P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 42

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 = 42

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	[5.981655, 5.981665]	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

123456789101112131415161718192021222324252627282930313233343536373839404142

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": [
    [
      "1.011640683",
      "1.012845727",
      "0.202341652"
    ],
    [
      "0.675745139",
      "0.879205572",
      "0.159162852"
    ],
    [
      "1.436269031",
      "1.718651330",
      "0.159291726"
    ],
    [
      "0.911627312",
      "0.665451380",
      "0.159162852"
    ],
    [
      "1.672342194",
      "1.504724059",
      "0.159291725"
    ],
    [
      "1.223097985",
      "0.721847544",
      "0.157372307"
    ],
    [
      "0.701286010",
      "1.194708618",
      "0.157372308"
    ],
    [
      "0.961299121",
      "1.365113231",
      "0.153504756"
    ],
    [
      "1.367148930",
      "0.997335972",
      "0.153504755"
    ],
    [
      "0.551467681",
      "0.279599743",
      "0.151378899"
    ],
    [
      "0.327121462",
      "0.482900161",
      "0.151378900"
    ],
    [
      "0.614995794",
      "0.575139887",
      "0.150912015"
    ],
    [
      "0.152793160",
      "0.934772364",
      "0.150285886"
    ],
    [
      "1.018273370",
      "0.150482395",
      "0.150285884"
    ],
    [
      "0.425361208",
      "1.060007835",
      "0.149676216"
    ],
    [
      "1.116140686",
      "0.434030011",
      "0.149676215"
    ],
    [
      "1.312026658",
      "0.208399085",
      "0.149122426"
    ],
    [
      "0.181586848",
      "1.232792939",
      "0.149122428"
    ],
    [
      "1.151959260",
      "1.837475395",
      "0.148849805"
    ],
    [
      "1.818507973",
      "1.233455248",
      "0.148849803"
    ],
    [
      "1.689342761",
      "0.497765307",
      "0.147103095"
    ],
    [
      "0.432506783",
      "1.636698181",
      "0.147103098"
    ],
    [
      "1.526702299",
      "0.742746029",
      "0.146950426"
    ],
    [
      "0.681498464",
      "1.792003535",
      "0.146353218"
    ],
    [
      "0.692273117",
      "1.498897862",
      "0.146950427"
    ],
    [
      "1.819449584",
      "0.760802988",
      "0.146353216"
    ],
    [
      "1.245472648",
      "1.270884136",
      "0.145884147"
    ],
    [
      "0.829758446",
      "0.374841731",
      "0.142758459"
    ],
    [
      "0.394578324",
      "0.769197837",
      "0.142758460"
    ],
    [
      "1.402059434",
      "0.483697851",
      "0.140524446"
    ],
    [
      "0.446715167",
      "1.349421785",
      "0.140524448"
    ],
    [
      "1.529812563",
      "1.241594637",
      "0.139960322"
    ],
    [
      "1.188405828",
      "1.550974193",
      "0.139960323"
    ],
    [
      "0.930653570",
      "1.654991638",
      "0.137989051"
    ],
    [
      "1.658635635",
      "0.995301172",
      "0.137989050"
    ],
    [
      "0.763716911",
      "0.136400325",
      "0.104659786"
    ],
    [
      "0.163773608",
      "0.680063262",
      "0.104659787"
    ],
    [
      "1.413386184",
      "1.456180192",
      "0.104175013"
    ],
    [
      "1.549853624",
      "0.290718125",
      "0.102548170"
    ],
    [
      "0.240156283",
      "1.477553461",
      "0.102548172"
    ],
    [
      "1.898307357",
      "0.995970411",
      "0.101683605"
    ],
    [
      "0.907786616",
      "1.893570945",
      "0.101683606"
    ]
  ]
}

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": 42
}

起步布局答案

{
  "circles": [
    [
      "1.011640683",
      "1.012845727",
      "0.202341652"
    ],
    [
      "0.675745139",
      "0.879205572",
      "0.159162852"
    ],
    [
      "1.436269031",
      "1.718651330",
      "0.159291726"
    ],
    [
      "0.911627312",
      "0.665451380",
      "0.159162852"
    ],
    [
      "1.672342194",
      "1.504724059",
      "0.159291725"
    ],
    [
      "1.223097985",
      "0.721847544",
      "0.157372307"
    ],
    [
      "0.701286010",
      "1.194708618",
      "0.157372308"
    ],
    [
      "0.961299121",
      "1.365113231",
      "0.153504756"
    ],
    [
      "1.367148930",
      "0.997335972",
      "0.153504755"
    ],
    [
      "0.551467681",
      "0.279599743",
      "0.151378899"
    ],
    [
      "0.327121462",
      "0.482900161",
      "0.151378900"
    ],
    [
      "0.614995794",
      "0.575139887",
      "0.150912015"
    ],
    [
      "0.152793160",
      "0.934772364",
      "0.150285886"
    ],
    [
      "1.018273370",
      "0.150482395",
      "0.150285884"
    ],
    [
      "0.425361208",
      "1.060007835",
      "0.149676216"
    ],
    [
      "1.116140686",
      "0.434030011",
      "0.149676215"
    ],
    [
      "1.312026658",
      "0.208399085",
      "0.149122426"
    ],
    [
      "0.181586848",
      "1.232792939",
      "0.149122428"
    ],
    [
      "1.151959260",
      "1.837475395",
      "0.148849805"
    ],
    [
      "1.818507973",
      "1.233455248",
      "0.148849803"
    ],
    [
      "1.689342761",
      "0.497765307",
      "0.147103095"
    ],
    [
      "0.432506783",
      "1.636698181",
      "0.147103098"
    ],
    [
      "1.526702299",
      "0.742746029",
      "0.146950426"
    ],
    [
      "0.681498464",
      "1.792003535",
      "0.146353218"
    ],
    [
      "0.692273117",
      "1.498897862",
      "0.146950427"
    ],
    [
      "1.819449584",
      "0.760802988",
      "0.146353216"
    ],
    [
      "1.245472648",
      "1.270884136",
      "0.145884147"
    ],
    [
      "0.829758446",
      "0.374841731",
      "0.142758459"
    ],
    [
      "0.394578324",
      "0.769197837",
      "0.142758460"
    ],
    [
      "1.402059434",
      "0.483697851",
      "0.140524446"
    ],
    [
      "0.446715167",
      "1.349421785",
      "0.140524448"
    ],
    [
      "1.529812563",
      "1.241594637",
      "0.139960322"
    ],
    [
      "1.188405828",
      "1.550974193",
      "0.139960323"
    ],
    [
      "0.930653570",
      "1.654991638",
      "0.137989051"
    ],
    [
      "1.658635635",
      "0.995301172",
      "0.137989050"
    ],
    [
      "0.763716911",
      "0.136400325",
      "0.104659786"
    ],
    [
      "0.163773608",
      "0.680063262",
      "0.104659787"
    ],
    [
      "1.413386184",
      "1.456180192",
      "0.104175013"
    ],
    [
      "1.549853624",
      "0.290718125",
      "0.102548170"
    ],
    [
      "0.240156283",
      "1.477553461",
      "0.102548172"
    ],
    [
      "1.898307357",
      "0.995970411",
      "0.101683605"
    ],
    [
      "0.907786616",
      "1.893570945",
      "0.101683606"
    ]
  ]
}

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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