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多米诺骨牌装入正方形 · n = 33

将 n 个 1 × 2 的多米诺骨牌装入正方形容器，允许边界接触但内部不得重叠，最小化容器正方形的边长。

子题n = 33
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[8.707106781, 8.70712)Joshua WolkErich Friedman ↗
本站纪录8.70710679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 1 × 2 单位骨牌规范折算。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多米诺骨牌的短边长，对应容器正方形边长 = 1/radius。x、y 为骨牌中心坐标；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 为骨牌短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
多米诺骨牌装入正方形 n = 33 的当前纪录构型，8.70710679
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（20 次易主）
  1. 邱仲普AI · OpenAI: GPT-5.6 Sol
    8.707106866→8.70710679
  2. zzzcyAI · OpenAI: GPT-6 Astra
    8.962364103→8.707106866
  3. NUE_13人工
    8.962390128→8.962364103
  4. NUE_13人工
    8.964564797→8.962390128
  5. NUE_13人工
    8.965576204→8.964564797
  6. NUE_13人工
    8.966032393→8.965576204
  7. NUE_13人工
    8.966515803→8.966032393
  8. NUE_13人工
    8.966702733→8.966515803
  9. NUE_13人工
    8.966960749→8.966702733
  10. NUE_13人工
    8.967832681→8.966960749
  11. NUE_13人工
    8.969183411→8.967832681
  12. NUE_13人工
    8.969974911→8.969183411
  13. NUE_13人工
    8.970922835→8.969974911
  14. NUE_13人工
    8.972426889→8.970922835
  15. NUE_13人工
    8.973465114→8.972426889
  16. NUE_13人工
    8.974283384→8.973465114
  17. NUE_13人工
    8.974479336→8.974283384
  18. NUE_13人工
    8.976255381→8.974479336
  19. NUE_13人工
    8.983319467→8.976255381
  20. NUE_13人工
    9.00000001→8.983319467
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多米诺骨牌的短边长，对应容器正方形边长 = 1/radius。x、y 为骨牌中心坐标；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": "p90-n33-v1",
  "placements": [
    {
      "quarter": 3,
      "turn": "-0.001014400",
      "x": "0.057680269",
      "y": "0.882428698"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.230310766",
      "y": "0.942575644"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.402714397",
      "y": "0.885151288"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.597277690",
      "y": "0.885151288"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.769956078",
      "y": "0.942575644"
    },
    {
      "quarter": 3,
      "turn": "-0.000000068",
      "x": "0.942575238",
      "y": "0.885151280"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.230015537",
      "y": "0.827726932"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.769948431",
      "y": "0.827726932"
    },
    {
      "quarter": 0,
      "turn": "0.000162785",
      "x": "0.114908760",
      "y": "0.709960636"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.425756440",
      "y": "0.712878220"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.655453864",
      "y": "0.712878220"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.827726932",
      "y": "0.655453863"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.942575644",
      "y": "0.655228958"
    },
    {
      "quarter": 3,
      "turn": "0.414213558",
      "x": "0.332067678",
      "y": "0.553083610"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.540605151",
      "y": "0.598029508"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.712878220",
      "y": "0.540605152"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848713",
      "y": "0.516819204"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.827726932",
      "y": "0.425756439"
    },
    {
      "quarter": 0,
      "turn": "0.204531327",
      "x": "0.519951474",
      "y": "0.442531743"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.401970492"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.287121780",
      "y": "0.344546136"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.942575644",
      "y": "0.345543094"
    },
    {
      "quarter": 0,
      "turn": "-0.414213563",
      "x": "0.687684491",
      "y": "0.312315509"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.287121780"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.459394848",
      "y": "0.287121781"
    },
    {
      "quarter": 0,
      "turn": "-0.007363648",
      "x": "0.884293070",
      "y": "0.173267665"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.172273068"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.287121780",
      "y": "0.114848712"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.459394848",
      "y": "0.172273069"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.631667916",
      "y": "0.114848712"
    },
    {
      "quarter": 0,
      "turn": "-0.002350911",
      "x": "0.833756985",
      "y": "0.057985753"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.057424356"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.459394848",
      "y": "0.057424357"
    }
  ],
  "radius": "0.114848712"
}
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

{
  "iid": "p90-n33-v1",
  "placements": [
    {
      "quarter": 3,
      "turn": "-0.001014400",
      "x": "0.057680269",
      "y": "0.882428698"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.230310766",
      "y": "0.942575644"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.402714397",
      "y": "0.885151288"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.597277690",
      "y": "0.885151288"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.769956078",
      "y": "0.942575644"
    },
    {
      "quarter": 3,
      "turn": "-0.000000068",
      "x": "0.942575238",
      "y": "0.885151280"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.230015537",
      "y": "0.827726932"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.769948431",
      "y": "0.827726932"
    },
    {
      "quarter": 0,
      "turn": "0.000162785",
      "x": "0.114908760",
      "y": "0.709960636"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.425756440",
      "y": "0.712878220"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.655453864",
      "y": "0.712878220"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.827726932",
      "y": "0.655453863"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.942575644",
      "y": "0.655228958"
    },
    {
      "quarter": 3,
      "turn": "0.414213558",
      "x": "0.332067678",
      "y": "0.553083610"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.540605151",
      "y": "0.598029508"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.712878220",
      "y": "0.540605152"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848713",
      "y": "0.516819204"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.827726932",
      "y": "0.425756439"
    },
    {
      "quarter": 0,
      "turn": "0.204531327",
      "x": "0.519951474",
      "y": "0.442531743"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.401970492"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.287121780",
      "y": "0.344546136"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.942575644",
      "y": "0.345543094"
    },
    {
      "quarter": 0,
      "turn": "-0.414213563",
      "x": "0.687684491",
      "y": "0.312315509"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.287121780"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.459394848",
      "y": "0.287121781"
    },
    {
      "quarter": 0,
      "turn": "-0.007363648",
      "x": "0.884293070",
      "y": "0.173267665"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.172273068"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.287121780",
      "y": "0.114848712"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.459394848",
      "y": "0.172273069"
    },
    {
      "quarter": 3,
      "turn": "0.000000000",
      "x": "0.631667916",
      "y": "0.114848712"
    },
    {
      "quarter": 0,
      "turn": "-0.002350911",
      "x": "0.833756985",
      "y": "0.057985753"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.114848712",
      "y": "0.057424356"
    },
    {
      "quarter": 0,
      "turn": "0.000000000",
      "x": "0.459394848",
      "y": "0.057424357"
    }
  ],
  "radius": "0.114848712"
}

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多米诺骨牌的短边长，对应容器正方形边长 = 1/radius。x、y 为骨牌中心坐标；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

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