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P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 69

文献中也称circles in a rectanglepacking equal circles in a 2:1 rectangle

在 2 × 1 的长方形内放置 n 个互不重叠的等圆，使它们的共同半径尽可能大。

子题n = 69
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[0.085860123235459322002145865519, 0.085860123235459322002145865521]Eckard Specht2010-08-10Packomania ↗
本站纪录0.086009769condor纪录详情 ↓

数值超出外部纪录区间 · 待审核。

严格定义

  • 容器宽 2、高 1 的长方形 [0, 2] × [0, 1]，左下角为原点 (0, 0)，右上角为 (2, 1)
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心坐标 centers
  • 约束每个圆完全位于长方形内；任意两圆内部不重叠，允许相切
  • 目标最大化共同半径 radius
挑战这个纪录 ↗

当前展示：本站纪录

1y0
0x2
已验证构造r = 0.086009769

帮助理解

交错之字层与行转换

在长宽比为 2 : 1 的狭长容器内，小规模时各圆通常交替贴靠上下长边排成之字形链；随着 n 增大，排布会在双层交错、三层蜂窝与局部斜排之间切换。除了完美契合 2 : 1 比例的偶数网格（如 2 × 1 或 4 × 2）外，整齐的轴对齐网格通常会浪费横向或纵向空间。

已知证明与开放前沿

n = 3..6 的最优半径可通过之字形链的初等几何下界严格证明（详见各子题的已知最优说明）；更大的 n 引用 Packomania 的 1 × 0.5 长方形表（本题是它放大两倍，半径也乘 2），表中只列出最优已知值，没有一项标为已证明最优。

2:1 长方形内的等圆装箱 n = 69 的当前纪录构型，0.086009769
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condor
解题方式AI · OpenAI: GPT-6.1 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暂无求解笔记（点击展开）＋

纪录保持者暂未分享求解过程。

历史纪录（5 次易主）
  1. condorAI · OpenAI: GPT-6.1 Sol
    0.085861206→0.086009769
  2. condorAI · OpenAI: GPT-6 Astra
    0.085860123→0.085861206
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.085860122→0.085860123
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.083647196→0.085860122
  5. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.066666666→0.083647196
ANSWER FORMAT

答案怎么写

容器是宽 2、高 1 的长方形，左下角是原点 (0, 0)，右上角是 (2, 1)。坐标和长度用同一个单位，直接写成小数，例如 "0.25"，最多九位小数。

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.25"）。

当前第一名的答案

{
  "centers": [
    [
      "0.924695976",
      "0.532929643"
    ],
    [
      "1.010705746",
      "0.383956352"
    ],
    [
      "1.075304024",
      "0.616043648"
    ],
    [
      "0.838686206",
      "0.383956352"
    ],
    [
      "1.161313794",
      "0.467070357"
    ],
    [
      "1.247323564",
      "0.616043648"
    ],
    [
      "0.666666667",
      "0.383956352"
    ],
    [
      "1.247323564",
      "0.318097066"
    ],
    [
      "0.580656897",
      "0.532929643"
    ],
    [
      "0.838686206",
      "0.681902934"
    ],
    [
      "1.333333333",
      "0.467070357"
    ],
    [
      "0.494647127",
      "0.383956352"
    ],
    [
      "0.989294254",
      "0.765016939"
    ],
    [
      "0.494647127",
      "0.681902934"
    ],
    [
      "0.924695976",
      "0.234983061"
    ],
    [
      "1.419343103",
      "0.318097066"
    ],
    [
      "0.408637357",
      "0.532929643"
    ],
    [
      "1.505352873",
      "0.467070357"
    ],
    [
      "0.752676436",
      "0.234983061"
    ],
    [
      "1.333333333",
      "0.765016939"
    ],
    [
      "0.430048849",
      "0.913990230"
    ],
    [
      "1.591362643",
      "0.616043648"
    ],
    [
      "1.569951151",
      "0.234983061"
    ],
    [
      "1.505352873",
      "0.765016939"
    ],
    [
      "1.333333333",
      "0.169123775"
    ],
    [
      "0.258029309",
      "0.616043648"
    ],
    [
      "1.655960921",
      "0.383956352"
    ],
    [
      "0.494647127",
      "0.086009770"
    ],
    [
      "1.591362643",
      "0.913990230"
    ],
    [
      "0.322627588",
      "0.383956352"
    ],
    [
      "1.182725285",
      "0.086009770"
    ],
    [
      "0.903284485",
      "0.913990230"
    ],
    [
      "0.236617818",
      "0.234983061"
    ],
    [
      "0.172019539",
      "0.765016939"
    ],
    [
      "1.913990230",
      "0.234983061"
    ],
    [
      "1.075304024",
      "0.913990230"
    ],
    [
      "0.086499220",
      "0.086461052"
    ],
    [
      "0.172019539",
      "0.467070357"
    ],
    [
      "1.763382182",
      "0.913990230"
    ],
    [
      "1.913990230",
      "0.532929643"
    ],
    [
      "0.838686206",
      "0.086009770"
    ],
    [
      "1.247323564",
      "0.913990230"
    ],
    [
      "1.655960921",
      "0.086009770"
    ],
    [
      "1.827980461",
      "0.681902934"
    ],
    [
      "1.827980461",
      "0.086009770"
    ],
    [
      "0.666666667",
      "0.086009770"
    ],
    [
      "1.419343103",
      "0.913990230"
    ],
    [
      "1.483941382",
      "0.086009770"
    ],
    [
      "0.086009770",
      "0.318097066"
    ],
    [
      "1.913990230",
      "0.830876225"
    ],
    [
      "0.086009770",
      "0.913990230"
    ],
    [
      "1.741970691",
      "0.532929643"
    ],
    [
      "1.419343103",
      "0.616043648"
    ],
    [
      "0.580656897",
      "0.234983061"
    ],
    [
      "1.096715515",
      "0.234983061"
    ],
    [
      "0.258029309",
      "0.913990230"
    ],
    [
      "0.408637357",
      "0.234983061"
    ],
    [
      "0.752676436",
      "0.830876225"
    ],
    [
      "0.602068388",
      "0.913990230"
    ],
    [
      "0.322627588",
      "0.086009770"
    ],
    [
      "0.752676436",
      "0.532929643"
    ],
    [
      "1.741970691",
      "0.234983061"
    ],
    [
      "0.086009770",
      "0.616043648"
    ],
    [
      "1.677372412",
      "0.765016939"
    ],
    [
      "0.666666667",
      "0.681902934"
    ],
    [
      "1.010705746",
      "0.086009770"
    ],
    [
      "1.161313794",
      "0.765016939"
    ],
    [
      "0.344039079",
      "0.765016939"
    ],
    [
      "1.827980461",
      "0.383956352"
    ]
  ],
  "radius": "0.086009769"
}
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": 69
}

