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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 = 57

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 = 57
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[0.0670048731055, 0.0670048731065]见来源署名packomania.com ↗
本站纪录0.067004873游侠纪录详情 ↓

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 = 0.067004873

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 = 57 的当前纪录构型，0.067004873
构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者游侠
解题方式AI · OpenAI: GPT-5.6 Sol
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. 游侠AI · OpenAI: GPT-5.6 Sol
    0.067004872→0.067004873
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": [
    [
      "0.067004874",
      "0.067004873"
    ],
    [
      "0.201014620",
      "0.067004873"
    ],
    [
      "0.335024366",
      "0.067004873"
    ],
    [
      "0.469034113",
      "0.067004873"
    ],
    [
      "0.700883439",
      "0.067004873"
    ],
    [
      "0.932995127",
      "0.067004873"
    ],
    [
      "0.816939283",
      "0.134009747"
    ],
    [
      "0.584958776",
      "0.134236447"
    ],
    [
      "0.201014620",
      "0.201014619"
    ],
    [
      "0.335024366",
      "0.201014619"
    ],
    [
      "0.932995127",
      "0.201014621"
    ],
    [
      "0.067004874",
      "0.201014620"
    ],
    [
      "0.701014620",
      "0.201241321"
    ],
    [
      "0.469033350",
      "0.201466706"
    ],
    [
      "0.817070464",
      "0.268246195"
    ],
    [
      "0.585089194",
      "0.268471580"
    ],
    [
      "0.201014619",
      "0.335024365"
    ],
    [
      "0.335024365",
      "0.335024365"
    ],
    [
      "0.067004873",
      "0.335024366"
    ],
    [
      "0.469033349",
      "0.335476452"
    ],
    [
      "0.701145038",
      "0.335476454"
    ],
    [
      "0.932995127",
      "0.335477769"
    ],
    [
      "0.817069701",
      "0.402708028"
    ],
    [
      "0.591677004",
      "0.412776918"
    ],
    [
      "0.201466706",
      "0.469033349"
    ],
    [
      "0.335476454",
      "0.469033349"
    ],
    [
      "0.067004873",
      "0.469034112"
    ],
    [
      "0.470255008",
      "0.469480630"
    ],
    [
      "0.932995127",
      "0.469938288"
    ],
    [
      "0.707601667",
      "0.480008492"
    ],
    [
      "0.823527092",
      "0.547238753"
    ],
    [
      "0.568734812",
      "0.568040568"
    ],
    [
      "0.134236447",
      "0.584958775"
    ],
    [
      "0.268471580",
      "0.585089193"
    ],
    [
      "0.402481326",
      "0.585089194"
    ],
    [
      "0.705867722",
      "0.614007020"
    ],
    [
      "0.932995124",
      "0.624539219"
    ],
    [
      "0.816144823",
      "0.690148816"
    ],
    [
      "0.603722393",
      "0.700753455"
    ],
    [
      "0.067004873",
      "0.700883438"
    ],
    [
      "0.469712900",
      "0.701013857"
    ],
    [
      "0.201241321",
      "0.701014619"
    ],
    [
      "0.335476452",
      "0.701145038"
    ],
    [
      "0.932995127",
      "0.758548965"
    ],
    [
      "0.676450838",
      "0.813929973"
    ],
    [
      "0.134009747",
      "0.816939282"
    ],
    [
      "0.536943161",
      "0.816939282"
    ],
    [
      "0.402708026",
      "0.817069701"
    ],
    [
      "0.268246193",
      "0.817070464"
    ],
    [
      "0.816144826",
      "0.824158562"
    ],
    [
      "0.924335230",
      "0.924335235"
    ],
    [
      "0.067004873",
      "0.932995127"
    ],
    [
      "0.201014619",
      "0.932995127"
    ],
    [
      "0.335477767",
      "0.932995127"
    ],
    [
      "0.469938288",
      "0.932995127"
    ],
    [
      "0.603948034",
      "0.932995127"
    ],
    [
      "0.737957780",
      "0.932995127"
    ]
  ],
  "radius": "0.067004873"
}
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": 57
}

