P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 29

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 = 29

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	[4.934, 4.935]	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

1234567891011121314151617181920212223242526272829

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": [
    [
      "0.670209874",
      "0.617405927",
      "0.214647629"
    ],
    [
      "1.329898038",
      "0.617498918",
      "0.214647617"
    ],
    [
      "1.741142226",
      "1.286218990",
      "0.205510787"
    ],
    [
      "0.258777173",
      "1.286010119",
      "0.205510813"
    ],
    [
      "1.517717382",
      "1.617571433",
      "0.194130429"
    ],
    [
      "0.482108666",
      "1.617425543",
      "0.194130447"
    ],
    [
      "1.731785335",
      "0.649232684",
      "0.188490611"
    ],
    [
      "0.268313609",
      "0.649026393",
      "0.188490635"
    ],
    [
      "1.530002360",
      "0.960616885",
      "0.182557214"
    ],
    [
      "0.470008822",
      "0.960467497",
      "0.182557231"
    ],
    [
      "1.354195517",
      "1.279406187",
      "0.181495892"
    ],
    [
      "0.645725826",
      "1.279306360",
      "0.181495903"
    ],
    [
      "1.286282259",
      "0.228340574",
      "0.176947266"
    ],
    [
      "0.713935381",
      "0.228259875",
      "0.176947275"
    ],
    [
      "0.826171087",
      "0.973205156",
      "0.173832731"
    ],
    [
      "1.173836542",
      "0.973254153",
      "0.173832726"
    ],
    [
      "0.999961556",
      "1.273116635",
      "0.172793900"
    ],
    [
      "1.000081496",
      "0.422130155",
      "0.168690228"
    ],
    [
      "1.163543652",
      "1.567085293",
      "0.163623510"
    ],
    [
      "0.836296630",
      "1.567039191",
      "0.163623514"
    ],
    [
      "0.999881285",
      "1.842903723",
      "0.157096268"
    ],
    [
      "1.855726339",
      "0.956884159",
      "0.143188151"
    ],
    [
      "0.144285870",
      "0.956642960",
      "0.143188173"
    ],
    [
      "1.574635527",
      "0.360491349",
      "0.140245788"
    ],
    [
      "0.425544840",
      "0.360329343",
      "0.140245802"
    ],
    [
      "1.000039187",
      "0.722271128",
      "0.131450746"
    ],
    [
      "1.000123148",
      "0.126719969",
      "0.126719960"
    ],
    [
      "1.280690841",
      "1.829792125",
      "0.124019224"
    ],
    [
      "0.719075428",
      "1.829713018",
      "0.124019231"
    ]
  ]
}

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": 29
}

起步布局答案

{
  "circles": [
    [
      "0.670209874",
      "0.617405927",
      "0.214647629"
    ],
    [
      "1.329898038",
      "0.617498918",
      "0.214647617"
    ],
    [
      "1.741142226",
      "1.286218990",
      "0.205510787"
    ],
    [
      "0.258777173",
      "1.286010119",
      "0.205510813"
    ],
    [
      "1.517717382",
      "1.617571433",
      "0.194130429"
    ],
    [
      "0.482108666",
      "1.617425543",
      "0.194130447"
    ],
    [
      "1.731785335",
      "0.649232684",
      "0.188490611"
    ],
    [
      "0.268313609",
      "0.649026393",
      "0.188490635"
    ],
    [
      "1.530002360",
      "0.960616885",
      "0.182557214"
    ],
    [
      "0.470008822",
      "0.960467497",
      "0.182557231"
    ],
    [
      "1.354195517",
      "1.279406187",
      "0.181495892"
    ],
    [
      "0.645725826",
      "1.279306360",
      "0.181495903"
    ],
    [
      "1.286282259",
      "0.228340574",
      "0.176947266"
    ],
    [
      "0.713935381",
      "0.228259875",
      "0.176947275"
    ],
    [
      "0.826171087",
      "0.973205156",
      "0.173832731"
    ],
    [
      "1.173836542",
      "0.973254153",
      "0.173832726"
    ],
    [
      "0.999961556",
      "1.273116635",
      "0.172793900"
    ],
    [
      "1.000081496",
      "0.422130155",
      "0.168690228"
    ],
    [
      "1.163543652",
      "1.567085293",
      "0.163623510"
    ],
    [
      "0.836296630",
      "1.567039191",
      "0.163623514"
    ],
    [
      "0.999881285",
      "1.842903723",
      "0.157096268"
    ],
    [
      "1.855726339",
      "0.956884159",
      "0.143188151"
    ],
    [
      "0.144285870",
      "0.956642960",
      "0.143188173"
    ],
    [
      "1.574635527",
      "0.360491349",
      "0.140245788"
    ],
    [
      "0.425544840",
      "0.360329343",
      "0.140245802"
    ],
    [
      "1.000039187",
      "0.722271128",
      "0.131450746"
    ],
    [
      "1.000123148",
      "0.126719969",
      "0.126719960"
    ],
    [
      "1.280690841",
      "1.829792125",
      "0.124019224"
    ],
    [
      "0.719075428",
      "1.829713018",
      "0.124019231"
    ]
  ]
}

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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