P12 · 装箱与覆盖 · 经典问题 · 难

2:1 长方形内的等圆装箱 · n = 71

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 = 71

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	[0.084786483236749812260700356137, 0.084786483236749812260700356139]	Eckard Specht2011-03-19	Packomania ↗
本站纪录	0.084786483	NUE_13	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768697071

0x2

已验证构造r = 0.084786483

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。

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NUE_13

解题方式AI · DeepSeek: DeepSeek V4.1 Flash

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（3 次易主）

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.084786482→0.084786483
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.082854058→0.084786482
2026-10-17
NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
0.066666666→0.082854058
2026-10-17

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": [
    [
      "0.084786483",
      "0.084786483"
    ],
    [
      "0.372488976",
      "0.084786483"
    ],
    [
      "0.651840318",
      "0.084786483"
    ],
    [
      "0.821413284",
      "0.084786483"
    ],
    [
      "0.990986250",
      "0.084786483"
    ],
    [
      "1.160559216",
      "0.084786483"
    ],
    [
      "1.330132183",
      "0.084786483"
    ],
    [
      "1.499705149",
      "0.084786483"
    ],
    [
      "1.669278115",
      "0.084786483"
    ],
    [
      "1.838851082",
      "0.084786483"
    ],
    [
      "0.228637730",
      "0.174574065"
    ],
    [
      "0.512164646",
      "0.180941036"
    ],
    [
      "0.906199766",
      "0.231640980"
    ],
    [
      "1.075772733",
      "0.231640980"
    ],
    [
      "1.245345699",
      "0.231640980"
    ],
    [
      "1.414918666",
      "0.231640980"
    ],
    [
      "1.584491632",
      "0.231640980"
    ],
    [
      "0.736602921",
      "0.231654765"
    ],
    [
      "1.915213517",
      "0.236192459"
    ],
    [
      "0.084786483",
      "0.264361646"
    ],
    [
      "1.751191972",
      "0.282950470"
    ],
    [
      "0.401838708",
      "0.312007731"
    ],
    [
      "0.596927239",
      "0.327809323"
    ],
    [
      "0.234470432",
      "0.344046690"
    ],
    [
      "0.821413283",
      "0.378495477"
    ],
    [
      "0.990986249",
      "0.378495477"
    ],
    [
      "1.160559216",
      "0.378495477"
    ],
    [
      "1.330132182",
      "0.378495477"
    ],
    [
      "1.620711942",
      "0.397300511"
    ],
    [
      "1.915213517",
      "0.405765426"
    ],
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      "0.086853799",
      "0.433922011"
    ],
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      "0.681737610",
      "0.474650026"
    ],
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      "1.469807855",
      "0.474650026"
    ],
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      "0.512164646",
      "0.474677611"
    ],
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      "0.474678947"
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      "1.765549592",
      "0.485488073"
    ],
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      "0.906199766",
      "0.525349974"
    ],
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      "1.075772733",
      "0.525349974"
    ],
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      "0.525349974"
    ],
    [
      "1.915213517",
      "0.575338392"
    ],
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      "1.601758537",
      "0.600641196"
    ],
    [
      "0.084786483",
      "0.603482375"
    ],
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      "0.253643290",
      "0.619050603"
    ],
    [
      "0.766524093",
      "0.621504523"
    ],
    [
      "1.385021372",
      "0.621504523"
    ],
    [
      "0.595208132",
      "0.622524729"
    ],
    [
      "0.425553624",
      "0.622571838"
    ],
    [
      "1.765549592",
      "0.655061039"
    ],
    [
      "0.990986250",
      "0.672204471"
    ],
    [
      "1.160559216",
      "0.672204471"
    ],
    [
      "1.915050809",
      "0.745842188"
    ],
    [
      "0.155732406",
      "0.757500774"
    ],
    [
      "0.336605234",
      "0.766943493"
    ],
    [
      "0.681737609",
      "0.768359020"
    ],
    [
      "0.851310577",
      "0.768359020"
    ],
    [
      "1.300234889",
      "0.768359020"
    ],
    [
      "1.469807855",
      "0.768359020"
    ],
    [
      "1.639380822",
      "0.768359020"
    ],
    [
      "0.510421648",
      "0.769379227"
    ],
    [
      "1.075772732",
      "0.819058968"
    ],
    [
      "0.084786483",
      "0.911519172"
    ],
    [
      "1.898791126",
      "0.914990533"
    ],
    [
      "1.728914747",
      "0.915015317"
    ],
    [
      "0.254319202",
      "0.915213517"
    ],
    [
      "0.423892169",
      "0.915213517"
    ],
    [
      "0.596951127",
      "0.915213517"
    ],
    [
      "0.766524093",
      "0.915213516"
    ],
    [
      "0.936097059",
      "0.915213517"
    ],
    [
      "1.215448405",
      "0.915213517"
    ],
    [
      "1.385021372",
      "0.915213517"
    ],
    [
      "1.554594338",
      "0.915213517"
    ]
  ],
  "radius": "0.084786483"
}

