P90 · 装箱与覆盖 · 经典问题 · 难

多米诺骨牌装入正方形 · n = 26

把 n 个1 × 2 的多米诺骨牌装入正方形，允许接触但内部不能重叠，最小化容器正方形边长。

子题n = 26

目标最小化容器正方形边长

已知最好（未证明）7.82842来源精度区间：7.82842–7.82843（区间内视为追平）本站展示证书 7.828427303Erich Friedman 编录；Found by Maurizio Morandi in August 2026. 原式：s = 5+2√2 = 7.82842+。

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,quarter},…]}，x、y、turn 用最多九位小数的十进制字符串。radius 是零件的短边长（直角边、多米诺的短边、方格边长），容器正方形边长 = 1/radius。x、y 是零件外接矩形的中心；quarter 是 0–3 的整数，先绕中心精确转 quarter 个直角，再转 turn = tan(θ/2) 给出的有理角。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/r（r 是零件短边在单位正方形里的长度），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

7.828427303

容器正方形边长

已追平已知最好

答案来源Maurizio Morandi

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,quarter},…]}，x、y、turn 用最多九位小数的十进制字符串。radius 是零件的短边长（直角边、多米诺的短边、方格边长），容器正方形边长 = 1/radius。x、y 是零件外接矩形的中心；quarter 是 0–3 的整数，先绕中心精确转 quarter 个直角，再转 turn = tan(θ/2) 给出的有理角。坐标和 turn 限于 ±4，0 < radius ≤ 4。

当前第一名的答案

{
  "placements": [
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.319348954",
      "y": "0.872260419"
    },
    {
      "turn": "0.000000000",
      "x": "0.872260414",
      "y": "0.936130204"
    },
    {
      "turn": "0.000000000",
      "x": "0.872260419",
      "y": "0.680651042"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.936130208",
      "y": "0.405340250"
    },
    {
      "turn": "0.000000000",
      "x": "0.127739581",
      "y": "0.319348954"
    },
    {
      "turn": "0.000000000",
      "x": "0.596605478",
      "y": "0.936130208"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.936130208",
      "y": "0.127739590"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.063869792",
      "y": "0.872260414"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.191609373",
      "y": "0.872260419"
    },
    {
      "turn": "0.000000000",
      "x": "0.872260414",
      "y": "0.808390623"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.063869792",
      "y": "0.600483047"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.191609373",
      "y": "0.616781255"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.808390627",
      "y": "0.383218749"
    },
    {
      "turn": "0.000000000",
      "x": "0.127739586",
      "y": "0.191609373"
    },
    {
      "turn": "0.000000000",
      "x": "0.127739586",
      "y": "0.063869792"
    },
    {
      "turn": "0.000000000",
      "x": "0.616781255",
      "y": "0.808390627"
    },
    {
      "turn": "0.000000000",
      "x": "0.383218745",
      "y": "0.191609373"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.680651046",
      "y": "0.127739581"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.808390627",
      "y": "0.127739582"
    },
    {
      "turn": "0.000000000",
      "x": "0.411778666",
      "y": "0.063869792"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.390967448",
      "y": "0.661944020"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.661944015",
      "y": "0.609032552"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.454837237",
      "y": "0.545162761"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.545162761",
      "y": "0.454837237"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.609032552",
      "y": "0.338055980"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.338055980",
      "y": "0.390967448"
    }
  ],
  "radius": "0.127739578"
}

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": [
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.319348954",
      "y": "0.872260419"
    },
    {
      "turn": "0.000000000",
      "x": "0.872260414",
      "y": "0.936130204"
    },
    {
      "turn": "0.000000000",
      "x": "0.872260419",
      "y": "0.680651042"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.936130208",
      "y": "0.405340250"
    },
    {
      "turn": "0.000000000",
      "x": "0.127739581",
      "y": "0.319348954"
    },
    {
      "turn": "0.000000000",
      "x": "0.596605478",
      "y": "0.936130208"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.936130208",
      "y": "0.127739590"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.063869792",
      "y": "0.872260414"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.191609373",
      "y": "0.872260419"
    },
    {
      "turn": "0.000000000",
      "x": "0.872260414",
      "y": "0.808390623"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.063869792",
      "y": "0.600483047"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.191609373",
      "y": "0.616781255"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.808390627",
      "y": "0.383218749"
    },
    {
      "turn": "0.000000000",
      "x": "0.127739586",
      "y": "0.191609373"
    },
    {
      "turn": "0.000000000",
      "x": "0.127739586",
      "y": "0.063869792"
    },
    {
      "turn": "0.000000000",
      "x": "0.616781255",
      "y": "0.808390627"
    },
    {
      "turn": "0.000000000",
      "x": "0.383218745",
      "y": "0.191609373"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.680651046",
      "y": "0.127739581"
    },
    {
      "quarter": 1,
      "turn": "0.000000000",
      "x": "0.808390627",
      "y": "0.127739582"
    },
    {
      "turn": "0.000000000",
      "x": "0.411778666",
      "y": "0.063869792"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.390967448",
      "y": "0.661944020"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.661944015",
      "y": "0.609032552"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.454837237",
      "y": "0.545162761"
    },
    {
      "quarter": 3,
      "turn": "-0.414213562",
      "x": "0.545162761",
      "y": "0.454837237"
    },
    {
      "quarter": 1,
      "turn": "-0.414213562",
      "x": "0.609032552",
      "y": "0.338055980"
    },
    {
      "quarter": 2,
      "turn": "-0.414213562",
      "x": "0.338055980",
      "y": "0.390967448"
    }
  ],
  "radius": "0.127739578"
}

DISCUSSION

讨论区

聊思路、贴方法、问为什么卡住。所有登录用户都可以发帖；发言公开署名，与纪录使用同一个名字，署名后的 #编号是账号注册序号，冒不了名。新发言经自动审核后公开。

还没有帖子。第一个聊聊这道题的思路？