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P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 33

在半径为 1 的圆内放置 n 个互不重叠的圆，每个圆的半径各自随意，使所有半径之和尽可能大。

子题n = 33
目标最大化 半径之和

纪录对比

半径之和 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[5.277475, 5.277485]见来源署名Erich Friedman ↗
本站纪录暂无选手纪录——

严格定义

  • 容器圆心 (1, 1)、半径 1 的闭圆盘；圆可以与容器边界相切
  • 提交恰好 n 个圆，每个是 [x, y, r]，r > 0，全部数字最多九位小数
  • 约束每个圆完整落在圆盘内：(x − 1)² + (y − 1)² ≤ (1 − r)²；任意两圆内部不相交：圆心距 ≥ 半径之和。全部按整数精确判定，相切允许
  • 分数半径之和，精确整数求和后按九位小数显示；越大越好S=Σr
  • 目标在所有合法放置中最大化半径之和maxSΣr
放大来摆，然后提交 ↗

当前展示：起步布局

2y0
0x2
已验证构造计分是全部半径之和

帮助理解

哪里有优化空间

和正方形里一样，等圆是最差的策略之一：一个大圆压在圆心附近，或者几个大圆围成一圈，再让小圆钻进它们和圆周之间的缝隙。圆盘没有角，所以“靠墙的大圆”换成了“沿圆周的一圈”，大小的取舍换了一种几何。

前沿在哪里

Erich Friedman 的“最大化总周长”表列到 n = 50：n ≤ 4 和 6–10 是 Elser 与 Cantrell 的闭式构型，n = 5 是 Haowei Lin 2026 年 7 月的改进，n = 33–50 是 Jonathan Viquerat 2026 年 9 月新加的行。表还在被改写，没有一行有最优性证明；本站展示的是按图重建并精确验算的构型。

查看来源
单位圆内圆的半径之和 n = 33 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

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。
ANSWER FORMAT

答案怎么写

容器是圆心 (1, 1)、半径 1 的圆盘，所以坐标都在 [0, 2] 之内，不会出现负数。坐标与半径共用一套单位，直接写成小数，最多九位。

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是圆心 (1, 1)、半径 1 的闭圆盘。

起步布局答案

{
  "circles": [
    [
      "1.419809311",
      "1.336226869",
      "0.205669672"
    ],
    [
      "0.465651781",
      "0.938679091",
      "0.205669701"
    ],
    [
      "0.523676447",
      "1.660513299",
      "0.185652441"
    ],
    [
      "0.866421697",
      "1.803317371",
      "0.185652431"
    ],
    [
      "1.563001853",
      "0.407518609",
      "0.182684097"
    ],
    [
      "1.024237764",
      "0.183043580",
      "0.182684111"
    ],
    [
      "0.666715657",
      "0.252439836",
      "0.181510750"
    ],
    [
      "1.765482307",
      "0.710238830",
      "0.181510721"
    ],
    [
      "0.811523795",
      "1.452363397",
      "0.169569288"
    ],
    [
      "1.430813286",
      "1.710675485",
      "0.168940595"
    ],
    [
      "0.192030877",
      "1.194539401",
      "0.168940626"
    ],
    [
      "1.787110653",
      "1.272080347",
      "0.167190960"
    ],
    [
      "0.252557513",
      "0.632711880",
      "0.167190997"
    ],
    [
      "1.127600541",
      "1.566916545",
      "0.166625495"
    ],
    [
      "0.507616754",
      "1.308601604",
      "0.166625509"
    ],
    [
      "0.900776096",
      "0.508514049",
      "0.165416174"
    ],
    [
      "1.418843165",
      "0.724365676",
      "0.165416161"
    ],
    [
      "1.588294793",
      "1.007024071",
      "0.164143577"
    ],
    [
      "0.580756367",
      "0.587235194",
      "0.164143601"
    ],
    [
      "1.223867963",
      "0.462693009",
      "0.160908698"
    ],
    [
      "1.067305622",
      "1.248315365",
      "0.157630868"
    ],
    [
      "0.776292848",
      "1.127065492",
      "0.157630874"
    ],
    [
      "1.268458140",
      "1.008084933",
      "0.155694834"
    ],
    [
      "0.805220789",
      "0.815078023",
      "0.155694844"
    ],
    [
      "1.104184198",
      "0.749946888",
      "0.150280946"
    ],
    [
      "1.335177186",
      "0.195538962",
      "0.128506277"
    ],
    [
      "0.133636938",
      "0.903151694",
      "0.128240543"
    ],
    [
      "1.678828145",
      "1.546952419",
      "0.128240515"
    ],
    [
      "1.173890212",
      "1.856190960",
      "0.126329143"
    ],
    [
      "0.269625746",
      "1.479431105",
      "0.126329160"
    ],
    [
      "0.386699426",
      "0.374370399",
      "0.123900695"
    ],
    [
      "1.876084602",
      "0.994919792",
      "0.123900668"
    ],
    [
      "1.004482996",
      "0.989240499",
      "0.108952080"
    ]
  ]
}
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
}