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": 71
}

当前第一名的答案

{
  "centers": [
    [
      "0.084786483",
      "0.084786483"
    ],
    [
      "0.372488976",
      "0.084786483"
    ],
    [
      "0.651840318",
      "0.084786483"
    ],
    [
      "0.821413284",
      "0.084786483"
    ],
    [
      "0.990986250",
      "0.084786483"
    ],
    [
      "1.160559216",
      "0.084786483"
    ],
    [
      "1.330132183",
      "0.084786483"
    ],
    [
      "1.499705149",
      "0.084786483"
    ],
    [
      "1.669278115",
      "0.084786483"
    ],
    [
      "1.838851082",
      "0.084786483"
    ],
    [
      "0.228637730",
      "0.174574065"
    ],
    [
      "0.512164646",
      "0.180941036"
    ],
    [
      "0.906199766",
      "0.231640980"
    ],
    [
      "1.075772733",
      "0.231640980"
    ],
    [
      "1.245345699",
      "0.231640980"
    ],
    [
      "1.414918666",
      "0.231640980"
    ],
    [
      "1.584491632",
      "0.231640980"
    ],
    [
      "0.736602921",
      "0.231654765"
    ],
    [
      "1.915213517",
      "0.236192459"
    ],
    [
      "0.084786483",
      "0.264361646"
    ],
    [
      "1.751191972",
      "0.282950470"
    ],
    [
      "0.401838708",
      "0.312007731"
    ],
    [
      "0.596927239",
      "0.327809323"
    ],
    [
      "0.234470432",
      "0.344046690"
    ],
    [
      "0.821413283",
      "0.378495477"
    ],
    [
      "0.990986249",
      "0.378495477"
    ],
    [
      "1.160559216",
      "0.378495477"
    ],
    [
      "1.330132182",
      "0.378495477"
    ],
    [
      "1.620711942",
      "0.397300511"
    ],
    [
      "1.915213517",
      "0.405765426"
    ],
    [
      "0.086853799",
      "0.433922011"
    ],
    [
      "0.681737610",
      "0.474650026"
    ],
    [
      "1.469807855",
      "0.474650026"
    ],
    [
      "0.512164646",
      "0.474677611"
    ],
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      "0.342591680",
      "0.474678947"
    ],
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      "0.906199766",
      "0.525349974"
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      "0.622524729"
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      "0.425553624",
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      "0.768359020"
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      "1.300234889",
      "0.768359020"
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      "1.469807855",
      "0.768359020"
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      "1.639380822",
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      "0.510421648",
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      "0.819058968"
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      "0.911519172"
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      "0.914990533"
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      "1.728914747",
      "0.915015317"
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      "0.915213517"
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      "0.423892169",
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      "0.596951127",
      "0.915213517"
    ],
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      "0.766524093",
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      "0.936097059",
      "0.915213517"
    ],
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      "1.215448405",
      "0.915213517"
    ],
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      "1.385021372",
      "0.915213517"
    ],
    [
      "1.554594338",
      "0.915213517"
    ]
  ],
  "radius": "0.084786483"
}

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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