P89 · 装箱与覆盖 · 经典问题 · 难

等腰直角三角形装入正方形 · n = 24

把 n 个直角边为 1 的等腰直角三角形（七巧板的小三角形）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 = 24

目标最小化容器正方形边长

纪录对比

容器正方形边长 · 越小越好

纪录	数值 / 区间	作者 / 持有人	来源
外部已知最好	[3.51127, 3.51128)	Joshua Wolk2026-09	Erich Friedman ↗
本站纪录	3.511269885	邱仲普	纪录详情 ↓

数值超出外部纪录区间 · 待审核。

严格定义

容器编辑器与证书将容器统一归一化为单位正方形 [0,1]²；公开得分按文献的单位直角边长规范折算。允许接触边界。
提交提交 {radius, placements:[{x,y,turn,quarter},…]}，其中 x、y、turn 为最多九位小数的十进制字符串。radius 为单位正方形容器内零件的直角边长，对应容器正方形边长 = 1/radius。x、y 为零件轴对齐外接框的中心；quarter 为 0–3 的整数，表示先绕中心旋转 quarter 个直角，再旋转由 turn = tan(θ/2) 给出的有理角。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。每个图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，可行性检验不设浮点容差。
目标容器正方形边长 = 1/r（其中 r 为零件直角边在单位正方形容器内的长度），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。

查看来源

构型与历史

纪录详情

查看构型、求解笔记与纪录历史。

纪录保持者邱仲普

解题方式AI · OpenAI: GPT-5.6 Sol

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）

邱仲普AI · OpenAI: GPT-5.6 Sol
3.511269909→3.511269885
2026-10-08
邱仲普AI · OpenAI: GPT-5.6 Sol
3.511269922→3.511269909
2026-10-08
zzzcyAI · OpenAI: GPT-6 Astra
3.515298854→3.511269922
2026-10-04
😰人工
3.517000007→3.515298854
2026-09-22

ANSWER FORMAT

答案怎么写

单位正方形 [0,1]²

提交 {radius, placements:[{x,y,turn,quarter},…]}，其中 x、y、turn 为最多九位小数的十进制字符串。radius 为单位正方形容器内零件的直角边长，对应容器正方形边长 = 1/radius。x、y 为零件轴对齐外接框的中心；quarter 为 0–3 的整数，表示先绕中心旋转 quarter 个直角，再旋转由 turn = tan(θ/2) 给出的有理角。坐标与 turn 的范围限于 [−4, 4]，且 0 < radius ≤ 4。

当前第一名的答案

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.414213566",
      "x": "0.402764134",
      "y": "0.798617933"
    },
    {
      "quarter": 2,
      "turn": "-0.000000005",
      "x": "0.857601373",
      "y": "0.857601372"
    },
    {
      "quarter": 0,
      "turn": "-0.000000003",
      "x": "0.142398638",
      "y": "0.142398626"
    },
    {
      "quarter": 2,
      "turn": "0.000000002",
      "x": "0.572804115",
      "y": "0.857601372"
    },
    {
      "quarter": 2,
      "turn": "-0.000000021",
      "x": "0.857601368",
      "y": "0.572804104"
    },
    {
      "quarter": 0,
      "turn": "-0.000000033",
      "x": "0.142398641",
      "y": "0.427195916"
    },
    {
      "quarter": 0,
      "turn": "0.000000021",
      "x": "0.427195880",
      "y": "0.142398631"
    },
    {
      "quarter": 1,
      "turn": "0.414213508",
      "x": "0.000000019",
      "y": "0.784797258"
    },
    {
      "quarter": 2,
      "turn": "0.414213443",
      "x": "0.201382066",
      "y": "0.583415150"
    },
    {
      "quarter": 0,
      "turn": "0.414213558",
      "x": "0.604146206",
      "y": "0.611056539"
    },
    {
      "quarter": 0,
      "turn": "0.414213566",
      "x": "0.201382068",
      "y": "0.999999998"
    },
    {
      "quarter": 1,
      "turn": "0.414213530",
      "x": "0.402764124",
      "y": "0.409674470"
    },
    {
      "quarter": 2,
      "turn": "0.414213561",
      "x": "0.798617934",
      "y": "0.402764135"
    },
    {
      "quarter": 0,
      "turn": "0.414213561",
      "x": "0.798617934",
      "y": "0.402764134"
    },
    {
      "quarter": 3,
      "turn": "0.414213559",
      "x": "0.597235867",
      "y": "0.201382071"
    },
    {
      "quarter": 1,
      "turn": "0.414213561",
      "x": "0.597235868",
      "y": "0.201382069"
    },
    {
      "quarter": 0,
      "turn": "-0.414213564",
      "x": "0.999999999",
      "y": "0.201382068"
    },
    {
      "quarter": 3,
      "turn": "-0.414213566",
      "x": "0.798617933",
      "y": "0.000000002"
    },
    {
      "quarter": 0,
      "turn": "0.414213422",
      "x": "0.201382068",
      "y": "0.583415142"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.857601371",
      "y": "0.857601364"
    },
    {
      "quarter": 2,
      "turn": "-0.000000025",
      "x": "0.142398618",
      "y": "0.142398663"
    },
    {
      "quarter": 3,
      "turn": "0.414213378",
      "x": "0.395853856",
      "y": "0.375122784"
    },
    {
      "quarter": 2,
      "turn": "0.414213548",
      "x": "0.617966870",
      "y": "0.611056543"
    },
    {
      "quarter": 0,
      "turn": "0.414213483",
      "x": "0.388943462",
      "y": "0.798617905"
    }
  ],
  "radius": "0.284797248"
}

