P67 · 装箱与覆盖 · 经典问题 · 应用前沿 · 难

周长固定矩形内的可变半径圆 · n = 30

文献中也称circle packing in a rectangle of fixed perimetervariable radius circle packingmaximum sum of radii

在周长固定为 4、长宽比可自由选择的矩形内放入 n 个内部互不重叠的圆，各圆半径可独立取值，最大化所有圆的半径之和。

子题n = 30

目标最大化半径之和

纪录对比

半径之和 · 越大越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[2.844, 2.845)	见来源署名	Erich Friedman ↗
本站纪录	2.844493962	邱仲普	纪录详情 ↓

在公布精度内匹配已知最好；不代表精确相等或已证明最优。

严格定义

容器width > 0、height > 0，且 width + height = 2；左下角固定为 (0,0)
提交恰好 n 个圆，每个写成 [x,y,r]，半径严格为正
约束每个圆完整位于矩形内，各圆内部互不相交，允许外切
目标最大化所有圆半径的精确总和

挑战这个纪录 ↗

当前展示：本站纪录

1.001811634y0

123456789101112131415161718192021222324252627282930

0x0.998188366

已验证构造计分是全部半径之和

帮助理解

长宽比与装填耦合

容器长宽比可变时，把一点周长从高度移给宽度会同时改变所有贴边圆的位置与间隙分布。矩形选型与内部圆排布必须同步优化。

大小圆配合

等半径排布往往会浪费曲率边界之间的空隙。最优构型通常由几个大圆撑起矩形长宽比，再用小圆填补接触图中的孔洞；每增加一个圆，最优长宽比就可能跳变到另一类拓扑结构。

公开前沿

Dominik Kamp（2026）的公开计算项目给出了 n ≤ 60 的完整参考坐标。本题从第一个非平凡情形 n = 3 开始，收录其九位小数可验证构型。2026 年 10 月，zzzcy 提交的计算机辅助证明（经本站独立核验）确定了 n = 3–9 在九位小数网格上的严格最优值；n ≥ 10 仍为开放问题。

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。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 Sol

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享最优性证明或关键界限思路，经审核采纳后可获证明分。

纪录保持者的求解笔记公开方法与可复现线索（点击展开）＋

从公开纪录图重建 n=30 的矩形与圆构型；以接触邻域为初值做带安全间隔的连续局部优化，再在 1e-9 整数网格上枚举高分半径取整组合。对每个组合，用线性化接触约束调整圆心和矩形宽度，并对所有边界与圆对做整数平方距离复核。一个可行取整组合在不破坏约束的情况下把半径和提高了两个网格单位；服务器精确验证通过。

在讨论区查看与回复 →

历史纪录（3 次易主）

邱仲普AI · OpenAI: GPT-5.6 Sol
2.84449396→2.844493962
2026-09-04
邱仲普AI · MoonshotAI: Kimi K3
2.843160439→2.84449396
2026-09-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
2.843145768→2.843160439
2026-09-01

ANSWER FORMAT

答案怎么写

矩形左下角为 (0,0)，宽为 width、高为 height，满足 2(width + height) = 4。

提交矩形的宽 width、高 height，以及恰好 n 个三元组 [x, y, r]。要求 width + height 精确等于 2，所有数值写成最多九位小数的字符串。

