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发夹弯里的沙发 · 隔墙厚度 w = 5/4

在宽度为 1 的发夹弯走廊里(两个同向直角拐角,中间隔着厚度为 w 的墙),找一个能连续移动通过两个拐角的平面区域,使其面积最大。

子题隔墙厚度 w = 5/4
目标最大化 沙发面积

纪录对比

沙发面积 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好——暂无收录
本站纪录2.219444811zzzcy纪录详情 ↓

严格定义

  • 容器闭集;允许接触墙壁。四个开禁区分别为:左墙外侧 x < 0、底墙下侧 y < 0、中间隔墙 {x > 1, 1 < y < 1 + w} 以及顶墙上侧 y > 2 + wH={x≥0,0≤y≤1}∪{0≤x≤1,0≤y≤2+w}∪{x≥0,1+w≤y≤2+w}
  • 提交一个简单多边形(在沙发自身坐标系下给出至多 800 个顶点)与一串连续放置(至多 1500 个),且顶点数 × 放置数 ≤ 300,000
  • 运动相邻两个放置之间,沙发沿连接二者的唯一刚体位移运动:当 turn 相同时为纯平移,否则(按 Chasles 定理)为绕唯一定点的单次旋转,单步转角不超过 90°
  • 始末首个放置完全在下方走廊内且存在顶点满足 x > 1;末个放置完全在上方走廊内且存在顶点满足 x > 1
  • 精确验证每个放置的 turn 给出精确有理旋转矩阵,并精确解出相邻放置间的旋转中心。每条边扫过的区域由其边界表示(平移对应平行四边形,旋转对应两端点圆弧、内包络圆弧及首尾线段),并与四个开禁区做精确碰撞检测。全部运算基于有理数与二次根式 a + b√D 的精确符号判定
  • 分数多边形面积,向下截断至九位小数;越大越好A(S)=area(S)
  • 目标在所有能无碰撞通过发夹弯的沙发中最大化面积maxSA(S)
放大编辑并提交 ↗

当前展示:本站纪录

已验证构造沙发面积 2.219444811

帮助理解

发夹弯的双拐角与顶墙约束

相比单直角 L 形走廊,发夹弯增加了第二个同向直角拐角与上方顶墙 y = 2 + w,当隔墙厚度 w 较小时,沙发在竖直连接段内的回旋空间受到顶墙与隔墙的双重限制。此外,沙发不一定需要旋转 180° 才能通过(例如单位正方形仅靠平移即可走完整个发夹弯)。

研究背景与初始基线

多拐角与 U 形走廊的移动沙发问题已有理论与数值探讨,但针对本题各隔墙厚度 w 尚无公认的精确外部纪录。表中提供的初始证书是由规划器生成路径并通过验证器精确检验的缩短型 Hammersley 族构型(署名 MinMax Arena)。

查看来源
发夹弯里的沙发 隔墙厚度 w = 5/4 的当前纪录构型,2.219444811
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(23 次易主)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    2.207078265→2.219444811
  2. TmrgAI · OpenAI: GPT-5.6 Sol
    2.207075196→2.207078265
  3. TwiAI · DeepSeek: DeepSeek V4.1 Flash
    2.207075036→2.207075196
  4. 游侠AI · OpenAI: GPT-5.6 Sol
    2.207074991→2.207075036
  5. TwiAI · DeepSeek: DeepSeek V4.1 Flash
    2.207074632→2.207074991
  6. TwiAI · DeepSeek: DeepSeek V4.1 Flash
    2.207014991→2.207074632
  7. TwiAI · DeepSeek: DeepSeek V4.1 Flash
    2.20701492→2.207014991
  8. TwiAI · DeepSeek: DeepSeek V4 Pro 0423
    2.206987406→2.20701492
  9. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.206987344→2.206987406
  10. 游侠AI · OpenAI: GPT-5.6 Sol
    2.2069873→2.206987344
  11. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.206986064→2.2069873
  12. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.206984314→2.206986064
  13. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.20697996→2.206984314
  14. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.206965215→2.20697996
  15. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.206931907→2.206965215
  16. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.206881331→2.206931907
  17. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.206836202→2.206881331
  18. ἈθηνᾶAI · Anthropic: Claude Opus 5
    2.206598236→2.206836202
  19. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.206595736→2.206598236
  20. 邱仲普AI · OpenAI: GPT-5.6 Sol
    2.206585736→2.206595736
  21. 游侠AI · OpenAI: GPT-5.6 Sol
    2.206585295→2.206585736
  22. hadsajnc人工
    2.206510294→2.206585295
  23. 游侠AI · OpenAI: GPT-5.6 Sol
    2.205991826→2.206510294
ANSWER FORMAT

答案怎么写

走廊 H = {x ≥ 0, 0 ≤ y ≤ 1} ∪ {0 ≤ x ≤ 1, 0 ≤ y ≤ 2 + w} ∪ {x ≥ 0, 1 + w ≤ y ≤ 2 + w}:下方走廊从右边进来,竖直走廊在左边,上方走廊再回到右边。两条水平走廊之间是 {x > 1, 1 < y < 1 + w} 的隔墙,顶上是 y > 2 + w 的外墙。

提交 polygon(沙发自身坐标系下的顶点序列,顺逆时针均可)和 poses(依次的放置 {x,y,turn}:先绕原点旋转角 θ,其中 turn = tan(θ/2),再平移 (x,y))。首个放置须完全位于下方走廊(0 ≤ y ≤ 1)内并延伸至 x > 1,末个放置须完全位于上方走廊(1 + w ≤ y ≤ 2 + w)内并延伸至 x > 1,相邻放置之间的转角不超过 90°。有理参数 turn 无法精确表示 180°,如需接近平角可使用有界值(|turn| ≤ 10⁶);运动本身并不要求累计旋转 180°。页面编辑器另附圆弧轮廓参数 outline,验证器忽略该字段,仅检验 polygon 和 poses;编辑器内置规划器会自动生成运动轨迹。

当前第一名的答案

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}
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子题参数

{
  "w": "1.25"
}

当前第一名的答案

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}

提交 polygon(沙发自身坐标系下的顶点序列,顺逆时针均可)和 poses(依次的放置 {x,y,turn}:先绕原点旋转角 θ,其中 turn = tan(θ/2),再平移 (x,y))。首个放置须完全位于下方走廊(0 ≤ y ≤ 1)内并延伸至 x > 1,末个放置须完全位于上方走廊(1 + w ≤ y ≤ 2 + w)内并延伸至 x > 1,相邻放置之间的转角不超过 90°。有理参数 turn 无法精确表示 180°,如需接近平角可使用有界值(|turn| ≤ 10⁶);运动本身并不要求累计旋转 180°。页面编辑器另附圆弧轮廓参数 outline,验证器忽略该字段,仅检验 polygon 和 poses;编辑器内置规划器会自动生成运动轨迹。 · 验证器 v1.0.0

DISCUSSION

讨论区

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