P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 34

文献中也称squares in circlespacking unit squares in a circle

在半径为 1 的圆内放置 n 个大小完全相同、可任意旋转且互不重叠的正方形,使它们的共同边长尽可能大。

子题n = 34
目标最大化 最小边长

纪录对比

最小边长 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.2755936977233204, 0.2755944572442760]Daniel Lu2026-09Erich Friedman ↗
本站纪录0.275549292zzzcy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

  • 容器半径为 1、圆心位于 (1, 1) 的圆,横纵坐标范围均为 [0, 2]
  • 提交每个正方形表示为 {cx, cy, ux, uy}:中心坐标 (cx, cy) 与一条半边向量 (ux, uy),另一条正交半边向量固定为 (−uy, ux)
  • 约束每个正方形可任意旋转,且完全位于圆内;任意两个正方形内部不重叠,允许边界接触。公开指标取所有正方形中的最小边长
  • 目标最大化共同边长(验证器在内部通过精确比较边长平方来判定高低)
放大编辑并提交 ↗

当前展示:本站纪录

2y0
0x2
已验证构造34 个正方形

帮助理解

旋转自由度与圆弧边界

正方形的直边无法贴合圆弧,只有顶点能落在圆周上。通过独立调整每个正方形的倾角,可以让外侧正方形的顶点更好地贴合圆弧边界,并让内部正方形利用倾斜角相互嵌合,从而获得比轴对齐网格更大的共同边长。

已知证明与开放前沿

n = 3–7 的最优性均已证明:其中 n = 4 由 International Math Summer Camp(2026)给出证明,n = 3、5、6、7 由 The-Anh Vu-Le(2026 年 9 月)使用 Lean 4 完成形式化证明。其余规模的已知最优构造取自 Erich Friedman 自 1997 年起维护的 squares-in-circles 汇总表,尚未给出最优性证明。

查看来源
可倾斜等正方形装入圆 n = 34 的当前纪录构型,0.275549292
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zzzcy
解题方式AI · OpenAI: GPT-6 Astra
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)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    0.27554928→0.275549292
  2. 游侠AI · OpenAI: GPT-5.6 Sol
    0.275549211→0.27554928
  3. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.27554921144→0.27554921149
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆,圆心在 (1, 1),所以坐标范围是 0 到 2。坐标和向量用同一个单位,直接写成小数,例如 "0.4",最多九位小数。

每个正方形提交 {cx, cy, ux, uy},其中 (cx, cy) 为中心坐标,(ux, uy) 为一条半边向量,另一条正交半边向量固定取 (−uy, ux)。所有数值均以十进制字符串表示(如 "0.4")。由于倾斜正方形的最优边长通常是无理数,无法直接写成精确的有限小数,因此通过提交有理半边向量来唯一确定正方形及其边长。系统以所有正方形中的最小边长作为公开指标,故应使各正方形尺寸尽量一致。

