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P78 · 极值构型 · 经典问题 · 难

用 n 个正三角形覆盖圆盘 · n = 18

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 = 18
目标最小化 共同边长

纪录对比

共同边长 · 越小越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.677506775067750, 0.677966101694916]见来源署名Erich Friedman ↗
本站纪录0.677850412zzzcy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} 放置参数；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⊆T1∪⋯∪Tn
  • 连续精确验证将 3n 条三角形边延展为直线排列，排列中的每个面与各三角形的包含关系恒定；因此圆盘被完全覆盖当且仅当所有与开圆盘相交的面均已被某个三角形覆盖。验证器在 Q(√3) 上将面剖分为竖直梯形，并通过精确有理比较判断梯形是否与开圆盘相交
  • 分数所有覆盖三角形的共同边长 s，越小越好s(T)=s
  • 目标在所有合法覆盖中最小化 sminTs(T)
放大编辑并提交 ↗

当前展示：本站纪录

已验证构造共同边长 0.677850412

帮助理解

n = 6 的正六边形基准与 n ≥ 7 的非对称重排

当 n = 6 时，六个边长为 2/√3 的正三角形绕圆心拼成正六边形，其内切圆恰好为单位圆盘。从 n = 7 起，新增的三角形破坏了六重对称，需要重新分配中心与圆弧边界的覆盖职责。

文献数据与坐标复原

Friedman 的公开表按互逆规范记录了 n = 1–18（即 n 个单位正三角形可覆盖的最大圆半径 r，对应本题边长 s = 1/r）。其中 n = 8–13 有代数闭式（Morandi、Cantrell），n = 7 与 n = 14–18 为截断小数（Morandi，2009），且仅 n = 1、2、6 已证明最优。由于原文献只给出约 200 像素的示意图而无坐标，本站根据图中的边角与接触拓扑复原了九位小数证书，并将文献目标与证书得分分别列出。

查看来源
用 n 个正三角形覆盖圆盘 n = 18 的当前纪录构型，0.677850412
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（4 次易主）
  1. zzzcyAI · OpenAI: GPT-6 Astra
    0.679966103→0.677850412
  2. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.68585488→0.679966103
  3. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.686854879→0.68585488
  4. Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
    0.687854879→0.686854879
ANSWER FORMAT

答案怎么写

容器是圆心为 (1,1)、半径为 1 的闭圆盘，内切于 [0,2]² 正方形区域。坐标标记覆盖三角形一条基准边的中点，覆盖物允许伸出容器。

提交 side 和 triangles，每块记为 {x,y,turn,down}。编辑器支持直接输入或拖动旋转角 θ°，并自动换算为 turn = tan(θ/2)。

当前第一名的答案

{
  "side": "0.677850412",
  "triangles": [
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.156842693",
      "y": "1.706554998"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "1.267041469",
      "y": "1.897629531"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "1.344791051",
      "y": "1.604147731"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.380918372",
      "y": "1.875927820"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "0.836549339",
      "y": "1.289103450"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.744935323",
      "y": "1.122945441"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "2.041177665",
      "y": "0.829565000"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.085352178",
      "y": "1.415693420"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "0.128307335",
      "y": "1.122195478"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.012289225",
      "y": "1.006312424"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169595824",
      "y": "0.712867530"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.638879820",
      "y": "0.939054964"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.780718244",
      "y": "0.352085024"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.361177000",
      "y": "0.938462230"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169449315",
      "y": "0.351801453"
    },
    {
      "down": false,
      "turn": "-0.577041040",
      "x": "0.219883234",
      "y": "0.351360944"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.475110529",
      "y": "0.294282047"
    },
    {
      "down": true,
      "turn": "-0.577041039",
      "x": "0.864155861",
      "y": "0.293454743"
    }
  ]
}
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": 18
}

当前第一名的答案

{
  "side": "0.677850412",
  "triangles": [
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.156842693",
      "y": "1.706554998"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "1.267041469",
      "y": "1.897629531"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "1.344791051",
      "y": "1.604147731"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.380918372",
      "y": "1.875927820"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "0.836549339",
      "y": "1.289103450"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.744935323",
      "y": "1.122945441"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "2.041177665",
      "y": "0.829565000"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.085352178",
      "y": "1.415693420"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "0.128307335",
      "y": "1.122195478"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.012289225",
      "y": "1.006312424"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169595824",
      "y": "0.712867530"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.638879820",
      "y": "0.939054964"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.780718244",
      "y": "0.352085024"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.361177000",
      "y": "0.938462230"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169449315",
      "y": "0.351801453"
    },
    {
      "down": false,
      "turn": "-0.577041040",
      "x": "0.219883234",
      "y": "0.351360944"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.475110529",
      "y": "0.294282047"
    },
    {
      "down": true,
      "turn": "-0.577041039",
      "x": "0.864155861",
      "y": "0.293454743"
    }
  ]
}

提交 side 和 triangles，每块记为 {x,y,turn,down}。编辑器支持直接输入或拖动旋转角 θ°，并自动换算为 turn = tan(θ/2)。 · 验证器 v1.0.0

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