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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 = 32

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 = 32
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[5.191, 5.192]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 = 32 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": [
    [
      "0.999893251",
      "0.478199959",
      "0.210467962"
    ],
    [
      "1.811558224",
      "0.995971334",
      "0.188431776"
    ],
    [
      "0.188440225",
      "0.996303577",
      "0.188431806"
    ],
    [
      "1.721501689",
      "0.629508991",
      "0.188933865"
    ],
    [
      "0.278346754",
      "0.629804345",
      "0.188933892"
    ],
    [
      "1.598262061",
      "1.552965790",
      "0.185329110"
    ],
    [
      "0.401964457",
      "1.553210731",
      "0.185329131"
    ],
    [
      "0.706803838",
      "1.761587671",
      "0.183924163"
    ],
    [
      "1.293508067",
      "1.761467531",
      "0.183924153"
    ],
    [
      "0.706143328",
      "0.228903535",
      "0.174808202"
    ],
    [
      "1.293541114",
      "0.228783342",
      "0.174808192"
    ],
    [
      "1.369547276",
      "0.564127808",
      "0.169041843"
    ],
    [
      "0.630274415",
      "0.564279101",
      "0.169041855"
    ],
    [
      "0.527697805",
      "0.885739404",
      "0.168387673"
    ],
    [
      "1.472255454",
      "0.885546069",
      "0.168387657"
    ],
    [
      "0.722354862",
      "1.414021532",
      "0.163989697"
    ],
    [
      "1.277814716",
      "1.413907810",
      "0.163989688"
    ],
    [
      "0.466251504",
      "1.211122918",
      "0.162746859"
    ],
    [
      "1.533834955",
      "1.210904367",
      "0.162746842"
    ],
    [
      "1.000119076",
      "1.581408606",
      "0.160311691"
    ],
    [
      "0.843557434",
      "0.810094460",
      "0.156403741"
    ],
    [
      "1.156364905",
      "0.810030435",
      "0.156403736"
    ],
    [
      "0.758594876",
      "1.103286951",
      "0.148851026"
    ],
    [
      "1.241447480",
      "1.103188109",
      "0.148851019"
    ],
    [
      "1.000056169",
      "1.274148595",
      "0.146948325"
    ],
    [
      "0.999822792",
      "0.133866010",
      "0.133865991"
    ],
    [
      "1.808810755",
      "1.316030668",
      "0.131639348"
    ],
    [
      "0.191318735",
      "1.316361810",
      "0.131639370"
    ],
    [
      "1.000178346",
      "1.870860136",
      "0.129139845"
    ],
    [
      "1.574555284",
      "0.346566691",
      "0.129891464"
    ],
    [
      "0.425177362",
      "0.346801889",
      "0.129891479"
    ],
    [
      "1.000004348",
      "1.021004313",
      "0.106195960"
    ]
  ]
}
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": 32
}

起步布局答案

{
  "circles": [
    [
      "0.999893251",
      "0.478199959",
      "0.210467962"
    ],
    [
      "1.811558224",
      "0.995971334",
      "0.188431776"
    ],
    [
      "0.188440225",
      "0.996303577",
      "0.188431806"
    ],
    [
      "1.721501689",
      "0.629508991",
      "0.188933865"
    ],
    [
      "0.278346754",
      "0.629804345",
      "0.188933892"
    ],
    [
      "1.598262061",
      "1.552965790",
      "0.185329110"
    ],
    [
      "0.401964457",
      "1.553210731",
      "0.185329131"
    ],
    [
      "0.706803838",
      "1.761587671",
      "0.183924163"
    ],
    [
      "1.293508067",
      "1.761467531",
      "0.183924153"
    ],
    [
      "0.706143328",
      "0.228903535",
      "0.174808202"
    ],
    [
      "1.293541114",
      "0.228783342",
      "0.174808192"
    ],
    [
      "1.369547276",
      "0.564127808",
      "0.169041843"
    ],
    [
      "0.630274415",
      "0.564279101",
      "0.169041855"
    ],
    [
      "0.527697805",
      "0.885739404",
      "0.168387673"
    ],
    [
      "1.472255454",
      "0.885546069",
      "0.168387657"
    ],
    [
      "0.722354862",
      "1.414021532",
      "0.163989697"
    ],
    [
      "1.277814716",
      "1.413907810",
      "0.163989688"
    ],
    [
      "0.466251504",
      "1.211122918",
      "0.162746859"
    ],
    [
      "1.533834955",
      "1.210904367",
      "0.162746842"
    ],
    [
      "1.000119076",
      "1.581408606",
      "0.160311691"
    ],
    [
      "0.843557434",
      "0.810094460",
      "0.156403741"
    ],
    [
      "1.156364905",
      "0.810030435",
      "0.156403736"
    ],
    [
      "0.758594876",
      "1.103286951",
      "0.148851026"
    ],
    [
      "1.241447480",
      "1.103188109",
      "0.148851019"
    ],
    [
      "1.000056169",
      "1.274148595",
      "0.146948325"
    ],
    [
      "0.999822792",
      "0.133866010",
      "0.133865991"
    ],
    [
      "1.808810755",
      "1.316030668",
      "0.131639348"
    ],
    [
      "0.191318735",
      "1.316361810",
      "0.131639370"
    ],
    [
      "1.000178346",
      "1.870860136",
      "0.129139845"
    ],
    [
      "1.574555284",
      "0.346566691",
      "0.129891464"
    ],
    [
      "0.425177362",
      "0.346801889",
      "0.129891479"
    ],
    [
      "1.000004348",
      "1.021004313",
      "0.106195960"
    ]
  ]
}

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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