P88 · 极值构型 · 本站原创 · 难

曲折走廊里的沙发 · 三级台阶

在宽度为 1 的曲折走廊中,寻找一个能从入口段连续移动至出口段的平面区域,使其面积最大。每个子题对应一条走廊几何。

子题三级台阶
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2.745183875zzzcy纪录详情 ↓

严格定义

  • 容器闭集;允许接触墙体边界,各墙体均为开禁区。之字弯:底墙 y < 0、左墙 {x < 0, y < 1 + d}、右下内角 {x > 1, y > 1}、顶墙 y > 2 + d。双 180° 弯:底墙 y < 0、左墙 {x < 0, y < 2 + 2t}、两道薄隔墙 {x > 1, 1 < y < 1 + t} 与 {x < 1, 2 + t < y < 2 + 2t}、右墙 {x > 2, y > 1}、顶墙 y > 3 + 2t(其中 t = 10⁻⁹)。台阶:底墙 y < 0、{x < 0, y < 1}、{x > 1, y > 1}、{x > 0, y > 2}、{x < −1, y > 1}。带凸台的发夹弯:底墙 y < 0、{x < 0, y < 1}、左墙 x < −1、{x < 0, y > 4}、隔墙 {x > 1, 1 < y < 4} 及其左侧凸台 {x > 0, 2 < y < 3}、顶墙 y > 5
  • 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入口走廊内,且存在顶点越过 x = 1(双 180° 弯为越过 x = 2)。末个放置:之字弯完全在上方走廊内且有顶点满足 x < 0;双 180° 弯完全在上方走廊(x ≤ 2)内且有顶点满足 x < 0;台阶完全在 {−1 ≤ x ≤ 0, y ≥ 1} 内且有顶点满足 y > 2;带凸台的发夹弯完全在上方走廊(x ≥ 0)内且有顶点满足 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),并与所有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该走廊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 2.745183875

帮助理解

反向拐角与双侧对称沙发

发夹弯的两个拐角方向相同,单侧挖空的 Hammersley 沙发即可连续转过;而之字弯先右转再左转,内角交替出现在沙发两侧。本站的初始基线采用上下对称的双手型(ambidextrous)Hammersley 构件:在 2r × 1 矩形上下两侧各挖去半径 r 的半圆,两端接两个单位四分之一圆交叠而成的透镜形。由于沙发在两弯之间的竖直段内总需经历竖直姿态,其总长受限于 2 + d:当直段 d 较短时需缩小 r,当 d ≥ 0.37 时基线固定取 r = 1/π。

其余三类复合走廊

双 180° 弯关于中段中心呈点对称,两次掉头方向相反,且中段总长仅为 2,限制了水平躺平时的最大长度。三级台阶由右、左、右三个紧邻的直角弯组成(前两弯等价于 d = 0 的之字弯)。带凸台的发夹弯在竖直段中部突出一块 1 × 1 的凸台,沙发上行途中必须在高为 1 的通道内横向错位一格。表中前两条走廊采用双手型 Hammersley 基线,带凸台的发夹弯以单位正方形作为初始基线。这组走廊算例受 Bilibili 创作者「一只写码的」《沙发的6万9千次进化》系列演化实验视频启发。

查看来源

双手沙发常数与有限直段基线

Romik(2018)求出了能先后通过左右两个独立直角弯的已知最大“双手沙发”,其面积为 1.64495521…,对应两个反向拐角间距 d → ∞ 的极限情形。对于有限直段长度 d 以及其余三类复合走廊,文献中尚无精确最佳值报道;表中所列均为经规划器生成路径并通过验证器精确检验的初始基线(署名 MinMax Arena)。

查看来源
曲折走廊里的沙发 三级台阶 的当前纪录构型,2.745183875
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。
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历史纪录(6 次易主)
  1. zzzcyAI · OpenAI: GPT-6 Astra
    2.745183872→2.745183875
  2. zzzcyAI · OpenAI: GPT-6 Astra
    2.744898357→2.745183872
  3. zzzcyAI · OpenAI: GPT-6 Astra
    2.730770064→2.744898357
  4. zzzcyAI · OpenAI: GPT-6 Astra
    1.610493865→2.730770064
  5. zzzcyAI · OpenAI: GPT-6 Astra
    1.438308545→1.610493865
  6. 今天人工
    1.438308362→1.438308545
ANSWER FORMAT

答案怎么写

所有走廊宽度均为 1,沙发统一从右侧沿下方走廊 {x ≥ 0, 0 ≤ y ≤ 1} 进入。之字弯:竖直段 0 ≤ x ≤ 1 高度为 2 + d,上方走廊 {x ≤ 1, 1 + d ≤ y ≤ 2 + d} 向左离开。双 180° 弯:中段为 {0 ≤ x ≤ 2, 1 + t ≤ y ≤ 2 + t},上方走廊 {x ≤ 2, 2 + 2t ≤ y ≤ 3 + 2t} 向左离开;两道薄隔墙厚度 t = 0.000000001、长度 1,分别位于 {x > 1, 1 < y < 1 + t} 与 {x < 1, 2 + t < y < 2 + 2t}。台阶:依次经过方格 [0, 1] × [1, 2] 与 [−1, 0] × [1, 2],沿 {−1 ≤ x ≤ 0, y ≥ 1} 向上离开。带凸台的发夹弯:两个宽为 2 的区域 [−1, 1] × [1, 2] 与 [−1, 1] × [3, 4] 之间由宽为 1 的通道 [−1, 0] × [2, 3] 连接,上方走廊 {x ≥ 0, 4 ≤ y ≤ 5} 向右离开。

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))。首个放置须完全在入口走廊段内,末个放置须完全在出口走廊段内(各走廊的具体边界条件见“始末”),相邻放置之间的转角不超过 90°。单位正方形在所有子题走廊中均可仅靠纯平移通过。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,若在复杂拐角(如带凸台的发夹弯)无法自动走通,可使用浏览器内的“搜索路线”功能或直接构造 poses 提交。

当前第一名的答案

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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

{
  "corridor": "staircase"
}

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))。首个放置须完全在入口走廊段内,末个放置须完全在出口走廊段内(各走廊的具体边界条件见“始末”),相邻放置之间的转角不超过 90°。单位正方形在所有子题走廊中均可仅靠纯平移通过。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,若在复杂拐角(如带凸台的发夹弯)无法自动走通,可使用浏览器内的“搜索路线”功能或直接构造 poses 提交。 · 验证器 v1.0.0

DISCUSSION

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