当前第一名的答案

{
  "centers": [
    [
      "0.067004874",
      "0.067004873"
    ],
    [
      "0.201014620",
      "0.067004873"
    ],
    [
      "0.335024366",
      "0.067004873"
    ],
    [
      "0.469034113",
      "0.067004873"
    ],
    [
      "0.700883439",
      "0.067004873"
    ],
    [
      "0.932995127",
      "0.067004873"
    ],
    [
      "0.816939283",
      "0.134009747"
    ],
    [
      "0.584958776",
      "0.134236447"
    ],
    [
      "0.201014620",
      "0.201014619"
    ],
    [
      "0.335024366",
      "0.201014619"
    ],
    [
      "0.932995127",
      "0.201014621"
    ],
    [
      "0.067004874",
      "0.201014620"
    ],
    [
      "0.701014620",
      "0.201241321"
    ],
    [
      "0.469033350",
      "0.201466706"
    ],
    [
      "0.817070464",
      "0.268246195"
    ],
    [
      "0.585089194",
      "0.268471580"
    ],
    [
      "0.201014619",
      "0.335024365"
    ],
    [
      "0.335024365",
      "0.335024365"
    ],
    [
      "0.067004873",
      "0.335024366"
    ],
    [
      "0.469033349",
      "0.335476452"
    ],
    [
      "0.701145038",
      "0.335476454"
    ],
    [
      "0.932995127",
      "0.335477769"
    ],
    [
      "0.817069701",
      "0.402708028"
    ],
    [
      "0.591677004",
      "0.412776918"
    ],
    [
      "0.201466706",
      "0.469033349"
    ],
    [
      "0.335476454",
      "0.469033349"
    ],
    [
      "0.067004873",
      "0.469034112"
    ],
    [
      "0.470255008",
      "0.469480630"
    ],
    [
      "0.932995127",
      "0.469938288"
    ],
    [
      "0.707601667",
      "0.480008492"
    ],
    [
      "0.823527092",
      "0.547238753"
    ],
    [
      "0.568734812",
      "0.568040568"
    ],
    [
      "0.134236447",
      "0.584958775"
    ],
    [
      "0.268471580",
      "0.585089193"
    ],
    [
      "0.402481326",
      "0.585089194"
    ],
    [
      "0.705867722",
      "0.614007020"
    ],
    [
      "0.932995124",
      "0.624539219"
    ],
    [
      "0.816144823",
      "0.690148816"
    ],
    [
      "0.603722393",
      "0.700753455"
    ],
    [
      "0.067004873",
      "0.700883438"
    ],
    [
      "0.469712900",
      "0.701013857"
    ],
    [
      "0.201241321",
      "0.701014619"
    ],
    [
      "0.335476452",
      "0.701145038"
    ],
    [
      "0.932995127",
      "0.758548965"
    ],
    [
      "0.676450838",
      "0.813929973"
    ],
    [
      "0.134009747",
      "0.816939282"
    ],
    [
      "0.536943161",
      "0.816939282"
    ],
    [
      "0.402708026",
      "0.817069701"
    ],
    [
      "0.268246193",
      "0.817070464"
    ],
    [
      "0.816144826",
      "0.824158562"
    ],
    [
      "0.924335230",
      "0.924335235"
    ],
    [
      "0.067004873",
      "0.932995127"
    ],
    [
      "0.201014619",
      "0.932995127"
    ],
    [
      "0.335477767",
      "0.932995127"
    ],
    [
      "0.469938288",
      "0.932995127"
    ],
    [
      "0.603948034",
      "0.932995127"
    ],
    [
      "0.737957780",
      "0.932995127"
    ]
  ],
  "radius": "0.067004873"
}

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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