P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 34

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 = 34

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	[5.360435, 5.360445]	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

12345678910111213141516171819202122232425262728293031323334

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": [
    [
      "1.255654784",
      "0.920195108",
      "0.201433794"
    ],
    [
      "1.582741185",
      "1.572355598",
      "0.183190218"
    ],
    [
      "1.153652641",
      "0.197772459",
      "0.183190253"
    ],
    [
      "0.579922085",
      "0.296174502",
      "0.180344104"
    ],
    [
      "1.054915799",
      "1.817814223",
      "0.180344066"
    ],
    [
      "1.777492444",
      "1.269022454",
      "0.177280374"
    ],
    [
      "0.701655130",
      "1.767589400",
      "0.176469096"
    ],
    [
      "1.486380982",
      "0.336448131",
      "0.177280397"
    ],
    [
      "0.317966369",
      "0.538444776",
      "0.176469126"
    ],
    [
      "1.428103928",
      "1.252014525",
      "0.172521861"
    ],
    [
      "0.175823729",
      "1.076826171",
      "0.172250770"
    ],
    [
      "1.208713194",
      "0.549197354",
      "0.172521878"
    ],
    [
      "0.278477817",
      "1.405677877",
      "0.172250762"
    ],
    [
      "0.915536531",
      "1.504377065",
      "0.162685716"
    ],
    [
      "0.643603605",
      "0.633241464",
      "0.162685736"
    ],
    [
      "0.896207890",
      "0.428466392",
      "0.162493836"
    ],
    [
      "1.239776010",
      "1.529085088",
      "0.162493811"
    ],
    [
      "1.517148448",
      "0.671652217",
      "0.159332757"
    ],
    [
      "1.612103897",
      "0.975841581",
      "0.159332750"
    ],
    [
      "0.495987700",
      "1.157332109",
      "0.157879770"
    ],
    [
      "1.098747887",
      "1.242431152",
      "0.156973574"
    ],
    [
      "0.943296782",
      "0.744444350",
      "0.156973585"
    ],
    [
      "0.602402494",
      "1.451443289",
      "0.154890903"
    ],
    [
      "0.416189845",
      "0.854912194",
      "0.154890916"
    ],
    [
      "0.795202449",
      "1.218466111",
      "0.147516418"
    ],
    [
      "0.707288993",
      "0.936835852",
      "0.147516424"
    ],
    [
      "1.814308141",
      "0.745806431",
      "0.146939557"
    ],
    [
      "1.362554487",
      "1.792707219",
      "0.128317436"
    ],
    [
      "0.847212599",
      "0.141812113",
      "0.128317466"
    ],
    [
      "0.422769808",
      "1.661870807",
      "0.121781542"
    ],
    [
      "0.148747825",
      "0.784043136",
      "0.121781557"
    ],
    [
      "1.729408127",
      "0.498309056",
      "0.114714724"
    ],
    [
      "1.885282167",
      "0.997650921",
      "0.114714716"
    ],
    [
      "0.955827174",
      "1.013789019",
      "0.112662393"
    ]
  ]
}

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": 34
}

起步布局答案

{
  "circles": [
    [
      "1.255654784",
      "0.920195108",
      "0.201433794"
    ],
    [
      "1.582741185",
      "1.572355598",
      "0.183190218"
    ],
    [
      "1.153652641",
      "0.197772459",
      "0.183190253"
    ],
    [
      "0.579922085",
      "0.296174502",
      "0.180344104"
    ],
    [
      "1.054915799",
      "1.817814223",
      "0.180344066"
    ],
    [
      "1.777492444",
      "1.269022454",
      "0.177280374"
    ],
    [
      "0.701655130",
      "1.767589400",
      "0.176469096"
    ],
    [
      "1.486380982",
      "0.336448131",
      "0.177280397"
    ],
    [
      "0.317966369",
      "0.538444776",
      "0.176469126"
    ],
    [
      "1.428103928",
      "1.252014525",
      "0.172521861"
    ],
    [
      "0.175823729",
      "1.076826171",
      "0.172250770"
    ],
    [
      "1.208713194",
      "0.549197354",
      "0.172521878"
    ],
    [
      "0.278477817",
      "1.405677877",
      "0.172250762"
    ],
    [
      "0.915536531",
      "1.504377065",
      "0.162685716"
    ],
    [
      "0.643603605",
      "0.633241464",
      "0.162685736"
    ],
    [
      "0.896207890",
      "0.428466392",
      "0.162493836"
    ],
    [
      "1.239776010",
      "1.529085088",
      "0.162493811"
    ],
    [
      "1.517148448",
      "0.671652217",
      "0.159332757"
    ],
    [
      "1.612103897",
      "0.975841581",
      "0.159332750"
    ],
    [
      "0.495987700",
      "1.157332109",
      "0.157879770"
    ],
    [
      "1.098747887",
      "1.242431152",
      "0.156973574"
    ],
    [
      "0.943296782",
      "0.744444350",
      "0.156973585"
    ],
    [
      "0.602402494",
      "1.451443289",
      "0.154890903"
    ],
    [
      "0.416189845",
      "0.854912194",
      "0.154890916"
    ],
    [
      "0.795202449",
      "1.218466111",
      "0.147516418"
    ],
    [
      "0.707288993",
      "0.936835852",
      "0.147516424"
    ],
    [
      "1.814308141",
      "0.745806431",
      "0.146939557"
    ],
    [
      "1.362554487",
      "1.792707219",
      "0.128317436"
    ],
    [
      "0.847212599",
      "0.141812113",
      "0.128317466"
    ],
    [
      "0.422769808",
      "1.661870807",
      "0.121781542"
    ],
    [
      "0.148747825",
      "0.784043136",
      "0.121781557"
    ],
    [
      "1.729408127",
      "0.498309056",
      "0.114714724"
    ],
    [
      "1.885282167",
      "0.997650921",
      "0.114714716"
    ],
    [
      "0.955827174",
      "1.013789019",
      "0.112662393"
    ]
  ]
}

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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