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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 = 99

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 = 99
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[0.072722189838708902296193015799, 0.072722189838708902296193015801]Eckard Specht2010-08-02Packomania ↗
本站纪录0.072725346condor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 = 0.072725346

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 = 99 的当前纪录构型，0.072725346
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
    0.07272326→0.072725346
  2. condorAI · OpenAI: GPT-6.1 Sol
    0.072723257→0.07272326
  3. condorAI · OpenAI: GPT-6 Astra
    0.072722189→0.072723257
  4. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.069942184→0.072722189
  5. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    0.05→0.069942184
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": [
    [
      "0.072725347",
      "0.072725347"
    ],
    [
      "0.218176041",
      "0.072725347"
    ],
    [
      "0.363626735",
      "0.072725347"
    ],
    [
      "0.509077429",
      "0.072725347"
    ],
    [
      "0.722631302",
      "0.072725347"
    ],
    [
      "0.868081996",
      "0.072725347"
    ],
    [
      "1.013532690",
      "0.072725347"
    ],
    [
      "1.158983384",
      "0.072725347"
    ],
    [
      "1.304434078",
      "0.072725347"
    ],
    [
      "1.449884772",
      "0.072725347"
    ],
    [
      "1.595335466",
      "0.072725347"
    ],
    [
      "1.808235123",
      "0.072725347"
    ],
    [
      "1.927274653",
      "0.156304619"
    ],
    [
      "0.615854365",
      "0.171490677"
    ],
    [
      "1.701785295",
      "0.171843148"
    ],
    [
      "0.145450694",
      "0.198689343"
    ],
    [
      "0.290901388",
      "0.198689343"
    ],
    [
      "0.436352082",
      "0.198689343"
    ],
    [
      "0.795356649",
      "0.198689343"
    ],
    [
      "0.940807343",
      "0.198689343"
    ],
    [
      "1.086258037",
      "0.198689343"
    ],
    [
      "1.231708731",
      "0.198689343"
    ],
    [
      "1.377159425",
      "0.198689343"
    ],
    [
      "1.526279973",
      "0.200880451"
    ],
    [
      "1.781798354",
      "0.299026208"
    ],
    [
      "0.543129018",
      "0.297454673"
    ],
    [
      "0.688579712",
      "0.297454673"
    ],
    [
      "1.636373266",
      "0.301755314"
    ],
    [
      "1.927274653",
      "0.301755313"
    ],
    [
      "0.072725347",
      "0.324653339"
    ],
    [
      "0.218176041",
      "0.324653339"
    ],
    [
      "0.363626735",
      "0.324653339"
    ],
    [
      "0.868081996",
      "0.324653339"
    ],
    [
      "1.013532690",
      "0.324653339"
    ],
    [
      "1.158983384",
      "0.324653339"
    ],
    [
      "1.304434078",
      "0.324653339"
    ],
    [
      "1.449884772",
      "0.324653339"
    ],
    [
      "0.470403671",
      "0.423418669"
    ],
    [
      "0.615854365",
      "0.423418669"
    ],
    [
      "0.761305059",
      "0.423418669"
    ],
    [
      "1.556661708",
      "0.423418669"
    ],
    [
      "1.709098613",
      "0.427719310"
