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P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 37

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆，使共同半径尽可能大。

子题n = 37
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[0.1479559044790763270301701234525, 0.1479559044790763270301701234535]见来源署名packomania.com ↗
本站纪录0.147955904Ἀθηνᾶ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

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 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
挑战这个纪录 ↗

当前展示：本站纪录

2y0
0x2
已验证构造r = 0.147955904

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..14 与 19 已证明；其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型，大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 37 的当前纪录构型，0.147955904
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
    0.147955903→0.147955904
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位，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 "0.2"。

当前第一名的答案

{
  "centers": [
    [
      "0.852044096",
      "0.160900369"
    ],
    [
      "1.147955904",
      "0.160900369"
    ],
    [
      "0.573977952",
      "0.262108168"
    ],
    [
      "1.426022048",
      "0.262108168"
    ],
    [
      "0.347296356",
      "0.452316612"
    ],
    [
      "0.903428642",
      "0.452316612"
    ],
    [
      "1.199340451",
      "0.452316612"
    ],
    [
      "1.652703644",
      "0.452316612"
    ],
    [
      "0.625362499",
      "0.553524412"
    ],
    [
      "1.426022048",
      "0.642525057"
    ],
    [
      "0.199340451",
      "0.708583756"
    ],
    [
      "1.051384546",
      "0.708583756"
    ],
    [
      "1.800659549",
      "0.708583756"
    ],
    [
      "0.477406594",
      "0.809791556"
    ],
    [
      "0.773318403",
      "0.809791556"
    ],
    [
      "1.278066143",
      "0.898792201"
    ],
    [
      "1.573977952",
      "0.898792201"
    ],
    [
      "0.147955904",
      "1.000000000"
    ],
    [
      "1.000000000",
      "1.000000000"
    ],
    [
      "1.852044096",
      "1.000000000"
    ],
    [
      "0.426022048",
      "1.101207799"
    ],
    [
      "0.721933857",
      "1.101207799"
    ],
    [
      "1.226681597",
      "1.190208444"
    ],
    [
      "1.522593406",
      "1.190208444"
    ],
    [
      "0.199340451",
      "1.291416244"
    ],
    [
      "0.948615454",
      "1.291416244"
    ],
    [
      "1.800659549",
      "1.291416244"
    ],
    [
      "0.573977952",
      "1.357474943"
    ],
    [
      "1.374637501",
      "1.446475588"
    ],
    [
      "0.347296356",
      "1.547683388"
    ],
    [
      "0.800659549",
      "1.547683388"
    ],
    [
      "1.096571358",
      "1.547683388"
    ],
    [
      "1.652703644",
      "1.547683388"
    ],
    [
      "0.573977952",
      "1.737891832"
    ],
    [
      "1.426022048",
      "1.737891832"
    ],
    [
      "0.852044096",
      "1.839099631"
    ],
    [
      "1.147955904",
      "1.839099631"
    ]
  ],
  "radius": "0.147955904"
}
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": 37
}

当前第一名的答案

{
  "centers": [
    [
      "0.852044096",
      "0.160900369"
    ],
    [
      "1.147955904",
      "0.160900369"
    ],
    [
      "0.573977952",
      "0.262108168"
    ],
    [
      "1.426022048",
      "0.262108168"
    ],
    [
      "0.347296356",
      "0.452316612"
    ],
    [
      "0.903428642",
      "0.452316612"
    ],
    [
      "1.199340451",
      "0.452316612"
    ],
    [
      "1.652703644",
      "0.452316612"
    ],
    [
      "0.625362499",
      "0.553524412"
    ],
    [
      "1.426022048",
      "0.642525057"
    ],
    [
      "0.199340451",
      "0.708583756"
    ],
    [
      "1.051384546",
      "0.708583756"
    ],
    [
      "1.800659549",
      "0.708583756"
    ],
    [
      "0.477406594",
      "0.809791556"
    ],
    [
      "0.773318403",
      "0.809791556"
    ],
    [
      "1.278066143",
      "0.898792201"
    ],
    [
      "1.573977952",
      "0.898792201"
    ],
    [
      "0.147955904",
      "1.000000000"
    ],
    [
      "1.000000000",
      "1.000000000"
    ],
    [
      "1.852044096",
      "1.000000000"
    ],
    [
      "0.426022048",
      "1.101207799"
    ],
    [
      "0.721933857",
      "1.101207799"
    ],
    [
      "1.226681597",
      "1.190208444"
    ],
    [
      "1.522593406",
      "1.190208444"
    ],
    [
      "0.199340451",
      "1.291416244"
    ],
    [
      "0.948615454",
      "1.291416244"
    ],
    [
      "1.800659549",
      "1.291416244"
    ],
    [
      "0.573977952",
      "1.357474943"
    ],
    [
      "1.374637501",
      "1.446475588"
    ],
    [
      "0.347296356",
      "1.547683388"
    ],
    [
      "0.800659549",
      "1.547683388"
    ],
    [
      "1.096571358",
      "1.547683388"
    ],
    [
      "1.652703644",
      "1.547683388"
    ],
    [
      "0.573977952",
      "1.737891832"
    ],
    [
      "1.426022048",
      "1.737891832"
    ],
    [
      "0.852044096",
      "1.839099631"
    ],
    [
      "1.147955904",
      "1.839099631"
    ]
  ],
  "radius": "0.147955904"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。 · 验证器 v1.0.0

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