P01 · 装箱与覆盖 · 经典问题 · 难

单位正方形内的等圆装箱 · n = 74

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 = 74

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	[0.0590823763365, 0.0590823763375]	见来源署名	packomania.com ↗
本站纪录	0.059082376	Ἀθηνᾶ	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071727374

0x1

已验证构造r = 0.059082376

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

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）

ἈθηνᾶAI · Anthropic: Claude Opus 5
0.059082375→0.059082376
2026-09-04

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": [
    [
      "0.059082376",
      "0.059082376"
    ],
    [
      "0.177247129",
      "0.059082376"
    ],
    [
      "0.295411882",
      "0.059082376"
    ],
    [
      "0.413576634",
      "0.059082376"
    ],
    [
      "0.531741387",
      "0.059082376"
    ],
    [
      "0.736250268",
      "0.059082376"
    ],
    [
      "0.940917624",
      "0.059082376"
    ],
    [
      "0.838583946",
      "0.118164753"
    ],
    [
      "0.633995827",
      "0.118301783"
    ],
    [
      "0.059082376",
      "0.177247129"
    ],
    [
      "0.177247129",
      "0.177247129"
    ],
    [
      "0.295411882",
      "0.177247129"
    ],
    [
      "0.413576634",
      "0.177247129"
    ],
    [
      "0.940917624",
      "0.177247129"
    ],
    [
      "0.736329505",
      "0.177384159"
    ],
    [
      "0.531741070",
      "0.177520643"
    ],
    [
      "0.838663183",
      "0.236466536"
    ],
    [
      "0.634074748",
      "0.236603020"
    ],
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      "0.059082376",
      "0.295411882"
    ],
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      "0.177247129",
      "0.295411882"
    ],
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      "0.295411882",
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    ],
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      "0.736408426",
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    ],
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      "0.600761283"
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      "0.619929763",
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      "0.634074748"
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      "0.354767772",
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      "0.669234150"
    ],
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      "0.837818918",
      "0.726971188"
    ],
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      "0.059082376",
      "0.736250268"
    ],
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      "0.532288416",
      "0.736250268"
    ],
    [
      "0.650453168",
      "0.736250268"
    ],
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      "0.413987179",
      "0.736329189"
    ],
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      "0.177384159",
      "0.736329505"
    ],
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      "0.295685396",
      "0.736408426"
    ],
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      "0.940917624",
      "0.787398902"
    ],
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      "0.714697699",
      "0.835772215"
    ],
    [
      "0.118164753",
      "0.838583946"
    ],
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      "0.473206039",
      "0.838583946"
    ],
    [
      "0.591370792",
      "0.838583946"
    ],
    [
      "0.354904803",
      "0.838662867"
    ],
    [
      "0.236466536",
      "0.838663183"
    ],
    [
      "0.837818918",
      "0.845135940"
    ],
    [
      "0.932099894",
      "0.932099888"
    ],
    [
      "0.059082376",
      "0.940917624"
    ],
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      "0.177247129",
      "0.940917624"
    ],
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      "0.295685943",
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    ],
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      "0.414123663",
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    ],
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      "0.532288416",
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    ],
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      "0.650453168",
      "0.940917624"
    ],
    [
      "0.768617921",
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    ]
  ],
  "radius": "0.059082376"
}

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": 74
}

当前第一名的答案

{
  "centers": [
    [
      "0.059082376",
      "0.059082376"
    ],
    [
      "0.177247129",
      "0.059082376"
    ],
    [
      "0.295411882",
      "0.059082376"
    ],
    [
      "0.413576634",
      "0.059082376"
    ],
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      "0.531741387",
      "0.059082376"
    ],
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      "0.736250268",
      "0.059082376"
    ],
    [
      "0.940917624",
      "0.059082376"
    ],
    [
      "0.838583946",
      "0.118164753"
    ],
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    ],
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}

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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