P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 27

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 = 27

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	[4.762, 4.763]	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

123456789101112131415161718192021222324252627

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": [
    [
      "0.792023039",
      "1.764233961",
      "0.207972245"
    ],
    [
      "1.207967525",
      "1.764236536",
      "0.207972238"
    ],
    [
      "1.765834544",
      "0.797996502",
      "0.207972246"
    ],
    [
      "1.557864575",
      "0.437776678",
      "0.207972255"
    ],
    [
      "0.442142530",
      "0.437769655",
      "0.207972274"
    ],
    [
      "0.234168014",
      "0.797986903",
      "0.207972271"
    ],
    [
      "0.630036529",
      "0.786398428",
      "0.188065825"
    ],
    [
      "1.369966210",
      "0.786403066",
      "0.188065814"
    ],
    [
      "0.999997375",
      "1.427198625",
      "0.188065810"
    ],
    [
      "1.000001353",
      "0.789941580",
      "0.181915963"
    ],
    [
      "0.818083422",
      "1.105028134",
      "0.181915961"
    ],
    [
      "1.181915340",
      "1.105030405",
      "0.181915956"
    ],
    [
      "1.539830720",
      "1.108429930",
      "0.176015567"
    ],
    [
      "0.460167979",
      "1.108423190",
      "0.176015582"
    ],
    [
      "0.636181670",
      "1.413292206",
      "0.176015575"
    ],
    [
      "1.363813243",
      "1.413296731",
      "0.176015565"
    ],
    [
      "0.823987725",
      "0.478277983",
      "0.176015586"
    ],
    [
      "1.176018893",
      "0.478280198",
      "0.176015581"
    ],
    [
      "1.000005223",
      "0.174807237",
      "0.174807236"
    ],
    [
      "0.285359564",
      "1.412591997",
      "0.174807229"
    ],
    [
      "1.714635328",
      "1.412600885",
      "0.174807208"
    ],
    [
      "1.540815583",
      "1.671977059",
      "0.137425561"
    ],
    [
      "0.688468680",
      "0.195647556",
      "0.137425587"
    ],
    [
      "0.459176106",
      "1.671970355",
      "0.137425572"
    ],
    [
      "1.852355330",
      "1.132382209",
      "0.137425563"
    ],
    [
      "1.311541496",
      "0.195651490",
      "0.137425580"
    ],
    [
      "0.147643037",
      "1.132371571",
      "0.137425582"
    ]
  ]
}

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": 27
}

起步布局答案

{
  "circles": [
    [
      "0.792023039",
      "1.764233961",
      "0.207972245"
    ],
    [
      "1.207967525",
      "1.764236536",
      "0.207972238"
    ],
    [
      "1.765834544",
      "0.797996502",
      "0.207972246"
    ],
    [
      "1.557864575",
      "0.437776678",
      "0.207972255"
    ],
    [
      "0.442142530",
      "0.437769655",
      "0.207972274"
    ],
    [
      "0.234168014",
      "0.797986903",
      "0.207972271"
    ],
    [
      "0.630036529",
      "0.786398428",
      "0.188065825"
    ],
    [
      "1.369966210",
      "0.786403066",
      "0.188065814"
    ],
    [
      "0.999997375",
      "1.427198625",
      "0.188065810"
    ],
    [
      "1.000001353",
      "0.789941580",
      "0.181915963"
    ],
    [
      "0.818083422",
      "1.105028134",
      "0.181915961"
    ],
    [
      "1.181915340",
      "1.105030405",
      "0.181915956"
    ],
    [
      "1.539830720",
      "1.108429930",
      "0.176015567"
    ],
    [
      "0.460167979",
      "1.108423190",
      "0.176015582"
    ],
    [
      "0.636181670",
      "1.413292206",
      "0.176015575"
    ],
    [
      "1.363813243",
      "1.413296731",
      "0.176015565"
    ],
    [
      "0.823987725",
      "0.478277983",
      "0.176015586"
    ],
    [
      "1.176018893",
      "0.478280198",
      "0.176015581"
    ],
    [
      "1.000005223",
      "0.174807237",
      "0.174807236"
    ],
    [
      "0.285359564",
      "1.412591997",
      "0.174807229"
    ],
    [
      "1.714635328",
      "1.412600885",
      "0.174807208"
    ],
    [
      "1.540815583",
      "1.671977059",
      "0.137425561"
    ],
    [
      "0.688468680",
      "0.195647556",
      "0.137425587"
    ],
    [
      "0.459176106",
      "1.671970355",
      "0.137425572"
    ],
    [
      "1.852355330",
      "1.132382209",
      "0.137425563"
    ],
    [
      "1.311541496",
      "0.195651490",
      "0.137425580"
    ],
    [
      "0.147643037",
      "1.132371571",
      "0.137425582"
    ]
  ]
}

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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