P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 36

把 n 个边长为 1 的等边三角形装入正方形，允许接触但内部不能重叠，最小化容器正方形边长。

子题n = 36

目标最小化容器正方形边长

已知最好（未证明）4.20791来源精度区间：4.20791–4.20792（区间内视为追平）本站展示证书 4.207918443Erich Friedman 编录；36. Found by Jake Loyd in June 2026. 原式：s = 4.20791+。

严格定义

容器编辑器及证书统一缩放到单位正方形 [0,1]²；仅是坐标约定，分数按文献的单位图形计算。边界接触允许。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。
约束恰好 n 个同样大的图形，完整位于容器内，任意两图形内部不重叠。多边形可独立旋转，不限直角。
精度输入小数精确表示有理数；正多边形的顶点用代数数定义，不截断成小数。可行性验算没有浮点容差。有限小数证书不等于连续最优性证明。
目标容器正方形边长 = 1/(2 sin(π/3) r)，越小越好。页面、排行榜和外部纪录使用相同单位，自动计算，无需手工换算。

放大来摆，然后提交 ↗

已验证构造已验证构造

帮助理解

外部来源与证书

外部目标按原始印刷精度换算，本站坐标证书另存。复现损失不降低文献目标；源表的构造不自动视为最优性证明。

查看来源

当前第一名

4.207918443

容器正方形边长

已追平已知最好

答案来源36. Found by Jake Loyd

解题方式公开参考构造

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。

纪录保持者的求解笔记暂无求解笔记（点击展开）＋

纪录保持者还没有分享求解过程。

ANSWER FORMAT

答案怎么写

单位正方形 [0,1]²

提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。

当前第一名的答案

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      "x": "0.513625333",
      "y": "0.931397164"
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    },
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      "x": "0.931397164",
      "y": "0.356470792"
    },
    {
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      "x": "0.440092266",
      "y": "0.343083182"
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    {
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      "x": "0.706111404",
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      "y": "0.237647195"
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      "x": "0.068602836",
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      "x": "0.590826538",
      "y": "0.137205672"
    },
    {
      "turn": "0.267923607",
      "x": "0.353172792",
      "y": "0.137205672"
    },
    {
      "turn": "-0.577350269",
      "x": "0.931397164",
      "y": "0.118823597"
    },
    {
      "turn": "-0.276511176",
      "x": "0.736245333",
      "y": "0.074342246"
    },
    {
      "turn": "-0.267974778",
      "x": "0.471999665",
      "y": "0.068608509"
    },
    {
      "turn": "-0.267974778",
      "x": "0.234345919",
      "y": "0.068608509"
    }
  ],
  "radius": "0.137205670"
}

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": 36,
  "sides": 5
}

当前第一名的答案

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      "y": "0.931397164"
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      "y": "0.862794328"
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      "x": "0.394801736",
      "y": "0.862794328"
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      "x": "0.137205672",
      "y": "0.762352805"
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      "y": "0.524705610"
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      "x": "0.629614083",
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      "x": "0.931397164",
      "y": "0.118823597"
    },
    {
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    {
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  ],
  "radius": "0.137205670"
}

提交 {radius, placements:[{x,y,turn},…]}，每个数用最多九位小数的十进制字符串。x、y 是中心坐标；圆的 turn 必须为 0，多边形 turn = tan(θ/2)。坐标和 turn 限于 ±4，0 < radius ≤ 4。 · 验证器 v1.0.0

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聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。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？