P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 37

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 = 37

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	[5.621715, 5.621725]	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 $S = Σ r$
目标在所有合法放置中最大化半径之和 $\max_{S} Σ r$

放大来摆，然后提交 ↗

当前展示：起步布局

2y0

12345678910111213141516171819202122232425262728293031323334353637

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

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": [
    [
      "1.617246323",
      "1.561313239",
      "0.165694554"
    ],
    [
      "0.382753699",
      "0.438686788",
      "0.165694589"
    ],
    [
      "1.787069046",
      "1.276745171",
      "0.165694556"
    ],
    [
      "0.212930955",
      "0.723254922",
      "0.165694588"
    ],
    [
      "1.153866278",
      "1.819994353",
      "0.165694558"
    ],
    [
      "0.846133848",
      "0.180005652",
      "0.165694586"
    ],
    [
      "0.822511729",
      "1.815207627",
      "0.165694563"
    ],
    [
      "1.177488443",
      "0.184792429",
      "0.165694581"
    ],
    [
      "0.366797239",
      "1.543249305",
      "0.165694573"
    ],
    [
      "0.205265367",
      "1.253894459",
      "0.165694580"
    ],
    [
      "1.794734705",
      "0.746105712",
      "0.165694563"
    ],
    [
      "1.633202901",
      "0.456750853",
      "0.165694570"
    ],
    [
      "0.992194475",
      "1.540328054",
      "0.157339512"
    ],
    [
      "1.007805642",
      "0.459672006",
      "0.157339526"
    ],
    [
      "0.535965024",
      "0.723076191",
      "0.157339529"
    ],
    [
      "1.464035030",
      "1.276923882",
      "0.157339508"
    ],
    [
      "0.528159462",
      "1.263404239",
      "0.157339522"
    ],
    [
      "1.471840635",
      "0.736595882",
      "0.157339516"
    ],
    [
      "1.000000045",
      "1.000000043",
      "0.155324629"
    ],
    [
      "0.840805619",
      "1.266758540",
      "0.155324627"
    ],
    [
      "1.159194482",
      "0.733241549",
      "0.155324630"
    ],
    [
      "1.151422462",
      "1.271245706",
      "0.155324623"
    ],
    [
      "0.848577625",
      "0.728754367",
      "0.155324635"
    ],
    [
      "1.310616890",
      "1.004487221",
      "0.155324624"
    ],
    [
      "0.689383191",
      "0.995512873",
      "0.155324633"
    ],
    [
      "0.683621943",
      "1.530147694",
      "0.151400908"
    ],
    [
      "1.316378193",
      "0.469852406",
      "0.151400913"
    ],
    [
      "1.617310417",
      "1.008917730",
      "0.151400902"
    ],
    [
      "1.300932305",
      "1.539065335",
      "0.151400899"
    ],
    [
      "0.382689637",
      "0.991082387",
      "0.151400919"
    ],
    [
      "0.699067775",
      "0.460934703",
      "0.151400922"
    ],
    [
      "1.431084752",
      "1.772209711",
      "0.115612131"
    ],
    [
      "1.884295598",
      "1.012774618",
      "0.115612135"
    ],
    [
      "0.115704419",
      "0.987225534",
      "0.115612154"
    ],
    [
      "0.568915318",
      "0.227790279",
      "0.115612157"
    ],
    [
      "0.546789220",
      "1.759435232",
      "0.115612141"
    ],
    [
      "1.453210965",
      "0.240564885",
      "0.115612147"
    ]
  ]
}

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": 37
}

起步布局答案

{
  "circles": [
    [
      "1.617246323",
      "1.561313239",
      "0.165694554"
    ],
    [
      "0.382753699",
      "0.438686788",
      "0.165694589"
    ],
    [
      "1.787069046",
      "1.276745171",
      "0.165694556"
    ],
    [
      "0.212930955",
      "0.723254922",
      "0.165694588"
    ],
    [
      "1.153866278",
      "1.819994353",
      "0.165694558"
    ],
    [
      "0.846133848",
      "0.180005652",
      "0.165694586"
    ],
    [
      "0.822511729",
      "1.815207627",
      "0.165694563"
    ],
    [
      "1.177488443",
      "0.184792429",
      "0.165694581"
    ],
    [
      "0.366797239",
      "1.543249305",
      "0.165694573"
    ],
    [
      "0.205265367",
      "1.253894459",
      "0.165694580"
    ],
    [
      "1.794734705",
      "0.746105712",
      "0.165694563"
    ],
    [
      "1.633202901",
      "0.456750853",
      "0.165694570"
    ],
    [
      "0.992194475",
      "1.540328054",
      "0.157339512"
    ],
    [
      "1.007805642",
      "0.459672006",
      "0.157339526"
    ],
    [
      "0.535965024",
      "0.723076191",
      "0.157339529"
    ],
    [
      "1.464035030",
      "1.276923882",
      "0.157339508"
    ],
    [
      "0.528159462",
      "1.263404239",
      "0.157339522"
    ],
    [
      "1.471840635",
      "0.736595882",
      "0.157339516"
    ],
    [
      "1.000000045",
      "1.000000043",
      "0.155324629"
    ],
    [
      "0.840805619",
      "1.266758540",
      "0.155324627"
    ],
    [
      "1.159194482",
      "0.733241549",
      "0.155324630"
    ],
    [
      "1.151422462",
      "1.271245706",
      "0.155324623"
    ],
    [
      "0.848577625",
      "0.728754367",
      "0.155324635"
    ],
    [
      "1.310616890",
      "1.004487221",
      "0.155324624"
    ],
    [
      "0.689383191",
      "0.995512873",
      "0.155324633"
    ],
    [
      "0.683621943",
      "1.530147694",
      "0.151400908"
    ],
    [
      "1.316378193",
      "0.469852406",
      "0.151400913"
    ],
    [
      "1.617310417",
      "1.008917730",
      "0.151400902"
    ],
    [
      "1.300932305",
      "1.539065335",
      "0.151400899"
    ],
    [
      "0.382689637",
      "0.991082387",
      "0.151400919"
    ],
    [
      "0.699067775",
      "0.460934703",
      "0.151400922"
    ],
    [
      "1.431084752",
      "1.772209711",
      "0.115612131"
    ],
    [
      "1.884295598",
      "1.012774618",
      "0.115612135"
    ],
    [
      "0.115704419",
      "0.987225534",
      "0.115612154"
    ],
    [
      "0.568915318",
      "0.227790279",
      "0.115612157"
    ],
    [
      "0.546789220",
      "1.759435232",
      "0.115612141"
    ],
    [
      "1.453210965",
      "0.240564885",
      "0.115612147"
    ]
  ]
}

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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