P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 49

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 = 49

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	[6.473305, 6.473315]	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

12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849

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": [
    [
      "0.364597422",
      "1.535418905",
      "0.169090955"
    ],
    [
      "0.172651546",
      "1.076839729",
      "0.169090974"
    ],
    [
      "1.677498670",
      "1.489013551",
      "0.164453052"
    ],
    [
      "1.127156050",
      "0.174185281",
      "0.164453105"
    ],
    [
      "0.815895610",
      "0.166992489",
      "0.146890429"
    ],
    [
      "1.464166958",
      "1.715782837",
      "0.146890373"
    ],
    [
      "1.827946507",
      "1.220730090",
      "0.143135254"
    ],
    [
      "1.423853876",
      "0.255308772",
      "0.143135288"
    ],
    [
      "1.374682884",
      "1.440937356",
      "0.142155362"
    ],
    [
      "0.948873624",
      "0.423632975",
      "0.142155399"
    ],
    [
      "1.544203816",
      "1.214540281",
      "0.140674943"
    ],
    [
      "0.243370579",
      "0.592450394",
      "0.140590457"
    ],
    [
      "1.229120658",
      "0.461772417",
      "0.140674969"
    ],
    [
      "0.759277650",
      "1.825007628",
      "0.140590414"
    ],
    [
      "0.410518140",
      "0.370286540",
      "0.137429478"
    ],
    [
      "1.034843348",
      "1.861866540",
      "0.137429427"
    ],
    [
      "1.058098209",
      "0.680554563",
      "0.137019657"
    ],
    [
      "1.268327427",
      "1.182814905",
      "0.137019640"
    ],
    [
      "0.787220235",
      "0.649535826",
      "0.135628538"
    ],
    [
      "1.100309786",
      "1.397540517",
      "0.135628513"
    ],
    [
      "0.517784429",
      "0.621046160",
      "0.135309304"
    ],
    [
      "0.931502810",
      "1.609463824",
      "0.135309271"
    ],
    [
      "1.162586203",
      "0.931947023",
      "0.135222715"
    ],
    [
      "0.998791404",
      "1.146764392",
      "0.134916578"
    ],
    [
      "0.894606666",
      "0.897855820",
      "0.134916586"
    ],
    [
      "0.665791504",
      "1.567074282",
      "0.133762041"
    ],
    [
      "0.627718353",
      "0.866663846",
      "0.133788290"
    ],
    [
      "1.326209965",
      "0.718457261",
      "0.133757978"
    ],
    [
      "0.361492760",
      "0.840071995",
      "0.133762065"
    ],
    [
      "0.833697286",
      "1.358769651",
      "0.133788273"
    ],
    [
      "1.429499981",
      "0.965228319",
      "0.133757970"
    ],
    [
      "0.733221223",
      "1.111664731",
      "0.132963072"
    ],
    [
      "1.195109708",
      "1.646647466",
      "0.130907219"
    ],
    [
      "0.572524608",
      "1.320105644",
      "0.130230803"
    ],
    [
      "0.676307508",
      "0.407173084",
      "0.130907259"
    ],
    [
      "0.471958275",
      "1.079841903",
      "0.130230810"
    ],
    [
      "1.587347418",
      "0.754156349",
      "0.129808321"
    ],
    [
      "1.744533837",
      "0.548329658",
      "0.129174683"
    ],
    [
      "1.844284536",
      "0.786644973",
      "0.129174675"
    ],
    [
      "1.490716811",
      "0.516639072",
      "0.126613071"
    ],
    [
      "1.688719477",
      "0.989688999",
      "0.126613057"
    ],
    [
      "0.125480701",
      "0.807336282",
      "0.104509457"
    ],
    [
      "0.523467325",
      "1.758168824",
      "0.104509432"
    ],
    [
      "0.170838498",
      "1.347059265",
      "0.101134643"
    ],
    [
      "1.269160166",
      "1.862070200",
      "0.096887479"
    ],
    [
      "0.574826330",
      "0.203231838",
      "0.096887518"
    ],
    [
      "0.352197290",
      "1.271148552",
      "0.095470157"
    ],
    [
      "1.907566478",
      "0.999025191",
      "0.092432998"
    ],
    [
      "1.637662264",
      "0.354194555",
      "0.092433013"
    ]
  ]
}

