P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 46

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 = 46

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	[6.260145, 6.260155]	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546

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": [
    [
      "1.003302083",
      "1.340622806",
      "0.175565189"
    ],
    [
      "0.670523398",
      "0.913513379",
      "0.175565196"
    ],
    [
      "1.655282044",
      "0.489442648",
      "0.169299471"
    ],
    [
      "1.493422010",
      "1.690403852",
      "0.151399528"
    ],
    [
      "0.451197074",
      "0.352745740",
      "0.151399547"
    ],
    [
      "1.817219657",
      "0.763879017",
      "0.149352548"
    ],
    [
      "1.428770509",
      "0.265318488",
      "0.149352555"
    ],
    [
      "0.427215892",
      "1.126559908",
      "0.147834503"
    ],
    [
      "1.704991545",
      "1.479073227",
      "0.147636089"
    ],
    [
      "0.707843633",
      "0.199269794",
      "0.147636106"
    ],
    [
      "0.737231584",
      "1.524453837",
      "0.147834498"
    ],
    [
      "1.226945736",
      "1.823027788",
      "0.146255830"
    ],
    [
      "0.257443160",
      "0.578706256",
      "0.146255846"
    ],
    [
      "0.756439970",
      "1.818087088",
      "0.146426352"
    ],
    [
      "0.147191743",
      "1.036139041",
      "0.146426362"
    ],
    [
      "0.494637064",
      "1.689171192",
      "0.145395630"
    ],
    [
      "0.700576192",
      "1.233293562",
      "0.145624056"
    ],
    [
      "0.208181973",
      "1.321516458",
      "0.145395634"
    ],
    [
      "1.524266198",
      "0.773777205",
      "0.143768080"
    ],
    [
      "1.347543416",
      "0.546959870",
      "0.143768083"
    ],
    [
      "0.993519409",
      "1.656512588",
      "0.140476033"
    ],
    [
      "0.361829733",
      "0.845761720",
      "0.140476044"
    ],
    [
      "0.537443766",
      "0.629284218",
      "0.138276170"
    ],
    [
      "0.987835745",
      "0.138880365",
      "0.138794452"
    ],
    [
      "1.832008163",
      "1.222345303",
      "0.138794439"
    ],
    [
      "1.246365177",
      "1.539159264",
      "0.138276159"
    ],
    [
      "1.244827153",
      "0.809245056",
      "0.137912852"
    ],
    [
      "0.477487560",
      "1.407111184",
      "0.137185249"
    ],
    [
      "1.140743586",
      "1.061823732",
      "0.135270925"
    ],
    [
      "1.408702933",
      "1.027350236",
      "0.134896862"
    ],
    [
      "1.073410891",
      "0.597014973",
      "0.134896867"
    ],
    [
      "0.974465681",
      "0.848412035",
      "0.135270927"
    ],
    [
      "1.304639326",
      "1.274212896",
      "0.133003132"
    ],
    [
      "0.808596809",
      "0.637560208",
      "0.133003140"
    ],
    [
      "1.160465704",
      "0.345971462",
      "0.130812343"
    ],
    [
      "1.673412163",
      "1.004319795",
      "0.130812335"
    ],
    [
      "0.904888563",
      "0.393696953",
      "0.129182651"
    ],
    [
      "1.564644736",
      "1.240470245",
      "0.129182641"
    ],
    [
      "0.899492836",
      "1.078309347",
      "0.106542432"
    ],
    [
      "0.990964580",
      "1.898464851",
      "0.101489717"
    ],
    [
      "0.126596650",
      "0.789079878",
      "0.101489727"
    ],
    [
      "0.290220534",
      "1.553018710",
      "0.100213034"
    ],
    [
      "1.462111601",
      "1.443082357",
      "0.097896005"
    ],
    [
      "0.683356116",
      "0.443577764",
      "0.097896014"
    ],
    [
      "1.224250400",
      "0.126275146",
      "0.097956008"
    ],
    [
      "1.902035947",
      "0.996188543",
      "0.097956000"
    ]
  ]
}