    ],
    [
      "1.854549307",
      "0.427719310"
    ],
    [
      "0.145450694",
      "0.450617335"
    ],
    [
      "0.290901388",
      "0.450617335"
    ],
    [
      "0.940807343",
      "0.450617335"
    ],
    [
      "1.086258037",
      "0.450617335"
    ],
    [
      "1.231708731",
      "0.450617335"
    ],
    [
      "1.377159424",
      "0.450617335"
    ],
    [
      "1.927274653",
      "0.553683306"
    ],
    [
      "0.397678324",
      "0.549382665"
    ],
    [
      "0.543129018",
      "0.549382665"
    ],
    [
      "0.688579712",
      "0.549382665"
    ],
    [
      "0.834030406",
      "0.549382665"
    ],
    [
      "1.483936360",
      "0.549382664"
    ],
    [
      "1.629387054",
      "0.549382664"
    ],
    [
      "1.781823960",
      "0.553683306"
    ],
    [
      "0.072725347",
      "0.576581331"
    ],
    [
      "0.218176041",
      "0.576581331"
    ],
    [
      "1.013532690",
      "0.576581331"
    ],
    [
      "1.158983384",
      "0.576581331"
    ],
    [
      "1.304434078",
      "0.576581331"
    ],
    [
      "1.854549306",
      "0.679647302"
    ],
    [
      "0.324952977",
      "0.675346661"
    ],
    [
      "0.470403671",
      "0.675346661"
    ],
    [
      "0.615854365",
      "0.675346661"
    ],
    [
      "0.761305059",
      "0.675346661"
    ],
    [
      "0.906755754",
      "0.675346661"
    ],
    [
      "1.411211014",
      "0.675346661"
    ],
    [
      "1.556661707",
      "0.675346660"
    ],
    [
      "1.702112401",
      "0.675346660"
    ],
    [
      "0.145450694",
      "0.702545327"
    ],
    [
      "1.086258037",
      "0.702545327"
    ],
    [
      "1.231708731",
      "0.702545327"
    ],
    [
      "0.252227630",
      "0.801310657"
    ],
    [
      "0.397678324",
      "0.801310657"
    ],
    [
      "0.543129018",
      "0.801310657"
    ],
    [
      "0.688579712",
      "0.801310657"
    ],
    [
      "0.834030407",
      "0.801310657"
    ],
    [
      "0.979481101",
      "0.801310657"
    ],
    [
      "1.483936361",
      "0.801310657"
    ],
    [
      "1.774837748",
      "0.801310657"
    ],
    [
      "1.927274653",
      "0.805611298"
    ],
    [
      "1.338485667",
      "0.801310657"
    ],
    [
      "1.629387054",
      "0.801310657"
    ],
    [
      "0.072725347",
      "0.828509323"
    ],
    [
      "1.158983384",
      "0.828509323"
    ],
    [
      "0.179502283",
      "0.927274653"
    ],
    [
      "0.324952977",
      "0.927274653"
    ],
    [
      "0.470403671",
      "0.927274653"
    ],
    [
      "0.615854365",
      "0.927274653"
    ],
    [
      "0.761305059",
      "0.927274653"
    ],
    [
      "0.906755754",
      "0.927274653"
    ],
    [
      "1.052206448",
      "0.927274653"
    ],
    [
      "1.265760320",
      "0.927274653"
    ],
    [
      "1.411211014",
      "0.927274653"
    ],
    [
      "1.556661708",
      "0.927274653"
    ],
    [
      "1.702112401",
      "0.927274653"
    ],
    [
      "1.847563095",
      "0.927274653"
    ]
  ],
  "radius": "0.072725346"
}
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": 99
}