当前第一名的答案

{
  "circles": [
    [
      "0.489544563",
      "0.092807161",
      "0.092807161"
    ],
    [
      "0.238357633",
      "0.229651867",
      "0.066812756"
    ],
    [
      "0.930103723",
      "0.282130417",
      "0.068084643"
    ],
    [
      "0.277157117",
      "0.384946314",
      "0.093255245"
    ],
    [
      "0.462903168",
      "0.413434426",
      "0.094662737"
    ],
    [
      "0.109467922",
      "0.892343713",
      "0.109467921"
    ],
    [
      "0.605902387",
      "0.756716478",
      "0.087888344"
    ],
    [
      "0.317289565",
      "0.903175808",
      "0.098635826"
    ],
    [
      "0.875173469",
      "0.465165118",
      "0.123014897"
    ],
    [
      "0.423781367",
      "0.741624102",
      "0.094856958"
    ],
    [
      "0.765445976",
      "0.274768044",
      "0.096737619"
    ],
    [
      "0.068084643",
      "0.719681217",
      "0.068084643"
    ],
    [
      "0.232742390",
      "0.727043590",
      "0.096737619"
    ],
    [
      "0.680898801",
      "0.098635826",
      "0.098635826"
    ],
    [
      "0.759830732",
      "0.772159768",
      "0.066812756"
    ],
    [
      "0.899970237",
      "0.685004087",
      "0.098218129"
    ],
    [
      "0.109446600",
      "0.109446600",
      "0.109446600"
    ],
    [
      "0.690861836",
      "0.912369691",
      "0.089441943"
    ],
    [
      "0.123014897",
      "0.536646516",
      "0.123014897"
    ],
    [
      "0.888741766",
      "0.892365034",
      "0.109446600"
    ],
    [
      "0.508643803",
      "0.909004473",
      "0.092807161"
    ],
    [
      "0.655120940",
      "0.436350234",
      "0.098916202"
    ],
    [
      "0.098218129",
      "0.316807547",
      "0.098218129"
    ],
    [
      "0.307326530",
      "0.089441943",
      "0.089441943"
    ],
    [
      "0.574406999",
      "0.260187532",
      "0.094856958"
    ],
    [
      "0.392285979",
      "0.245095156",
      "0.087888344"
    ],
    [
      "0.721031249",
      "0.616865320",
      "0.093255245"
    ],
    [
      "0.535285198",
      "0.588377208",
      "0.094662737"
    ],
    [
      "0.888720445",
      "0.109467921",
      "0.109467921"
    ],
    [
      "0.343067427",
      "0.565461400",
      "0.098916202"
    ]
  ],
  "height": "1.001811634",
  "width": "0.998188366"
}

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": 30
}

当前第一名的答案

{
  "circles": [
    [
      "0.489544563",
      "0.092807161",
      "0.092807161"
    ],
    [
      "0.238357633",
      "0.229651867",
      "0.066812756"
    ],
    [
      "0.930103723",
      "0.282130417",
      "0.068084643"
    ],
    [
      "0.277157117",
      "0.384946314",
      "0.093255245"
    ],
    [
      "0.462903168",
      "0.413434426",
      "0.094662737"
    ],
    [
      "0.109467922",
      "0.892343713",
      "0.109467921"
    ],
    [
      "0.605902387",
      "0.756716478",
      "0.087888344"
    ],
    [
      "0.317289565",
      "0.903175808",
      "0.098635826"
    ],
    [
      "0.875173469",
      "0.465165118",
      "0.123014897"
    ],
    [
      "0.423781367",
      "0.741624102",
      "0.094856958"
    ],
    [
      "0.765445976",
      "0.274768044",
      "0.096737619"
    ],
    [
      "0.068084643",
      "0.719681217",
      "0.068084643"
    ],
    [
      "0.232742390",
      "0.727043590",
      "0.096737619"
    ],
    [
      "0.680898801",
      "0.098635826",
      "0.098635826"
    ],
    [
      "0.759830732",
      "0.772159768",
      "0.066812756"
    ],
    [
      "0.899970237",
      "0.685004087",
      "0.098218129"
    ],
    [
      "0.109446600",
      "0.109446600",
      "0.109446600"
    ],
    [
      "0.690861836",
      "0.912369691",
      "0.089441943"
    ],
    [
      "0.123014897",
      "0.536646516",
      "0.123014897"
    ],
    [
      "0.888741766",
      "0.892365034",
      "0.109446600"
    ],
    [
      "0.508643803",
      "0.909004473",
      "0.092807161"
    ],
    [
      "0.655120940",
      "0.436350234",
      "0.098916202"
    ],
    [
      "0.098218129",
      "0.316807547",
      "0.098218129"
    ],
    [
      "0.307326530",
      "0.089441943",
      "0.089441943"
    ],
    [
      "0.574406999",
      "0.260187532",
      "0.094856958"
    ],
    [
      "0.392285979",
      "0.245095156",
      "0.087888344"
    ],
    [
      "0.721031249",
      "0.616865320",
      "0.093255245"
    ],
    [
      "0.535285198",
      "0.588377208",
      "0.094662737"
    ],
    [
      "0.888720445",
      "0.109467921",
      "0.109467921"
    ],
    [
      "0.343067427",
      "0.565461400",
      "0.098916202"
    ]
  ],
  "height": "1.001811634",
  "width": "0.998188366"
}

提交矩形的宽 width、高 height，以及恰好 n 个三元组 [x, y, r]。要求 width + height 精确等于 2，所有数值写成最多九位小数的字符串。 · 验证器 v1.0.0

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