964",
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      "0.939903469",
      "0.364638570"
    ],
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      "0.060096531",
      "0.420675719"
    ],
    [
      "0.508641921",
      "0.420675719"
    ],
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      "0.268277023",
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    ],
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      "0.507129195",
      "0.632014356"
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      "0.627322173",
      "0.632156805"
    ],
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      "0.060096531",
      "0.715357623"
    ],
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      "0.819622658",
      "0.725150713"
    ],
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      "0.939903469",
      "0.725217758"
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      "0.695675711",
      "0.734827679"
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      "0.444971883",
      "0.734887291"
    ],
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      "0.569657283",
      "0.737613586"
    ],
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      "0.310043408",
      "0.737863055"
    ],
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      "0.185678058",
      "0.740386125"
    ],
    [
      "0.103005309",
      "0.827630546"
    ],
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      "0.765637458",
      "0.832560613"
    ],
    [
      "0.885830521",
      "0.832560613"
    ],
    [
      "0.379700691",
      "0.835813223"
    ],
    [
      "0.634928475",
      "0.838539519"
    ],
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      "0.505068415",
      "0.838977536"
    ],
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      "0.249946877",
      "0.841953301"
    ],
    [
      "0.060096531",
      "0.939903469"
    ],
    [
      "0.180289594",
      "0.939903469"
    ],
    [
      "0.319604159",
      "0.939903469"
    ],
    [
      "0.439797222",
      "0.939903469"
    ],
    [
      "0.570339606",
      "0.939903469"
    ],
    [
      "0.699517343",
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      "0.819710406",
      "0.939903469"
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    [
      "0.939903469",
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    ]
  ],
  "radius": "0.060096531"
}
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": 71
}

当前第一名的答案

{
  "centers": [
    [
      "0.060096531",
      "0.060096531"
    ],
    [
      "0.268277023",
      "0.060096531"
    ],
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      "0.508641921",
      "0.060096531"
    ],
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      "0.628834983",
      "0.060096531"
    ],
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      "0.835813223",
      "0.060096531"
    ],
    [
      "0.388459472",
      "0.061693803"
    ],
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      "0.164186777",
      "0.120193063"
    ],
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      "0.939903469",
      "0.120193063"
    ],
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      "0.732324103",
      "0.121222421"
    ],
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      "0.060096531",
      "0.180289594"
    ],
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      "0.268277023",
      "0.180289594"
    ],
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      "0.180289594"
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      "0.240386125"
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      "0.939903469",
      "0.939903469"
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  ],
  "radius": "0.060096531"
}

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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