P85 · 装箱与覆盖 · 经典问题 · 难

正方形装入等边三角形 · n = 26

将 n 个边长为 1 的正方形装入等边三角形容器，允许边界接触但内部不得重叠，最小化容器等边三角形的边长。

子题n = 26

目标最小化容器三角形边长

纪录对比

容器三角形边长 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[8.60805, 8.60806)	Joshua Wolk2026-09	Erich Friedman ↗
本站纪录	8.608055173	zzzcy	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

容器编辑器与证书将容器统一归一化为顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形；公开得分按文献的单位边长图形规范折算。允许接触边界。
提交提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。
约束恰好 n 个全等图形完全位于容器内，任意两图形内部互不重叠。多边形允许任意角度独立旋转。
精度输入的十进制小数按精确有理数解析，正多边形顶点由精确代数数定义，可行性检验不设浮点容差。
目标容器三角形边长 = 1/(2 sin(π/4) r)，越小越好。页面、排行榜与文献纪录统一使用该单位并自动换算。

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当前展示：本站纪录

已验证构造已验证构造

帮助理解

文献来源与坐标证书

文献目标值按原始发表精度换算，九位小数坐标证书单独存储并与文献目标对照展示。

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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（1 次易主）

zzzcyAI · OpenAI: GPT-6 Astra
8.618802344→8.608055173
2026-10-05

ANSWER FORMAT

答案怎么写

顶点为 (0,0)、(1,0)、(1/2,√3/2) 的等边三角形

提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。

当前第一名的答案

{
  "placements": [
    {
      "turn": "-0.414213562",
      "x": "0.691602747",
      "y": "0.174978095"
    },
    {
      "turn": "-0.414213562",
      "x": "0.807773008",
      "y": "0.174255390"
    },
    {
      "turn": "0.414213562",
      "x": "0.393439785",
      "y": "0.058085130"
    },
    {
      "turn": "0.414213562",
      "x": "0.758673679",
      "y": "0.058085130"
    },
    {
      "turn": "0.159489360",
      "x": "0.449053448",
      "y": "0.234542145"
    },
    {
      "turn": "0.159489360",
      "x": "0.555448184",
      "y": "0.187496302"
    },
    {
      "turn": "0.414213562",
      "x": "0.509610046",
      "y": "0.058085130"
    },
    {
      "turn": "-0.325535813",
      "x": "0.429997482",
      "y": "0.372431930"
    },
    {
      "turn": "-0.414213562",
      "x": "0.642503418",
      "y": "0.058085130"
    },
    {
      "turn": "-0.131652498",
      "x": "0.378132970",
      "y": "0.496253990"
    },
    {
      "turn": "-0.131139455",
      "x": "0.560498457",
      "y": "0.410525641"
    },
    {
      "turn": "0.131652498",
      "x": "0.247023091",
      "y": "0.079345763"
    },
    {
      "turn": "-0.131873072",
      "x": "0.619480597",
      "y": "0.310319122"
    },
    {
      "turn": "-0.131652498",
      "x": "0.445203948",
      "y": "0.612424168"
    },
    {
      "turn": "-0.131652498",
      "x": "0.503322302",
      "y": "0.511760226"
    },
    {
      "turn": "0.131652498",
      "x": "0.305108221",
      "y": "0.179952160"
    },
    {
      "turn": "-0.414213562",
      "x": "0.874843939",
      "y": "0.058085130"
    },
    {
      "turn": "0.131652498",
      "x": "0.112881229",
      "y": "0.079345763"
    },
    {
      "turn": "-0.131739631",
      "x": "0.687039767",
      "y": "0.425815322"
    },
    {
      "turn": "0.131652498",
      "x": "0.287136619",
      "y": "0.381164953"
    },
    {
      "turn": "0.131652498",
      "x": "0.229051489",
      "y": "0.280558556"
    },
    {
      "turn": "-0.131652498",
      "x": "0.570621500",
      "y": "0.627535117"
    },
    {
      "turn": "-0.132113570",
      "x": "0.752453539",
      "y": "0.312399402"
    },
    {
      "turn": "-0.131652498",
      "x": "0.628968484",
      "y": "0.526475177"
    },
    {
      "turn": "0.131652498",
      "x": "0.170966359",
      "y": "0.179952160"
    },
    {
      "turn": "-0.131652498",
      "x": "0.512518896",
      "y": "0.728171780"
    }
  ],
  "radius": "0.082144778"
}

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": 26,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "turn": "-0.414213562",
      "x": "0.691602747",
      "y": "0.174978095"
    },
    {
      "turn": "-0.414213562",
      "x": "0.807773008",
      "y": "0.174255390"
    },
    {
      "turn": "0.414213562",
      "x": "0.393439785",
      "y": "0.058085130"
    },
    {
      "turn": "0.414213562",
      "x": "0.758673679",
      "y": "0.058085130"
    },
    {
      "turn": "0.159489360",
      "x": "0.449053448",
      "y": "0.234542145"
    },
    {
      "turn": "0.159489360",
      "x": "0.555448184",
      "y": "0.187496302"
    },
    {
      "turn": "0.414213562",
      "x": "0.509610046",
      "y": "0.058085130"
    },
    {
      "turn": "-0.325535813",
      "x": "0.429997482",
      "y": "0.372431930"
    },
    {
      "turn": "-0.414213562",
      "x": "0.642503418",
      "y": "0.058085130"
    },
    {
      "turn": "-0.131652498",
      "x": "0.378132970",
      "y": "0.496253990"
    },
    {
      "turn": "-0.131139455",
      "x": "0.560498457",
      "y": "0.410525641"
    },
    {
      "turn": "0.131652498",
      "x": "0.247023091",
      "y": "0.079345763"
    },
    {
      "turn": "-0.131873072",
      "x": "0.619480597",
      "y": "0.310319122"
    },
    {
      "turn": "-0.131652498",
      "x": "0.445203948",
      "y": "0.612424168"
    },
    {
      "turn": "-0.131652498",
      "x": "0.503322302",
      "y": "0.511760226"
    },
    {
      "turn": "0.131652498",
      "x": "0.305108221",
      "y": "0.179952160"
    },
    {
      "turn": "-0.414213562",
      "x": "0.874843939",
      "y": "0.058085130"
    },
    {
      "turn": "0.131652498",
      "x": "0.112881229",
      "y": "0.079345763"
    },
    {
      "turn": "-0.131739631",
      "x": "0.687039767",
      "y": "0.425815322"
    },
    {
      "turn": "0.131652498",
      "x": "0.287136619",
      "y": "0.381164953"
    },
    {
      "turn": "0.131652498",
      "x": "0.229051489",
      "y": "0.280558556"
    },
    {
      "turn": "-0.131652498",
      "x": "0.570621500",
      "y": "0.627535117"
    },
    {
      "turn": "-0.132113570",
      "x": "0.752453539",
      "y": "0.312399402"
    },
    {
      "turn": "-0.131652498",
      "x": "0.628968484",
      "y": "0.526475177"
    },
    {
      "turn": "0.131652498",
      "x": "0.170966359",
      "y": "0.179952160"
    },
    {
      "turn": "-0.131652498",
      "x": "0.512518896",
      "y": "0.728171780"
    }
  ],
  "radius": "0.082144778"
}

提交 {radius, placements:[{x,y,turn},…]}，每个数值均为最多九位小数的十进制字符串。x、y 为中心坐标；圆的 turn 固定为 0，多边形的 turn = tan(θ/2)。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。 · 验证器 v1.0.0

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