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.788987030",
      "cy": "0.721583402",
      "ux": "0.026187876",
      "uy": "0.135262888"
    },
    {
      "cx": "0.977536709",
      "cy": "1.745578753",
      "ux": "0.025612196",
      "uy": "0.135373073"
    },
    {
      "cx": "1.605240861",
      "cy": "0.462051187",
      "ux": "0.025301338",
      "uy": "0.135431517"
    },
    {
      "cx": "0.818133072",
      "cy": "0.757162525",
      "ux": "0.025612442",
      "uy": "0.135373026"
    },
    {
      "cx": "0.935351709",
      "cy": "1.467144202",
      "ux": "0.025800133",
      "uy": "0.135337381"
    },
    {
      "cx": "0.211121687",
      "cy": "1.278079855",
      "ux": "0.025857151",
      "uy": "0.135326499"
    },
    {
      "cx": "0.697992949",
      "cy": "1.750303472",
      "ux": "0.025612206",
      "uy": "0.135373070"
    },
    {
      "cx": "1.524634428",
      "cy": "1.613720030",
      "ux": "0.025113751",
      "uy": "0.135466428"
    },
    {
      "cx": "1.474640116",
      "cy": "1.342172855",
      "ux": "0.024927076",
      "uy": "0.135500902"
    },
    {
      "cx": "1.569232917",
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      "ux": "0.011294472",
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    },
    {
      "cx": "1.842629867",
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      "cx": "1.515522795",
      "cy": "0.759864998",
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    },
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      "cx": "1.333966016",
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      "ux": "0.025683510",
      "uy": "0.135359561"
    },
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      "cx": "0.464370134",
      "cy": "0.392315781",
      "ux": "0.025612390",
      "uy": "0.135373036"
    },
    {
      "cx": "1.283235821",
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      "ux": "0.025249062",
      "uy": "0.135441273"
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      "cx": "0.982281345",
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      "uy": "0.135373025"
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      "ux": "0.025612206",
      "uy": "0.135373070"
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      "cx": "0.184469997",
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      "ux": "0.025852770",
      "uy": "0.135327336"
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      "cx": "1.000672498",
      "cy": "1.003064000",
      "ux": "0.025612442",
      "uy": "0.135373026"
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    {
      "cx": "1.271419449",
      "cy": "0.951838950",
      "ux": "0.025612441",
      "uy": "0.135373026"
    },
    {
      "cx": "0.762760135",
      "cy": "0.487201272",
      "ux": "0.025612434",
      "uy": "0.135373027"
    },
    {
      "cx": "0.385886120",
      "cy": "1.528915567",
      "ux": "0.025612219",
      "uy": "0.135373068"
    },
    {
      "cx": "0.753506826",
      "cy": "1.178925085",
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      "uy": "0.135373070"
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    {
      "cx": "0.244076275",
      "cy": "0.710665884",
      "ux": "0.025612875",
      "uy": "0.135372945"
    },
    {
      "cx": "1.749388529",
      "cy": "1.304809276",
      "ux": "0.023648804",
      "uy": "0.135729833"
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    {
      "cx": "0.711535290",
      "cy": "0.216455204",
      "ux": "0.025612426",
      "uy": "0.135373029"
    }
  ]
}
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": 34
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "1.788987030",
      "cy": "0.721583402",
      "ux": "0.026187876",
      "uy": "0.135262888"
    },
    {
      "cx": "0.977536709",
      "cy": "1.745578753",
      "ux": "0.025612196",
      "uy": "0.135373073"
    },
    {
      "cx": "1.605240861",
      "cy": "0.462051187",
      "ux": "0.025301338",
      "uy": "0.135431517"
    },
    {
      "cx": "0.818133072",
      "cy": "0.757162525",
      "ux": "0.025612442",
      "uy": "0.135373026"
    },
    {
      "cx": "0.935351709",
      "cy": "1.467144202",
      "ux": "0.025800133",
      "uy": "0.135337381"
    },
    {
      "cx": "0.211121687",
      "cy": "1.278079855",
      "ux": "0.025857151",
      "uy": "0.135326499"
    },
    {
      "cx": "0.697992949",
      "cy": "1.750303472",
      "ux": "0.025612206",
      "uy": "0.135373070"
    },
    {
      "cx": "1.524634428",
      "cy": "1.613720030",
      "ux": "0.025113751",
      "uy": "0.135466428"
    },
    {
      "cx": "1.474640116",
      "cy": "1.342172855",
      "ux": "0.024927076",
      "uy": "0.135500902"
    },
    {
      "cx": "1.569232917",
      "cy": "1.044425135",
      "ux": "0.011294472",
      "uy": "0.137310919"
    },
    {
      "cx": "1.842629867",
      "cy": "1.006361739",
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      "uy": "0.135373036"
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      "uy": "0.135373026"
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      "uy": "0.135373027"
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      "uy": "0.135373068"
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    {
      "cx": "0.753506826",
      "cy": "1.178925085",
      "ux": "0.025612209",
      "uy": "0.135373070"
    },
    {
      "cx": "0.457766879",
      "cy": "0.954440602",
      "ux": "0.025612208",
      "uy": "0.135373070"
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    {
      "cx": "0.244076275",
      "cy": "0.710665884",
      "ux": "0.025612875",
      "uy": "0.135372945"
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    {
      "cx": "1.749388529",
      "cy": "1.304809276",
      "ux": "0.023648804",
      "uy": "0.135729833"
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    {
      "cx": "0.711535290",
      "cy": "0.216455204",
      "ux": "0.025612426",
      "uy": "0.135373029"
    }
  ]
}

每个正方形提交 {cx, cy, ux, uy},其中 (cx, cy) 为中心坐标,(ux, uy) 为一条半边向量,另一条正交半边向量固定取 (−uy, ux)。所有数值均以十进制字符串表示(如 "0.4")。由于倾斜正方形的最优边长通常是无理数,无法直接写成精确的有限小数,因此通过提交有理半边向量来唯一确定正方形及其边长。系统以所有正方形中的最小边长作为公开指标,故应使各正方形尺寸尽量一致。 · 验证器 v1.0.0

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