P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 43

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 = 43

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	[6.047095, 6.047105]	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

12345678910111213141516171819202122232425262728293031323334353637383940414243

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": [
    [
      "1.215414152",
      "1.785611256",
      "0.185390705"
    ],
    [
      "0.907961006",
      "0.718640428",
      "0.184098584"
    ],
    [
      "0.752256227",
      "1.479992528",
      "0.180719771"
    ],
    [
      "1.008853250",
      "0.157457050",
      "0.157410537"
    ],
    [
      "1.677715058",
      "1.502355906",
      "0.156401069"
    ],
    [
      "0.460319877",
      "1.648464725",
      "0.156340628"
    ],
    [
      "1.116801677",
      "0.451016384",
      "0.155367252"
    ],
    [
      "0.525608369",
      "0.301271086",
      "0.155447150"
    ],
    [
      "1.314425265",
      "0.213874230",
      "0.153325934"
    ],
    [
      "0.592895198",
      "0.601356023",
      "0.152089004"
    ],
    [
      "1.242569779",
      "0.730054911",
      "0.150704822"
    ],
    [
      "0.815922026",
      "0.397164075",
      "0.150293728"
    ],
    [
      "0.304247017",
      "0.511892782",
      "0.150105376"
    ],
    [
      "1.819354406",
      "1.232615116",
      "0.148265631"
    ],
    [
      "1.373881834",
      "1.492763439",
      "0.147583541"
    ],
    [
      "0.184006440",
      "1.246821323",
      "0.147494131"
    ],
    [
      "0.719363687",
      "1.805792647",
      "0.146736540"
    ],
    [
      "1.079723002",
      "1.482445505",
      "0.146756190"
    ],
    [
      "1.695786067",
      "0.504560913",
      "0.145846536"
    ],
    [
      "1.525243655",
      "1.241239705",
      "0.145971546"
    ],
    [
      "0.146162599",
      "0.956718804",
      "0.145066336"
    ],
    [
      "1.234810791",
      "1.235901413",
      "0.144510373"
    ],
    [
      "1.383164967",
      "0.988761932",
      "0.143737582"
    ],
    [
      "1.823064186",
      "0.765023861",
      "0.144051146"
    ],
    [
      "1.536252548",
      "0.745835772",
      "0.143401628"
    ],
    [
      "0.456069407",
      "1.349022591",
      "0.143131670"
    ],
    [
      "1.097290812",
      "0.984357827",
      "0.142170494"
    ],
    [
      "0.411305635",
      "1.067236430",
      "0.142187862"
    ],
    [
      "0.949505605",
      "1.225871136",
      "0.140971071"
    ],
    [
      "0.632914380",
      "0.891401926",
      "0.140704712"
    ],
    [
      "1.409790772",
      "0.491726430",
      "0.140436596"
    ],
    [
      "0.674444573",
      "1.168937041",
      "0.139920477"
    ],
    [
      "0.374118413",
      "0.790452193",
      "0.137083337"
    ],
    [
      "1.662995640",
      "0.995019637",
      "0.136163050"
    ],
    [
      "1.510969244",
      "1.719587832",
      "0.117449028"
    ],
    [
      "0.741913933",
      "0.147927376",
      "0.109698829"
    ],
    [
      "0.251744057",
      "1.490421863",
      "0.105348917"
    ],
    [
      "0.851645392",
      "1.001236490",
      "0.104054122"
    ],
    [
      "0.942140787",
      "1.691607680",
      "0.103598920"
    ],
    [
      "1.557130700",
      "0.297473013",
      "0.103373665"
    ],
    [
      "0.950630101",
      "1.896836621",
      "0.101805526"
    ],
    [
      "0.149473545",
      "0.710213288",
      "0.101461414"
    ],
    [
      "1.899572085",
      "0.997218716",
      "0.100423615"
    ]
  ]
}

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": 43
}

起步布局答案

{
  "circles": [
    [
      "1.215414152",
      "1.785611256",
      "0.185390705"
    ],
    [
      "0.907961006",
      "0.718640428",
      "0.184098584"
    ],
    [
      "0.752256227",
      "1.479992528",
      "0.180719771"
    ],
    [
      "1.008853250",
      "0.157457050",
      "0.157410537"
    ],
    [
      "1.677715058",
      "1.502355906",
      "0.156401069"
    ],
    [
      "0.460319877",
      "1.648464725",
      "0.156340628"
    ],
    [
      "1.116801677",
      "0.451016384",
      "0.155367252"
    ],
    [
      "0.525608369",
      "0.301271086",
      "0.155447150"
    ],
    [
      "1.314425265",
      "0.213874230",
      "0.153325934"
    ],
    [
      "0.592895198",
      "0.601356023",
      "0.152089004"
    ],
    [
      "1.242569779",
      "0.730054911",
      "0.150704822"
    ],
    [
      "0.815922026",
      "0.397164075",
      "0.150293728"
    ],
    [
      "0.304247017",
      "0.511892782",
      "0.150105376"
    ],
    [
      "1.819354406",
      "1.232615116",
      "0.148265631"
    ],
    [
      "1.373881834",
      "1.492763439",
      "0.147583541"
    ],
    [
      "0.184006440",
      "1.246821323",
      "0.147494131"
    ],
    [
      "0.719363687",
      "1.805792647",
      "0.146736540"
    ],
    [
      "1.079723002",
      "1.482445505",
      "0.146756190"
    ],
    [
      "1.695786067",
      "0.504560913",
      "0.145846536"
    ],
    [
      "1.525243655",
      "1.241239705",
      "0.145971546"
    ],
    [
      "0.146162599",
      "0.956718804",
      "0.145066336"
    ],
    [
      "1.234810791",
      "1.235901413",
      "0.144510373"
    ],
    [
      "1.383164967",
      "0.988761932",
      "0.143737582"
    ],
    [
      "1.823064186",
      "0.765023861",
      "0.144051146"
    ],
    [
      "1.536252548",
      "0.745835772",
      "0.143401628"
    ],
    [
      "0.456069407",
      "1.349022591",
      "0.143131670"
    ],
    [
      "1.097290812",
      "0.984357827",
      "0.142170494"
    ],
    [
      "0.411305635",
      "1.067236430",
      "0.142187862"
    ],
    [
      "0.949505605",
      "1.225871136",
      "0.140971071"
    ],
    [
      "0.632914380",
      "0.891401926",
      "0.140704712"
    ],
    [
      "1.409790772",
      "0.491726430",
      "0.140436596"
    ],
    [
      "0.674444573",
      "1.168937041",
      "0.139920477"
    ],
    [
      "0.374118413",
      "0.790452193",
      "0.137083337"
    ],
    [
      "1.662995640",
      "0.995019637",
      "0.136163050"
    ],
    [
      "1.510969244",
      "1.719587832",
      "0.117449028"
    ],
    [
      "0.741913933",
      "0.147927376",
      "0.109698829"
    ],
    [
      "0.251744057",
      "1.490421863",
      "0.105348917"
    ],
    [
      "0.851645392",
      "1.001236490",
      "0.104054122"
    ],
    [
      "0.942140787",
      "1.691607680",
      "0.103598920"
    ],
    [
      "1.557130700",
      "0.297473013",
      "0.103373665"
    ],
    [
      "0.950630101",
      "1.896836621",
      "0.101805526"
    ],
    [
      "0.149473545",
      "0.710213288",
      "0.101461414"
    ],
    [
      "1.899572085",
      "0.997218716",
      "0.100423615"
    ]
  ]
}

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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