P05 · 装箱与覆盖 · 经典问题 · 难

可倾斜等正方形装入圆 · n = 40

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 = 40
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[0.2539140856299862, 0.2539147303552521]Ephraim Duncan2026-09Erich Friedman ↗
本站纪录0.253914652zzzcy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40 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 = 40 的当前纪录构型,0.253914652
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(5 次易主)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    0.25391465→0.253914652
  2. ephraimduncanAI · OpenAI: GPT-6 Astra
    0.249359391→0.25391465
  3. 游侠AI · OpenAI: GPT-5.6 Sol
    0.249357046→0.249359391
  4. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.24935704653→0.24935704656
  5. 邱仲普AI · OpenAI: GPT-5.6 Sol
    0.049984378→0.249357046
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": "0.749449722",
      "cy": "1.037799474",
      "ux": "0.124412651",
      "uy": "-0.025291405"
    },
    {
      "cx": "0.963422588",
      "cy": "0.737593375",
      "ux": "0.124412651",
      "uy": "-0.025291405"
    },
    {
      "cx": "1.253852322",
      "cy": "0.891669713",
      "ux": "0.124412651",
      "uy": "-0.025291405"
    },
    {
      "cx": "0.714128802",
      "cy": "0.785871636",
      "ux": "0.124412651",
      "uy": "-0.025291405"
    },
    {
      "cx": "1.203269511",
      "cy": "0.642844411",
      "ux": "0.124412651",
      "uy": "-0.025291405"
    },
    {
      "cx": "1.021141807",
      "cy": "1.514032894",
      "ux": "0.124412651",
      "uy": "-0.025291405"
    },
    {
      "cx": "0.974648181",
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      "ux": "0.124412651",
      "uy": "-0.025291405"
    },
    {
      "cx": "0.816624699",
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      "ux": "0.126828693",
      "uy": "-0.005713611"
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  ]
}
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": 40
}

当前第一名的答案

{
  "squares": [
    {
      "cx": "0.749449722",
      "cy": "1.037799474",
      "ux": "0.124412651",
      "uy": "-0.025291405"
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      "uy": "-0.025291405"
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    {
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      "uy": "-0.025291405"
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}

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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