起步布局答案

{
  "circles": [
    [
      "1.419809311",
      "1.336226869",
      "0.205669672"
    ],
    [
      "0.465651781",
      "0.938679091",
      "0.205669701"
    ],
    [
      "0.523676447",
      "1.660513299",
      "0.185652441"
    ],
    [
      "0.866421697",
      "1.803317371",
      "0.185652431"
    ],
    [
      "1.563001853",
      "0.407518609",
      "0.182684097"
    ],
    [
      "1.024237764",
      "0.183043580",
      "0.182684111"
    ],
    [
      "0.666715657",
      "0.252439836",
      "0.181510750"
    ],
    [
      "1.765482307",
      "0.710238830",
      "0.181510721"
    ],
    [
      "0.811523795",
      "1.452363397",
      "0.169569288"
    ],
    [
      "1.430813286",
      "1.710675485",
      "0.168940595"
    ],
    [
      "0.192030877",
      "1.194539401",
      "0.168940626"
    ],
    [
      "1.787110653",
      "1.272080347",
      "0.167190960"
    ],
    [
      "0.252557513",
      "0.632711880",
      "0.167190997"
    ],
    [
      "1.127600541",
      "1.566916545",
      "0.166625495"
    ],
    [
      "0.507616754",
      "1.308601604",
      "0.166625509"
    ],
    [
      "0.900776096",
      "0.508514049",
      "0.165416174"
    ],
    [
      "1.418843165",
      "0.724365676",
      "0.165416161"
    ],
    [
      "1.588294793",
      "1.007024071",
      "0.164143577"
    ],
    [
      "0.580756367",
      "0.587235194",
      "0.164143601"
    ],
    [
      "1.223867963",
      "0.462693009",
      "0.160908698"
    ],
    [
      "1.067305622",
      "1.248315365",
      "0.157630868"
    ],
    [
      "0.776292848",
      "1.127065492",
      "0.157630874"
    ],
    [
      "1.268458140",
      "1.008084933",
      "0.155694834"
    ],
    [
      "0.805220789",
      "0.815078023",
      "0.155694844"
    ],
    [
      "1.104184198",
      "0.749946888",
      "0.150280946"
    ],
    [
      "1.335177186",
      "0.195538962",
      "0.128506277"
    ],
    [
      "0.133636938",
      "0.903151694",
      "0.128240543"
    ],
    [
      "1.678828145",
      "1.546952419",
      "0.128240515"
    ],
    [
      "1.173890212",
      "1.856190960",
      "0.126329143"
    ],
    [
      "0.269625746",
      "1.479431105",
      "0.126329160"
    ],
    [
      "0.386699426",
      "0.374370399",
      "0.123900695"
    ],
    [
      "1.876084602",
      "0.994919792",
      "0.123900668"
    ],
    [
      "1.004482996",
      "0.989240499",
      "0.108952080"
    ]
  ]
}

提交 circles：恰好 n 个三元组 [x, y, r]，每个数写成最多九位小数的十进制字符串，例如 "0.25"。容器是圆心 (1, 1)、半径 1 的闭圆盘。 · 验证器 v1.0.0

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