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": 49
}

起步布局答案

{
  "circles": [
    [
      "0.364597422",
      "1.535418905",
      "0.169090955"
    ],
    [
      "0.172651546",
      "1.076839729",
      "0.169090974"
    ],
    [
      "1.677498670",
      "1.489013551",
      "0.164453052"
    ],
    [
      "1.127156050",
      "0.174185281",
      "0.164453105"
    ],
    [
      "0.815895610",
      "0.166992489",
      "0.146890429"
    ],
    [
      "1.464166958",
      "1.715782837",
      "0.146890373"
    ],
    [
      "1.827946507",
      "1.220730090",
      "0.143135254"
    ],
    [
      "1.423853876",
      "0.255308772",
      "0.143135288"
    ],
    [
      "1.374682884",
      "1.440937356",
      "0.142155362"
    ],
    [
      "0.948873624",
      "0.423632975",
      "0.142155399"
    ],
    [
      "1.544203816",
      "1.214540281",
      "0.140674943"
    ],
    [
      "0.243370579",
      "0.592450394",
      "0.140590457"
    ],
    [
      "1.229120658",
      "0.461772417",
      "0.140674969"
    ],
    [
      "0.759277650",
      "1.825007628",
      "0.140590414"
    ],
    [
      "0.410518140",
      "0.370286540",
      "0.137429478"
    ],
    [
      "1.034843348",
      "1.861866540",
      "0.137429427"
    ],
    [
      "1.058098209",
      "0.680554563",
      "0.137019657"
    ],
    [
      "1.268327427",
      "1.182814905",
      "0.137019640"
    ],
    [
      "0.787220235",
      "0.649535826",
      "0.135628538"
    ],
    [
      "1.100309786",
      "1.397540517",
      "0.135628513"
    ],
    [
      "0.517784429",
      "0.621046160",
      "0.135309304"
    ],
    [
      "0.931502810",
      "1.609463824",
      "0.135309271"
    ],
    [
      "1.162586203",
      "0.931947023",
      "0.135222715"
    ],
    [
      "0.998791404",
      "1.146764392",
      "0.134916578"
    ],
    [
      "0.894606666",
      "0.897855820",
      "0.134916586"
    ],
    [
      "0.665791504",
      "1.567074282",
      "0.133762041"
    ],
    [
      "0.627718353",
      "0.866663846",
      "0.133788290"
    ],
    [
      "1.326209965",
      "0.718457261",
      "0.133757978"
    ],
    [
      "0.361492760",
      "0.840071995",
      "0.133762065"
    ],
    [
      "0.833697286",
      "1.358769651",
      "0.133788273"
    ],
    [
      "1.429499981",
      "0.965228319",
      "0.133757970"
    ],
    [
      "0.733221223",
      "1.111664731",
      "0.132963072"
    ],
    [
      "1.195109708",
      "1.646647466",
      "0.130907219"
    ],
    [
      "0.572524608",
      "1.320105644",
      "0.130230803"
    ],
    [
      "0.676307508",
      "0.407173084",
      "0.130907259"
    ],
    [
      "0.471958275",
      "1.079841903",
      "0.130230810"
    ],
    [
      "1.587347418",
      "0.754156349",
      "0.129808321"
    ],
    [
      "1.744533837",
      "0.548329658",
      "0.129174683"
    ],
    [
      "1.844284536",
      "0.786644973",
      "0.129174675"
    ],
    [
      "1.490716811",
      "0.516639072",
      "0.126613071"
    ],
    [
      "1.688719477",
      "0.989688999",
      "0.126613057"
    ],
    [
      "0.125480701",
      "0.807336282",
      "0.104509457"
    ],
    [
      "0.523467325",
      "1.758168824",
      "0.104509432"
    ],
    [
      "0.170838498",
      "1.347059265",
      "0.101134643"
    ],
    [
      "1.269160166",
      "1.862070200",
      "0.096887479"
    ],
    [
      "0.574826330",
      "0.203231838",
      "0.096887518"
    ],
    [
      "0.352197290",
      "1.271148552",
      "0.095470157"
    ],
    [
      "1.907566478",
      "0.999025191",
      "0.092432998"
    ],
    [
      "1.637662264",
      "0.354194555",
      "0.092433013"
    ]
  ]
}

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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