P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 33

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 = 33

目标最小化容器正方形边长

已知最好（未证明）4.05164来源精度区间：4.05164–4.05165（区间内视为追平）本站展示证书 4.051645123Erich Friedman 编录；33. Found by Jake Loyd in June 2026. 原式：s = 4.05164+。

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

4.051645123

容器正方形边长

已追平已知最好

答案来源33. 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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      "y": "0.928751129"
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      "y": "0.928751129"
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    },
    {
      "turn": "-0.572971036",
      "x": "0.927940501",
      "y": "0.751700176"
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    {
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      "x": "0.294567982",
      "y": "0.739548160"
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      "x": "0.570603140",
      "y": "0.715004518"
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      "x": "0.142497741",
      "y": "0.636124397"
    },
    {
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      "y": "0.619428003"
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    {
      "turn": "-0.566971486",
      "x": "0.713083458",
      "y": "0.574735085"
    },
    {
      "turn": "-0.165461141",
      "x": "0.520674786",
      "y": "0.521373305"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.512717733"
    },
    {
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      "x": "0.928751129",
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    {
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    {
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      "x": "0.430446624",
      "y": "0.356244353"
    },
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      "x": "0.307039960",
      "y": "0.284995482"
    },
    {
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      "x": "0.553853287",
      "y": "0.284995482"
    },
    {
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      "x": "0.071248871",
      "y": "0.265904405"
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      "x": "0.928751129",
      "y": "0.258545101"
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      "x": "0.740439983",
      "y": "0.142497741"
    },
    {
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      "x": "0.123406664",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.370219992",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.617033319",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.863846647",
      "y": "0.071248871"
    }
  ],
  "radius": "0.142497739"
}

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": 33,
  "sides": 5
}

当前第一名的答案

{
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      "x": "0.352367660",
      "y": "0.928751129"
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      "x": "0.599180987",
      "y": "0.928751129"
    },
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      "y": "0.928751129"
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      "y": "0.877995633"
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      "x": "0.475774324",
      "y": "0.857502259"
    },
    {
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      "x": "0.722587651",
      "y": "0.857502259"
    },
    {
      "turn": "0.000000000",
      "x": "0.071248871",
      "y": "0.759531060"
    },
    {
      "turn": "-0.572971036",
      "x": "0.927940501",
      "y": "0.751700176"
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      "x": "0.294567982",
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      "x": "0.570603140",
      "y": "0.715004518"
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      "x": "0.142497741",
      "y": "0.636124397"
    },
    {
      "turn": "0.003290665",
      "x": "0.855868611",
      "y": "0.628757938"
    },
    {
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      "x": "0.373770597",
      "y": "0.619428003"
    },
    {
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      "x": "0.713083458",
      "y": "0.574735085"
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      "x": "0.520674786",
      "y": "0.521373305"
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      "x": "0.071248871",
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      "x": "0.292795263",
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      "x": "0.857502259",
      "y": "0.381951764"
    },
    {
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      "x": "0.662846724",
      "y": "0.381208794"
    },
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      "x": "0.430446624",
      "y": "0.356244353"
    },
    {
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      "y": "0.284995482"
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    },
    {
      "turn": "-0.267949192",
      "x": "0.370219992",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.617033319",
      "y": "0.071248871"
    },
    {
      "turn": "-0.267949192",
      "x": "0.863846647",
      "y": "0.071248871"
    }
  ],
  "radius": "0.142497739"
}

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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