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": 24,
  "sides": 5
}

当前第一名的答案

{
  "placements": [
    {
      "quarter": 2,
      "turn": "0.414213566",
      "x": "0.402764134",
      "y": "0.798617933"
    },
    {
      "quarter": 2,
      "turn": "-0.000000005",
      "x": "0.857601373",
      "y": "0.857601372"
    },
    {
      "quarter": 0,
      "turn": "-0.000000003",
      "x": "0.142398638",
      "y": "0.142398626"
    },
    {
      "quarter": 2,
      "turn": "0.000000002",
      "x": "0.572804115",
      "y": "0.857601372"
    },
    {
      "quarter": 2,
      "turn": "-0.000000021",
      "x": "0.857601368",
      "y": "0.572804104"
    },
    {
      "quarter": 0,
      "turn": "-0.000000033",
      "x": "0.142398641",
      "y": "0.427195916"
    },
    {
      "quarter": 0,
      "turn": "0.000000021",
      "x": "0.427195880",
      "y": "0.142398631"
    },
    {
      "quarter": 1,
      "turn": "0.414213508",
      "x": "0.000000019",
      "y": "0.784797258"
    },
    {
      "quarter": 2,
      "turn": "0.414213443",
      "x": "0.201382066",
      "y": "0.583415150"
    },
    {
      "quarter": 0,
      "turn": "0.414213558",
      "x": "0.604146206",
      "y": "0.611056539"
    },
    {
      "quarter": 0,
      "turn": "0.414213566",
      "x": "0.201382068",
      "y": "0.999999998"
    },
    {
      "quarter": 1,
      "turn": "0.414213530",
      "x": "0.402764124",
      "y": "0.409674470"
    },
    {
      "quarter": 2,
      "turn": "0.414213561",
      "x": "0.798617934",
      "y": "0.402764135"
    },
    {
      "quarter": 0,
      "turn": "0.414213561",
      "x": "0.798617934",
      "y": "0.402764134"
    },
    {
      "quarter": 3,
      "turn": "0.414213559",
      "x": "0.597235867",
      "y": "0.201382071"
    },
    {
      "quarter": 1,
      "turn": "0.414213561",
      "x": "0.597235868",
      "y": "0.201382069"
    },
    {
      "quarter": 0,
      "turn": "-0.414213564",
      "x": "0.999999999",
      "y": "0.201382068"
    },
    {
      "quarter": 3,
      "turn": "-0.414213566",
      "x": "0.798617933",
      "y": "0.000000002"
    },
    {
      "quarter": 0,
      "turn": "0.414213422",
      "x": "0.201382068",
      "y": "0.583415142"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.857601371",
      "y": "0.857601364"
    },
    {
      "quarter": 2,
      "turn": "-0.000000025",
      "x": "0.142398618",
      "y": "0.142398663"
    },
    {
      "quarter": 3,
      "turn": "0.414213378",
      "x": "0.395853856",
      "y": "0.375122784"
    },
    {
      "quarter": 2,
      "turn": "0.414213548",
      "x": "0.617966870",
      "y": "0.611056543"
    },
    {
      "quarter": 0,
      "turn": "0.414213483",
      "x": "0.388943462",
      "y": "0.798617905"
    }
  ],
  "radius": "0.284797248"
}

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