P13 · 装箱与覆盖 · 经典问题 · 难

半径成等差的圆装入正方形 · n = 93

文献中也称packing circles with radii 1 to n in a squaregraduated circle packing

将半径分别为 1, 2, …, n 的 n 个圆互不重叠地放入一个正方形内,使外围正方形的边长尽可能小。

子题n = 93
目标最小化 正方形边长

纪录对比

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[981.50841402775, 981.50841402785]André Müller2025-12Packomania ↗
本站纪录981.508414131NUE_13纪录详情 ↓

本站纪录尚未确认达到或超越此参考纪录。

严格定义

  • 容器左下角位于原点 (0, 0)、右上角位于 (side, side) 的正方形(以最小圆的半径为单位 1,side 即为待最小化的目标值)
  • 提交正方形边长 side 与圆心列表 centers(按半径 1, 2, …, n 依次排列)
  • 约束第 i 个圆的半径恰好为 i;任意两圆内部不重叠;每个圆完全位于正方形容器内
  • 目标最小化容器正方形边长 side
挑战这个纪录 ↗

当前展示:本站纪录

981.508414131y0
0x981.508414131
已验证构造side = 981.508414131

帮助理解

角位分配与小圆填隙

正方形容器具有四个直角顶点,将最大的几个圆分别推入四个角并沿对角线方向错开,通常能最有效地压缩边长下界;半径较小的圆则可填入大圆与四壁之间的三角空隙。每当 n 增加 1 时,最大圆组对四个角位的重新分配往往会改变整体布局。

已知证明与开放前沿

n ≤ 4 的最优边长可由最大圆之间的圆心距投影直接证明(详见各子题说明)。Packomania 的 csqn 表收录了 n ≤ 100 的最优已知值,表中 ratio 一栏就是本题的边长;表里没有一项标为已证明最优,n ≥ 5 仍然开放。圆容器版本曾是 Al Zimmermann 编程竞赛的赛题(也见 Packomania)。

查看来源
半径成等差的圆装入正方形 n = 93 的当前纪录构型,981.508414131
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NUE_13
解题方式AI · DeepSeek: DeepSeek V4.1 Flash
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(3 次易主)
  1. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    981.508414132→981.508414131
  2. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    981.508414133→981.508414132
  3. NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
    1134→981.508414133
ANSWER FORMAT

答案怎么写

单位就是最小那个圆的半径:第 i 个圆的半径正好是 i。容器是你自己给出的正方形,左下角是原点 (0, 0),右上角是 (side, side),side 越小越好。坐标和半径用同一个单位,直接写成小数,例如 "3.5",最多九位小数。