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": 46
}

起步布局答案

{
  "circles": [
    [
      "1.003302083",
      "1.340622806",
      "0.175565189"
    ],
    [
      "0.670523398",
      "0.913513379",
      "0.175565196"
    ],
    [
      "1.655282044",
      "0.489442648",
      "0.169299471"
    ],
    [
      "1.493422010",
      "1.690403852",
      "0.151399528"
    ],
    [
      "0.451197074",
      "0.352745740",
      "0.151399547"
    ],
    [
      "1.817219657",
      "0.763879017",
      "0.149352548"
    ],
    [
      "1.428770509",
      "0.265318488",
      "0.149352555"
    ],
    [
      "0.427215892",
      "1.126559908",
      "0.147834503"
    ],
    [
      "1.704991545",
      "1.479073227",
      "0.147636089"
    ],
    [
      "0.707843633",
      "0.199269794",
      "0.147636106"
    ],
    [
      "0.737231584",
      "1.524453837",
      "0.147834498"
    ],
    [
      "1.226945736",
      "1.823027788",
      "0.146255830"
    ],
    [
      "0.257443160",
      "0.578706256",
      "0.146255846"
    ],
    [
      "0.756439970",
      "1.818087088",
      "0.146426352"
    ],
    [
      "0.147191743",
      "1.036139041",
      "0.146426362"
    ],
    [
      "0.494637064",
      "1.689171192",
      "0.145395630"
    ],
    [
      "0.700576192",
      "1.233293562",
      "0.145624056"
    ],
    [
      "0.208181973",
      "1.321516458",
      "0.145395634"
    ],
    [
      "1.524266198",
      "0.773777205",
      "0.143768080"
    ],
    [
      "1.347543416",
      "0.546959870",
      "0.143768083"
    ],
    [
      "0.993519409",
      "1.656512588",
      "0.140476033"
    ],
    [
      "0.361829733",
      "0.845761720",
      "0.140476044"
    ],
    [
      "0.537443766",
      "0.629284218",
      "0.138276170"
    ],
    [
      "0.987835745",
      "0.138880365",
      "0.138794452"
    ],
    [
      "1.832008163",
      "1.222345303",
      "0.138794439"
    ],
    [
      "1.246365177",
      "1.539159264",
      "0.138276159"
    ],
    [
      "1.244827153",
      "0.809245056",
      "0.137912852"
    ],
    [
      "0.477487560",
      "1.407111184",
      "0.137185249"
    ],
    [
      "1.140743586",
      "1.061823732",
      "0.135270925"
    ],
    [
      "1.408702933",
      "1.027350236",
      "0.134896862"
    ],
    [
      "1.073410891",
      "0.597014973",
      "0.134896867"
    ],
    [
      "0.974465681",
      "0.848412035",
      "0.135270927"
    ],
    [
      "1.304639326",
      "1.274212896",
      "0.133003132"
    ],
    [
      "0.808596809",
      "0.637560208",
      "0.133003140"
    ],
    [
      "1.160465704",
      "0.345971462",
      "0.130812343"
    ],
    [
      "1.673412163",
      "1.004319795",
      "0.130812335"
    ],
    [
      "0.904888563",
      "0.393696953",
      "0.129182651"
    ],
    [
      "1.564644736",
      "1.240470245",
      "0.129182641"
    ],
    [
      "0.899492836",
      "1.078309347",
      "0.106542432"
    ],
    [
      "0.990964580",
      "1.898464851",
      "0.101489717"
    ],
    [
      "0.126596650",
      "0.789079878",
      "0.101489727"
    ],
    [
      "0.290220534",
      "1.553018710",
      "0.100213034"
    ],
    [
      "1.462111601",
      "1.443082357",
      "0.097896005"
    ],
    [
      "0.683356116",
      "0.443577764",
      "0.097896014"
    ],
    [
      "1.224250400",
      "0.126275146",
      "0.097956008"
    ],
    [
      "1.902035947",
      "0.996188543",
      "0.097956000"
    ]
  ]
}

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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