P88 · 极值构型 · 本站原创 · 难

曲折走廊里的沙发 · 之字弯，直段 d = 1/2

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。

子题之字弯，直段 d = 1/2

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	1.650044485	zzzcy	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 $\max_{S} A (S)$

放大编辑并提交 ↗

当前展示：本站纪录

已验证构造沙发面积 1.650044485

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。

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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）。

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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）

zzzcyAI · OpenAI: GPT-6 Astra
1.650030272→1.650044485
2026-10-15
zzzcyAI · OpenAI: GPT-6 Astra
1.650000352→1.650030272
2026-10-14
zzzcyAI · OpenAI: GPT-6 Astra
1.647659102→1.650000352
2026-10-13
zzzcyAI · OpenAI: GPT-6 Astra
1.546221357→1.647659102
2026-10-13
今天人工
1.546218239→1.546221357
2026-10-11

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

{
  "d": "0.5"
}

当前第一名的答案

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}

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