当前第一名的答案

{
  "centers": [
    [
      "0.072725347",
      "0.072725347"
    ],
    [
      "0.218176041",
      "0.072725347"
    ],
    [
      "0.363626735",
      "0.072725347"
    ],
    [
      "0.509077429",
      "0.072725347"
    ],
    [
      "0.722631302",
      "0.072725347"
    ],
    [
      "0.868081996",
      "0.072725347"
    ],
    [
      "1.013532690",
      "0.072725347"
    ],
    [
      "1.158983384",
      "0.072725347"
    ],
    [
      "1.304434078",
      "0.072725347"
    ],
    [
      "1.449884772",
      "0.072725347"
    ],
    [
      "1.595335466",
      "0.072725347"
    ],
    [
      "1.808235123",
      "0.072725347"
    ],
    [
      "1.927274653",
      "0.156304619"
    ],
    [
      "0.615854365",
      "0.171490677"
    ],
    [
      "1.701785295",
      "0.171843148"
    ],
    [
      "0.145450694",
      "0.198689343"
    ],
    [
      "0.290901388",
      "0.198689343"
    ],
    [
      "0.436352082",
      "0.198689343"
    ],
    [
      "0.795356649",
      "0.198689343"
    ],
    [
      "0.940807343",
      "0.198689343"
    ],
    [
      "1.086258037",
      "0.198689343"
    ],
    [
      "1.231708731",
      "0.198689343"
    ],
    [
      "1.377159425",
      "0.198689343"
    ],
    [
      "1.526279973",
      "0.200880451"
    ],
    [
      "1.781798354",
      "0.299026208"
    ],
    [
      "0.543129018",
      "0.297454673"
    ],
    [
      "0.688579712",
      "0.297454673"
    ],
    [
      "1.636373266",
      "0.301755314"
    ],
    [
      "1.927274653",
      "0.301755313"
    ],
    [
      "0.072725347",
      "0.324653339"
    ],
    [
      "0.218176041",
      "0.324653339"
    ],
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      "0.363626735",
      "0.324653339"
    ],
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      "0.868081996",
      "0.324653339"
    ],
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      "0.324653339"
    ],
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      "0.324653339"
    ],
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    ],
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      "0.423418669"
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      "0.427719310"
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      "0.145450694",
      "0.450617335"
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      "1.377159424",
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    ],
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      "1.927274653",
      "0.553683306"
    ],
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    ],
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      "0.543129018",
      "0.549382665"
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      "0.688579712",
      "0.549382665"
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      "0.549382665"
    ],
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      "0.549382664"
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      "0.549382664"
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      "0.553683306"
    ],
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      "0.576581331"
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      "0.576581331"
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      "1.013532690",
      "0.576581331"
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      "1.158983384",
      "0.576581331"
    ],
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      "1.304434078",
      "0.576581331"
    ],
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      "1.854549306",
      "0.679647302"
    ],
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      "0.324952977",
      "0.675346661"
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      "0.470403671",
      "0.675346661"
    ],
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      "0.615854365",
      "0.675346661"
    ],
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      "0.761305059",
      "0.675346661"
    ],
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      "0.675346661"
    ],
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      "1.411211014",
      "0.675346661"
    ],
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      "1.556661707",
      "0.675346660"
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    [
      "1.702112401",
      "0.675346660"
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    [
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      "0.702545327"
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      "0.702545327"
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      "0.702545327"
    ],
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      "0.801310657"
    ],
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      "0.397678324",
      "0.801310657"
    ],
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      "0.543129018",
      "0.801310657"
    ],
    [
      "0.688579712",
      "0.801310657"
    ],
    [
      "0.834030407",
      "0.801310657"
    ],
    [
      "0.979481101",
      "0.801310657"
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      "1.483936361",
      "0.801310657"
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      "1.774837748",
      "0.801310657"
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    [
      "1.927274653",
      "0.805611298"
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    [
      "1.338485667",
      "0.801310657"
    ],
    [
      "1.629387054",
      "0.801310657"
    ],
    [
      "0.072725347",
      "0.828509323"
    ],
    [
      "1.158983384",
      "0.828509323"
    ],
    [
      "0.179502283",
      "0.927274653"
    ],
    [
      "0.324952977",
      "0.927274653"
    ],
    [
      "0.470403671",
      "0.927274653"
    ],
    [
      "0.615854365",
      "0.927274653"
    ],
    [
      "0.761305059",
      "0.927274653"
    ],
    [
      "0.906755754",
      "0.927274653"
    ],
    [
      "1.052206448",
      "0.927274653"
    ],
    [
      "1.265760320",
      "0.927274653"
    ],
    [
      "1.411211014",
      "0.927274653"
    ],
    [
      "1.556661708",
      "0.927274653"
    ],
    [
      "1.702112401",
      "0.927274653"
    ],
    [
      "1.847563095",
      "0.927274653"
    ]
  ],
  "radius": "0.072725346"
}

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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本题n=8时用抽屉原理的证明NUE_13#313 条回复好！！！NUE_13#310 条回复