提交容器正方形边长 side 与各圆圆心坐标 centers(按半径 1, 2, …, n 的顺序排列)。所有数值均以十进制字符串表示(如 "3.5")。

当前第一名的答案

{
  "centers": [
    [
      "72.035750754",
      "121.952567887"
    ],
    [
      "291.465386623",
      "301.848199108"
    ],
    [
      "679.030771856",
      "551.213349472"
    ],
    [
      "831.679763959",
      "161.619905118"
    ],
    [
      "837.007103512",
      "431.067615410"
    ],
    [
      "278.849829550",
      "379.192013433"
    ],
    [
      "187.511272555",
      "746.937846886"
    ],
    [
      "8.632966753",
      "105.708901762"
    ],
    [
      "19.514145543",
      "281.653683653"
    ],
    [
      "611.196439272",
      "718.814006931"
    ],
    [
      "192.920275309",
      "11.413482896"
    ],
    [
      "968.129703490",
      "687.617205634"
    ],
    [
      "968.397506930",
      "586.806739989"
    ],
    [
      "877.654688389",
      "14.208818396"
    ],
    [
      "966.380444302",
      "109.919140409"
    ],
    [
      "16.188050963",
      "402.488153145"
    ],
    [
      "768.597338447",
      "964.400394979"
    ],
    [
      "320.553822374",
      "18.024926921"
    ],
    [
      "962.003305000",
      "254.206663273"
    ],
    [
      "474.766617796",
      "373.371339855"
    ],
    [
      "21.029081409",
      "817.538965438"
    ],
    [
      "959.508414131",
      "429.816036434"
    ],
    [
      "566.491007288",
      "418.108136509"
    ],
    [
      "607.357291679",
      "872.448105498"
    ],
    [
      "868.352341999",
      "825.718859121"
    ],
    [
      "522.899562394",
      "955.481769676"
    ],
    [
      "242.159464941",
      "597.524417006"
    ],
    [
      "373.914872127",
      "461.555742630"
    ],
    [
      "710.807452556",
      "531.359601559"
    ],
    [
      "525.307351887",
      "374.245334771"
    ],
    [
      "664.860492225",
      "207.312725915"
    ],
    [
      "464.654629578",
      "949.508414048"
    ],
    [
      "33.000000118",
      "143.740993837"
    ],
    [
      "765.654073748",
      "334.062340073"
    ],
    [
      "705.873551856",
      "35.000000060"
    ],
    [
      "264.097801815",
      "945.497019224"
    ],
    [
      "190.489725649",
      "944.508414089"
    ],
    [
      "150.569959207",
      "38.128215744"
    ],
    [
      "375.859325540",
      "528.746403359"
    ],
    [
      "941.508414036",
      "941.508414060"
    ],
    [
      "588.262652006",
      "940.452740502"
    ],
    [
      "776.597476862",
      "558.147750959"
    ],
    [
      "523.264770183",
      "43.001008067"
    ],
    [
      "937.508414104",
      "638.614108487"
    ],
    [
      "936.508414035",
      "856.655600314"
    ],
    [
      "157.418958416",
      "376.999110345"
    ],
    [
      "47.034314641",
      "599.524486250"
    ],
    [
      "614.297368787",
      "48.212659368"
    ],
    [
      "243.266484998",
      "336.314934256"
    ],
    [
      "931.251903740",
      "50.305283399"
    ],
    [
      "497.964388213",
      "297.999926135"
    ],
    [
      "552.000250852",
      "583.096389830"
    ],
    [
      "54.103689178",
      "227.111458083"
    ],
    [
      "423.148102780",
      "224.328198603"
    ],
    [
      "687.242642132",
      "123.050492412"
    ],
    [
      "324.807914018",
      "609.020285796"
    ],
    [
      "57.000000113",
      "57.000000062"
    ],
    [
      "157.009842693",
      "272.999915035"
    ],
    [
      "59.000000124",
      "339.004380736"
    ],
    [
      "155.057744855",
      "594.588142288"
    ],
    [
      "441.804403028",
      "608.113886166"
    ],
    [
      "235.671429459",
      "686.342256291"
    ],
    [
      "634.717354942",
      "789.563593248"
    ],
    [
      "830.076186669",
      "649.673787919"
    ],
    [
      "252.044464440",
      "65.000000028"
    ],
    [
      "915.508414131",
      "747.660187545"
    ],
    [
      "625.609757010",
      "487.117622915"
    ],
    [
      "837.201175550",
      "913.508414024"
    ],
    [
      "145.682974670",
      "146.506033516"
    ],
    [
      "532.392491768",
      "704.601116949"
    ],
    [
      "910.508414103",
      "179.298803209"
    ],
    [
      "697.035789687",
      "909.319242485"
    ],
    [
      "757.722084933",
      "440.789091622"
    ],
    [
      "367.330429103",
      "907.508414011"
    ],
    [
      "808.343059509",
      "75.000000102"
    ],
    [
      "770.524246950",
      "224.170096346"
    ],
    [
      "508.245570539",
      "849.604310694"
    ],
    [
      "775.469168022",
      "781.201349785"
    ],
    [
      "480.246662015",
      "473.495192347"
    ],
    [
      "80.000000088",
      "476.408893044"
    ],
    [
      "81.000000099",
      "900.508414090"
    ],
    [
      "375.111708267",
      "351.562253805"
    ],
    [
      "230.102843065",
      "831.235288726"
    ],
    [
      "285.494481855",
      "210.196750467"
    ],
    [
      "896.508414100",
      "516.303029657"
    ],
    [
      "401.577069587",
      "86.000000000"
    ],
    [
      "553.068970595",
      "169.541629925"
    ],
    [
      "678.192761602",
      "643.721881428"
    ],
    [
      "378.360422711",
      "744.036148370"
    ],
    [
      "90.000000000",
      "729.745420432"
    ],
    [
      "890.508414068",
      "340.328607193"
    ],
    [
      "637.924784616",
      "327.327173614"
    ],
    [
      "254.024745211",
      "477.906811983"
    ]
  ],
  "side": "981.508414131"
}
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": 93
}

