P83 · 装箱与覆盖 · 经典问题 · 难

等边三角形装入正方形 · n = 38

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 = 38

目标最小化容器正方形边长

已知最好（未证明）4.32008来源精度区间：4.32008–4.32009（区间内视为追平）本站展示证书 4.320080058Erich Friedman 编录；38. Found by Thomas Schadt in July 2026. 原式：s = 4.32008+。

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

4.320080058

容器正方形边长

已追平已知最好

答案来源38. Found by Thomas Schadt

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

{
  "placements": [
    {
      "turn": "0.265579306",
      "x": "0.828724740",
      "y": "0.932666816"
    },
    {
      "turn": "0.280889374",
      "x": "0.117313167",
      "y": "0.930412351"
    },
    {
      "turn": "0.280889374",
      "x": "0.586928464",
      "y": "0.930412351"
    },
    {
      "turn": "0.280889374",
      "x": "0.352120815",
      "y": "0.930412351"
    },
    {
      "turn": "-0.255092196",
      "x": "0.469433982",
      "y": "0.866395284"
    },
    {
      "turn": "-0.255092196",
      "x": "0.704241631",
      "y": "0.866395284"
    },
    {
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      "x": "0.234626333",
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    },
    {
      "turn": "-0.577350269",
      "x": "0.933178290",
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    },
    {
      "turn": "0.280889374",
      "x": "0.696428159",
      "y": "0.732525059"
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    {
      "turn": "0.280889374",
      "x": "0.228763983",
      "y": "0.732572027"
    },
    {
      "turn": "0.280889374",
      "x": "0.462714889",
      "y": "0.732551403"
    },
    {
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      "x": "0.345666639",
      "y": "0.667802686"
    },
    {
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    {
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      "x": "0.866356579",
      "y": "0.666234097"
    },
    {
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      "x": "0.114096990",
      "y": "0.663918965"
    },
    {
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      "x": "0.933178290",
      "y": "0.550495500"
    },
    {
      "turn": "0.280889374",
      "x": "0.347788437",
      "y": "0.534171626"
    },
    {
      "turn": "0.280889374",
      "x": "0.582719858",
      "y": "0.533968689"
    },
    {
      "turn": "0.280889374",
      "x": "0.117313167",
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    },
    {
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    {
      "turn": "-0.255092196",
      "x": "0.465101604",
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      "x": "0.233691448",
      "y": "0.464583977"
    },
    {
      "turn": "0.243573126",
      "x": "0.732151149",
      "y": "0.389990430"
    },
    {
      "turn": "0.000000000",
      "x": "0.066821710",
      "y": "0.348305624"
    },
    {
      "turn": "0.262873546",
      "x": "0.526539916",
      "y": "0.334102543"
    },
    {
      "turn": "-0.576786853",
      "x": "0.908684513",
      "y": "0.333159849"
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    {
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      "x": "0.642779008",
      "y": "0.267946570"
    },
    {
      "turn": "-0.577350269",
      "x": "0.133643421",
      "y": "0.232567026"
    },
    {
      "turn": "0.000422665",
      "x": "0.841960663",
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    },
    {
      "turn": "0.280889374",
      "x": "0.367707578",
      "y": "0.133604716"
    },
    {
      "turn": "0.267949192",
      "x": "0.602400950",
      "y": "0.133643421"
    },
    {
      "turn": "0.000000000",
      "x": "0.066821710",
      "y": "0.116828429"
    },
    {
      "turn": "-0.576786853",
      "x": "0.933080476",
      "y": "0.115682070"
    },
    {
      "turn": "-0.267949192",
      "x": "0.248752804",
      "y": "0.066821710"
    },
    {
      "turn": "-0.267949192",
      "x": "0.718139547",
      "y": "0.066821710"
    },
    {
      "turn": "-0.267949192",
      "x": "0.486662352",
      "y": "0.066821710"
    }
  ],
  "radius": "0.133643419"
}

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": 38,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "0.265579306",
      "x": "0.828724740",
      "y": "0.932666816"
    },
    {
      "turn": "0.280889374",
      "x": "0.117313167",
      "y": "0.930412351"
    },
    {
      "turn": "0.280889374",
      "x": "0.586928464",
      "y": "0.930412351"
    },
    {
      "turn": "0.280889374",
      "x": "0.352120815",
      "y": "0.930412351"
    },
    {
      "turn": "-0.255092196",
      "x": "0.469433982",
      "y": "0.866395284"
    },
    {
      "turn": "-0.255092196",
      "x": "0.704241631",
      "y": "0.866395284"
    },
    {
      "turn": "-0.255092196",
      "x": "0.234626333",
      "y": "0.866395284"
    },
    {
      "turn": "-0.577350269",
      "x": "0.933178290",
      "y": "0.781972695"
    },
    {
      "turn": "0.280889374",
      "x": "0.696428159",
      "y": "0.732525059"
    },
    {
      "turn": "0.280889374",
      "x": "0.228763983",
      "y": "0.732572027"
    },
    {
      "turn": "0.280889374",
      "x": "0.462714889",
      "y": "0.732551403"
    },
    {
      "turn": "-0.255092196",
      "x": "0.345666639",
      "y": "0.667802686"
    },
    {
      "turn": "-0.255092196",
      "x": "0.579503681",
      "y": "0.667573405"
    },
    {
      "turn": "-0.000000000",
      "x": "0.866356579",
      "y": "0.666234097"
    },
    {
      "turn": "-0.255092196",
      "x": "0.114096990",
      "y": "0.663918965"
    },
    {
      "turn": "0.577350269",
      "x": "0.933178290",
      "y": "0.550495500"
    },
    {
      "turn": "0.280889374",
      "x": "0.347788437",
      "y": "0.534171626"
    },
    {
      "turn": "0.280889374",
      "x": "0.582719858",
      "y": "0.533968689"
    },
    {
      "turn": "0.280889374",
      "x": "0.117313167",
      "y": "0.530314249"
    },
    {
      "turn": "-0.292238406",
      "x": "0.738220252",
      "y": "0.523578705"
    },
    {
      "turn": "-0.255092196",
      "x": "0.465101604",
      "y": "0.470154559"
    },
    {
      "turn": "-0.255092196",
      "x": "0.233691448",
      "y": "0.464583977"
    },
    {
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      "x": "0.732151149",
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    },
    {
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    {
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    {
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      "x": "0.908684513",
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    },
    {
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      "x": "0.288543577",
      "y": "0.332222259"
    },
    {
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      "x": "0.405856743",
      "y": "0.268205192"
    },
    {
      "turn": "-0.272627912",
      "x": "0.642779008",
      "y": "0.267946570"
    },
    {
      "turn": "-0.577350269",
      "x": "0.133643421",
      "y": "0.232567026"
    },
    {
      "turn": "0.000422665",
      "x": "0.841960663",
      "y": "0.217364806"
    },
    {
      "turn": "0.280889374",
      "x": "0.367707578",
      "y": "0.133604716"
    },
    {
      "turn": "0.267949192",
      "x": "0.602400950",
      "y": "0.133643421"
    },
    {
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      "x": "0.066821710",
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    },
    {
      "turn": "-0.576786853",
      "x": "0.933080476",
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    },
    {
      "turn": "-0.267949192",
      "x": "0.248752804",
      "y": "0.066821710"
    },
    {
      "turn": "-0.267949192",
      "x": "0.718139547",
      "y": "0.066821710"
    },
    {
      "turn": "-0.267949192",
      "x": "0.486662352",
      "y": "0.066821710"
    }
  ],
  "radius": "0.133643419"
}

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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