当前第一名的答案

{
  "centers": [
    [
      "0.924695976",
      "0.532929643"
    ],
    [
      "1.010705746",
      "0.383956352"
    ],
    [
      "1.075304024",
      "0.616043648"
    ],
    [
      "0.838686206",
      "0.383956352"
    ],
    [
      "1.161313794",
      "0.467070357"
    ],
    [
      "1.247323564",
      "0.616043648"
    ],
    [
      "0.666666667",
      "0.383956352"
    ],
    [
      "1.247323564",
      "0.318097066"
    ],
    [
      "0.580656897",
      "0.532929643"
    ],
    [
      "0.838686206",
      "0.681902934"
    ],
    [
      "1.333333333",
      "0.467070357"
    ],
    [
      "0.494647127",
      "0.383956352"
    ],
    [
      "0.989294254",
      "0.765016939"
    ],
    [
      "0.494647127",
      "0.681902934"
    ],
    [
      "0.924695976",
      "0.234983061"
    ],
    [
      "1.419343103",
      "0.318097066"
    ],
    [
      "0.408637357",
      "0.532929643"
    ],
    [
      "1.505352873",
      "0.467070357"
    ],
    [
      "0.752676436",
      "0.234983061"
    ],
    [
      "1.333333333",
      "0.765016939"
    ],
    [
      "0.430048849",
      "0.913990230"
    ],
    [
      "1.591362643",
      "0.616043648"
    ],
    [
      "1.569951151",
      "0.234983061"
    ],
    [
      "1.505352873",
      "0.765016939"
    ],
    [
      "1.333333333",
      "0.169123775"
    ],
    [
      "0.258029309",
      "0.616043648"
    ],
    [
      "1.655960921",
      "0.383956352"
    ],
    [
      "0.494647127",
      "0.086009770"
    ],
    [
      "1.591362643",
      "0.913990230"
    ],
    [
      "0.322627588",
      "0.383956352"
    ],
    [
      "1.182725285",
      "0.086009770"
    ],
    [
      "0.903284485",
      "0.913990230"
    ],
    [
      "0.236617818",
      "0.234983061"
    ],
    [
      "0.172019539",
      "0.765016939"
    ],
    [
      "1.913990230",
      "0.234983061"
    ],
    [
      "1.075304024",
      "0.913990230"
    ],
    [
      "0.086499220",
      "0.086461052"
    ],
    [
      "0.172019539",
      "0.467070357"
    ],
    [
      "1.763382182",
      "0.913990230"
    ],
    [
      "1.913990230",
      "0.532929643"
    ],
    [
      "0.838686206",
      "0.086009770"
    ],
    [
      "1.247323564",
      "0.913990230"
    ],
    [
      "1.655960921",
      "0.086009770"
    ],
    [
      "1.827980461",
      "0.681902934"
    ],
    [
      "1.827980461",
      "0.086009770"
    ],
    [
      "0.666666667",
      "0.086009770"
    ],
    [
      "1.419343103",
      "0.913990230"
    ],
    [
      "1.483941382",
      "0.086009770"
    ],
    [
      "0.086009770",
      "0.318097066"
    ],
    [
      "1.913990230",
      "0.830876225"
    ],
    [
      "0.086009770",
      "0.913990230"
    ],
    [
      "1.741970691",
      "0.532929643"
    ],
    [
      "1.419343103",
      "0.616043648"
    ],
    [
      "0.580656897",
      "0.234983061"
    ],
    [
      "1.096715515",
      "0.234983061"
    ],
    [
      "0.258029309",
      "0.913990230"
    ],
    [
      "0.408637357",
      "0.234983061"
    ],
    [
      "0.752676436",
      "0.830876225"
    ],
    [
      "0.602068388",
      "0.913990230"
    ],
    [
      "0.322627588",
      "0.086009770"
    ],
    [
      "0.752676436",
      "0.532929643"
    ],
    [
      "1.741970691",
      "0.234983061"
    ],
    [
      "0.086009770",
      "0.616043648"
    ],
    [
      "1.677372412",
      "0.765016939"
    ],
    [
      "0.666666667",
      "0.681902934"
    ],
    [
      "1.010705746",
      "0.086009770"
    ],
    [
      "1.161313794",
      "0.765016939"
    ],
    [
      "0.344039079",
      "0.765016939"
    ],
    [
      "1.827980461",
      "0.383956352"
    ]
  ],
  "radius": "0.086009769"
}

提交共同半径 radius 与各圆圆心坐标 centers，所有数值均以十进制字符串表示（如 "0.25"）。 · 验证器 v1.0.0

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