P78 · 极值构型 · 经典问题 · 难

用 n 个正三角形覆盖圆盘 · n = 17

用 n 个全等正三角形连续覆盖单位圆盘。允许重叠和伸出容器，最小化共同边长。

子题n = 17

目标最小化共同边长

已知最好（未证明）0.697350069735007来源精度区间：0.696864111498257–0.697350069735007（区间内视为追平）本站展示证书 0.707502586论文原值（n 个单位正三角形能盖住的最大圆的半径 r（s = 1/r））：1.434+Maurizio Morandi（2009 年 8 月）的排法，收录于 Friedman 的公开表，那里只印了截断的 r = 1.434+；换算成共同边长 s = 1/r，真值落在 0.696864111498257 与 0.697350069735007 之间，这里显示上端。落在区间内算持平，低于下端才算领先。本站从公开小图复原同一接触拓扑，九位证书严格验证到 0.707502586；它不冒充未公布的原坐标。最优性未知

严格定义

容器闭圆盘 D = {(x,y) : (x−1)² + (y−1)² ≤ 1}
提交共同边长 s 与恰好 n 个 {x,y,turn,down} 放置；每块可独立旋转和翻转
覆盖D 的每一点至少属于一个闭覆盖三角形 $D \subseteq T_{1} \cup \dots \cup T_{n}$
连续精确验证把 3n 条边延长成直线，每个面上各三角形的归属恒定，于是圆盘被盖住当且仅当没有一个与开圆盘相交的面是空的。面按竖直条带内的梯形逐一采样，全部算术在 Q(√3) 中；圆只在「梯形是否碰到开圆盘」这一步出现，而那是有理比较。不是像素采样，也没有任何 ε
分数所有覆盖三角形共享的边长 s，越小越好 $s (T) = s$
目标在全部合法覆盖中最小化 s $\min_{T} s (T)$

放大来摆，然后提交 ↗

已验证构造共同边长 0.707502586

帮助理解

六块正好，第七块起才有题

六个正三角形绕圆心拼成正六边形，其内切圆恰好是单位圆，边长 2/√3，没有任何余地。从第七块开始，多出来的那块该放哪、其余几块该怎么让开，就没有现成答案了。

前沿在哪里

Friedman 的公开表登记 n = 1–18，用的是倒过来的规范：n 个单位正三角形能盖住的最大圆半径 r，这里的边长 s = 1/r。n = 8–13 有闭式（Morandi、Cantrell），n = 7 与 14–18 只有截断的小数（Morandi，2009）；除 n = 1、2、6 外都没有证明最优。原页只给约 200 像素的小图而没有坐标，本站按图中的水平边、±60° 边和接触关系逐项复原，再用连续精确验证器修掉像素舍入留下的细缝。页面把文献目标和九位复原证书分开显示，复原误差不会被冒充为原作者的数据。

查看来源

当前第一名

0.707502586

共同边长

答案来源Maurizio Morandi

解题方式公开参考构造

挑战这个纪录 ↗

ANSWER FORMAT

答案怎么写

容器是圆心 (1,1)、半径 1 的闭圆盘，正好填满 [0,2]² 的板子。坐标标记覆盖三角形一条基准边的中点；覆盖物可以伸出容器。

提交 side 和 triangles；每块写 {x,y,turn,down}。页面编辑器可直接输入或拖动 θ°，并自动换算 turn = tan(θ/2)。

当前第一名的答案

{
  "side": "0.707502586",
  "triangles": [
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.285571790",
      "y": "2.000848638"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.719170916",
      "y": "1.999609970"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.607792025",
      "y": "1.337069630"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.655050679",
      "y": "1.338850865"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.395972967",
      "y": "1.341679245"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.344942020",
      "y": "1.344009033"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.004182636",
      "y": "1.268903304"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.998245550",
      "y": "1.270684539"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.966299854",
      "y": "0.658422482"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.350759417",
      "y": "0.658422482"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.611542254",
      "y": "0.670820872"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.644944572",
      "y": "0.658422482"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.010624050",
      "y": "0.670820872"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.035306287",
      "y": "0.654883430"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.389956135",
      "y": "0.660203717"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.313757350",
      "y": "-0.006591401"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.702821047",
      "y": "-0.006591401"
    }
  ]
}

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": 17
}

当前第一名的答案

{
  "side": "0.707502586",
  "triangles": [
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.285571790",
      "y": "2.000848638"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.719170916",
      "y": "1.999609970"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.607792025",
      "y": "1.337069630"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.655050679",
      "y": "1.338850865"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.395972967",
      "y": "1.341679245"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.344942020",
      "y": "1.344009033"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.004182636",
      "y": "1.268903304"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.998245550",
      "y": "1.270684539"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.966299854",
      "y": "0.658422482"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.350759417",
      "y": "0.658422482"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.611542254",
      "y": "0.670820872"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.644944572",
      "y": "0.658422482"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "1.010624050",
      "y": "0.670820872"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.035306287",
      "y": "0.654883430"
    },
    {
      "down": true,
      "turn": "0.000000000",
      "x": "0.389956135",
      "y": "0.660203717"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "1.313757350",
      "y": "-0.006591401"
    },
    {
      "down": false,
      "turn": "0.000000000",
      "x": "0.702821047",
      "y": "-0.006591401"
    }
  ]
}

提交 side 和 triangles；每块写 {x,y,turn,down}。页面编辑器可直接输入或拖动 θ°，并自动换算 turn = tan(θ/2)。 · 验证器 v1.0.0

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。发帖即公开署名，与纪录同一个名字；署名后的 #编号是账号的注册序号，冒不了名。发言资格与实绩绑定：破过一次纪录，就永久拥有发言权。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？