P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 22

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 = 22

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	[4.287, 4.288]	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

12345678910111213141516171819202122

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": [
    [
      "1.000000029",
      "1.000000026",
      "0.270210210"
    ],
    [
      "1.771199758",
      "1.175737197",
      "0.209030575"
    ],
    [
      "0.657060900",
      "1.712758999",
      "0.209030584"
    ],
    [
      "1.343438363",
      "1.712518576",
      "0.209030575"
    ],
    [
      "0.381422537",
      "0.507055266",
      "0.209030601"
    ],
    [
      "1.618232070",
      "0.506622131",
      "0.209030585"
    ],
    [
      "0.228923553",
      "1.176277370",
      "0.209030595"
    ],
    [
      "0.999723025",
      "0.209030646",
      "0.209030597"
    ],
    [
      "1.000167205",
      "1.477305147",
      "0.207094940"
    ],
    [
      "0.534624731",
      "0.893952624",
      "0.207094952"
    ],
    [
      "0.626932087",
      "1.297725611",
      "0.207094947"
    ],
    [
      "0.792754482",
      "0.570035489",
      "0.207094953"
    ],
    [
      "1.206944359",
      "0.569890436",
      "0.207094948"
    ],
    [
      "1.465300919",
      "0.893626676",
      "0.207094941"
    ],
    [
      "1.373276425",
      "1.297464203",
      "0.207094937"
    ],
    [
      "1.000295019",
      "1.842200007",
      "0.157799941"
    ],
    [
      "1.658642389",
      "1.524872507",
      "0.157799937"
    ],
    [
      "1.821018670",
      "0.812305352",
      "0.157799942"
    ],
    [
      "1.365151188",
      "0.241076070",
      "0.157799951"
    ],
    [
      "0.341725479",
      "1.525333778",
      "0.157799950"
    ],
    [
      "0.634317388",
      "0.241331999",
      "0.157799959"
    ],
    [
      "0.178850078",
      "0.812880476",
      "0.157799958"
    ]
  ]
}

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": 22
}

起步布局答案

{
  "circles": [
    [
      "1.000000029",
      "1.000000026",
      "0.270210210"
    ],
    [
      "1.771199758",
      "1.175737197",
      "0.209030575"
    ],
    [
      "0.657060900",
      "1.712758999",
      "0.209030584"
    ],
    [
      "1.343438363",
      "1.712518576",
      "0.209030575"
    ],
    [
      "0.381422537",
      "0.507055266",
      "0.209030601"
    ],
    [
      "1.618232070",
      "0.506622131",
      "0.209030585"
    ],
    [
      "0.228923553",
      "1.176277370",
      "0.209030595"
    ],
    [
      "0.999723025",
      "0.209030646",
      "0.209030597"
    ],
    [
      "1.000167205",
      "1.477305147",
      "0.207094940"
    ],
    [
      "0.534624731",
      "0.893952624",
      "0.207094952"
    ],
    [
      "0.626932087",
      "1.297725611",
      "0.207094947"
    ],
    [
      "0.792754482",
      "0.570035489",
      "0.207094953"
    ],
    [
      "1.206944359",
      "0.569890436",
      "0.207094948"
    ],
    [
      "1.465300919",
      "0.893626676",
      "0.207094941"
    ],
    [
      "1.373276425",
      "1.297464203",
      "0.207094937"
    ],
    [
      "1.000295019",
      "1.842200007",
      "0.157799941"
    ],
    [
      "1.658642389",
      "1.524872507",
      "0.157799937"
    ],
    [
      "1.821018670",
      "0.812305352",
      "0.157799942"
    ],
    [
      "1.365151188",
      "0.241076070",
      "0.157799951"
    ],
    [
      "0.341725479",
      "1.525333778",
      "0.157799950"
    ],
    [
      "0.634317388",
      "0.241331999",
      "0.157799959"
    ],
    [
      "0.178850078",
      "0.812880476",
      "0.157799958"
    ]
  ]
}

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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