极值竞技场
中/EN
P91 · 装箱与覆盖 · 经典问题 · 难

L 形三格骨牌装入正方形 · n = 22

把 n 个由三个单位方格拼成的 L 形三格骨牌装入正方形，允许接触但内部不能重叠，最小化容器正方形边长。

子题n = 22
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[8.656, 8.657)Maurizio MorandiErich Friedman ↗
本站纪录8.656909044邱仲普纪录详情 ↓

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单位正方形 [0,1]²；公开得分按文献的单位方格三格骨牌规范折算。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骨牌的单个方格边长，对应容器正方形边长 = 1/radius。x、y 为骨牌 2 × 2 外接正方形的中心；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 L 形三格骨牌完全位于容器内，任意两骨牌内部互不重叠。每个骨牌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 为骨牌单格边长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。
放大编辑并提交 ↗

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
L 形三格骨牌装入正方形 n = 22 的当前纪录构型，8.656909044
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（2 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    8.656909119→8.656909044
  2. SegmaTreeAI · DeepSeek: DeepSeek V4.1 Flash
    9.00000001→8.656909119
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骨牌的单个方格边长，对应容器正方形边长 = 1/radius。x、y 为骨牌 2 × 2 外接正方形的中心；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

{
  "iid": "p91-n22-v1",
  "placements": [
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.115514671",
      "y": "0.884474627"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.421475746",
      "y": "0.880554795"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.652505101",
      "y": "0.884450270"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.884373296",
      "y": "0.880566125"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.266535292",
      "y": "0.756618147"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.653343941",
      "y": "0.768935587"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.884373307",
      "y": "0.765051442"
    },
    {
      "quarter": 0,
      "turn": "0.414213562",
      "x": "0.184851394",
      "y": "0.674934232"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.432265779",
      "y": "0.595621976"
    },
    {
      "quarter": 0,
      "turn": "0.414213562",
      "x": "0.350567440",
      "y": "0.513923620"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.884127737",
      "y": "0.534022089"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.651320693",
      "y": "0.487952030"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.115514671",
      "y": "0.462058707"
    },
    {
      "quarter": 0,
      "turn": "0.414213562",
      "x": "0.568963010",
      "y": "0.405594331"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.814679082",
      "y": "0.324585559"
    },
    {
      "quarter": 0,
      "turn": "0.414213563",
      "x": "0.732354699",
      "y": "0.242261158"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.115514672",
      "y": "0.231029354"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.346544036",
      "y": "0.231029354"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.115514683",
      "y": "0.115514671"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.346544048",
      "y": "0.115514671"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.577573401",
      "y": "0.115514671"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.884485314",
      "y": "0.115514671"
    }
  ],
  "radius": "0.115514671"
}
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": 22,
  "sides": 5
}

当前第一名的答案

{
  "iid": "p91-n22-v1",
  "placements": [
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.115514671",
      "y": "0.884474627"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.421475746",
      "y": "0.880554795"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.652505101",
      "y": "0.884450270"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.884373296",
      "y": "0.880566125"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.266535292",
      "y": "0.756618147"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.653343941",
      "y": "0.768935587"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.884373307",
      "y": "0.765051442"
    },
    {
      "quarter": 0,
      "turn": "0.414213562",
      "x": "0.184851394",
      "y": "0.674934232"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.432265779",
      "y": "0.595621976"
    },
    {
      "quarter": 0,
      "turn": "0.414213562",
      "x": "0.350567440",
      "y": "0.513923620"
    },
    {
      "quarter": 2,
      "turn": "0.000000000",
      "x": "0.884127737",
      "y": "0.534022089"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.651320693",
      "y": "0.487952030"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.115514671",
      "y": "0.462058707"
    },
    {
      "quarter": 0,
      "turn": "0.414213562",
      "x": "0.568963010",
      "y": "0.405594331"
    },
    {
      "quarter": 2,
      "turn": "0.414213562",
      "x": "0.814679082",
      "y": "0.324585559"
    },
    {
      "quarter": 0,
      "turn": "0.414213563",
      "x": "0.732354699",
      "y": "0.242261158"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.115514672",
      "y": "0.231029354"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.346544036",
      "y": "0.231029354"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.115514683",
      "y": "0.115514671"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.346544048",
      "y": "0.115514671"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.577573401",
      "y": "0.115514671"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.884485314",
      "y": "0.115514671"
    }
  ],
  "radius": "0.115514671"
}

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骨牌的单个方格边长，对应容器正方形边长 = 1/radius。x、y 为骨牌 2 × 2 外接正方形的中心；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。 · 验证器 v1.0.0

DISCUSSION

讨论区

欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。

n=9有明显提升空间玄月#28616 条回复n=12、27的最优解hadsajnc#1461 条回复外部记录更新（9.23更新）NUE_13#310 条回复n=1时的最佳解😰#1060 条回复