当前第一名的答案

{
  "centers": [
    [
      "72.035750754",
      "121.952567887"
    ],
    [
      "291.465386623",
      "301.848199108"
    ],
    [
      "679.030771856",
      "551.213349472"
    ],
    [
      "831.679763959",
      "161.619905118"
    ],
    [
      "837.007103512",
      "431.067615410"
    ],
    [
      "278.849829550",
      "379.192013433"
    ],
    [
      "187.511272555",
      "746.937846886"
    ],
    [
      "8.632966753",
      "105.708901762"
    ],
    [
      "19.514145543",
      "281.653683653"
    ],
    [
      "611.196439272",
      "718.814006931"
    ],
    [
      "192.920275309",
      "11.413482896"
    ],
    [
      "968.129703490",
      "687.617205634"
    ],
    [
      "968.397506930",
      "586.806739989"
    ],
    [
      "877.654688389",
      "14.208818396"
    ],
    [
      "966.380444302",
      "109.919140409"
    ],
    [
      "16.188050963",
      "402.488153145"
    ],
    [
      "768.597338447",
      "964.400394979"
    ],
    [
      "320.553822374",
      "18.024926921"
    ],
    [
      "962.003305000",
      "254.206663273"
    ],
    [
      "474.766617796",
      "373.371339855"
    ],
    [
      "21.029081409",
      "817.538965438"
    ],
    [
      "959.508414131",
      "429.816036434"
    ],
    [
      "566.491007288",
      "418.108136509"
    ],
    [
      "607.357291679",
      "872.448105498"
    ],
    [
      "868.352341999",
      "825.718859121"
    ],
    [
      "522.899562394",
      "955.481769676"
    ],
    [
      "242.159464941",
      "597.524417006"
    ],
    [
      "373.914872127",
      "461.555742630"
    ],
    [
      "710.807452556",
      "531.359601559"
    ],
    [
      "525.307351887",
      "374.245334771"
    ],
    [
      "664.860492225",
      "207.312725915"
    ],
    [
      "464.654629578",
      "949.508414048"
    ],
    [
      "33.000000118",
      "143.740993837"
    ],
    [
      "765.654073748",
      "334.062340073"
    ],
    [
      "705.873551856",
      "35.000000060"
    ],
    [
      "264.097801815",
      "945.497019224"
    ],
    [
      "190.489725649",
      "944.508414089"
    ],
    [
      "150.569959207",
      "38.128215744"
    ],
    [
      "375.859325540",
      "528.746403359"
    ],
    [
      "941.508414036",
      "941.508414060"
    ],
    [
      "588.262652006",
      "940.452740502"
    ],
    [
      "776.597476862",
      "558.147750959"
    ],
    [
      "523.264770183",
      "43.001008067"
    ],
    [
      "937.508414104",
      "638.614108487"
    ],
    [
      "936.508414035",
      "856.655600314"
    ],
    [
      "157.418958416",
      "376.999110345"
    ],
    [
      "47.034314641",
      "599.524486250"
    ],
    [
      "614.297368787",
      "48.212659368"
    ],
    [
      "243.266484998",
      "336.314934256"
    ],
    [
      "931.251903740",
      "50.305283399"
    ],
    [
      "497.964388213",
      "297.999926135"
    ],
    [
      "552.000250852",
      "583.096389830"
    ],
    [
      "54.103689178",
      "227.111458083"
    ],
    [
      "423.148102780",
      "224.328198603"
    ],
    [
      "687.242642132",
      "123.050492412"
    ],
    [
      "324.807914018",
      "609.020285796"
    ],
    [
      "57.000000113",
      "57.000000062"
    ],
    [
      "157.009842693",
      "272.999915035"
    ],
    [
      "59.000000124",
      "339.004380736"
    ],
    [
      "155.057744855",
      "594.588142288"
    ],
    [
      "441.804403028",
      "608.113886166"
    ],
    [
      "235.671429459",
      "686.342256291"
    ],
    [
      "634.717354942",
      "789.563593248"
    ],
    [
      "830.076186669",
      "649.673787919"
    ],
    [
      "252.044464440",
      "65.000000028"
    ],
    [
      "915.508414131",
      "747.660187545"
    ],
    [
      "625.609757010",
      "487.117622915"
    ],
    [
      "837.201175550",
      "913.508414024"
    ],
    [
      "145.682974670",
      "146.506033516"
    ],
    [
      "532.392491768",
      "704.601116949"
    ],
    [
      "910.508414103",
      "179.298803209"
    ],
    [
      "697.035789687",
      "909.319242485"
    ],
    [
      "757.722084933",
      "440.789091622"
    ],
    [
      "367.330429103",
      "907.508414011"
    ],
    [
      "808.343059509",
      "75.000000102"
    ],
    [
      "770.524246950",
      "224.170096346"
    ],
    [
      "508.245570539",
      "849.604310694"
    ],
    [
      "775.469168022",
      "781.201349785"
    ],
    [
      "480.246662015",
      "473.495192347"
    ],
    [
      "80.000000088",
      "476.408893044"
    ],
    [
      "81.000000099",
      "900.508414090"
    ],
    [
      "375.111708267",
      "351.562253805"
    ],
    [
      "230.102843065",
      "831.235288726"
    ],
    [
      "285.494481855",
      "210.196750467"
    ],
    [
      "896.508414100",
      "516.303029657"
    ],
    [
      "401.577069587",
      "86.000000000"
    ],
    [
      "553.068970595",
      "169.541629925"
    ],
    [
      "678.192761602",
      "643.721881428"
    ],
    [
      "378.360422711",
      "744.036148370"
    ],
    [
      "90.000000000",
      "729.745420432"
    ],
    [
      "890.508414068",
      "340.328607193"
    ],
    [
      "637.924784616",
      "327.327173614"
    ],
    [
      "254.024745211",
      "477.906811983"
    ]
  ],
  "side": "981.508414131"
}

提交容器正方形边长 side 与各圆圆心坐标 centers(按半径 1, 2, …, n 的顺序排列)。所有数值均以十进制字符串表示(如 "3.5")。 · 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。

暂无讨论,欢迎发表第一篇帖子。