P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 39

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 = 39

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	[5.763365, 5.763375]	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

123456789101112131415161718192021222324252627282930313233343536373839

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": [
    [
      "1.332203895",
      "1.051695385",
      "0.193441828"
    ],
    [
      "0.979540466",
      "1.000116757",
      "0.162973447"
    ],
    [
      "1.693499471",
      "1.469485008",
      "0.162528992"
    ],
    [
      "1.168895297",
      "0.735789731",
      "0.162178868"
    ],
    [
      "1.205237813",
      "0.186839014",
      "0.161338358"
    ],
    [
      "1.464830008",
      "1.698414551",
      "0.161042419"
    ],
    [
      "1.085638703",
      "1.305916306",
      "0.160708803"
    ],
    [
      "0.969298163",
      "1.839203506",
      "0.160235075"
    ],
    [
      "0.883986983",
      "0.168490174",
      "0.160436058"
    ],
    [
      "1.642892224",
      "0.458375979",
      "0.159365125"
    ],
    [
      "0.657966457",
      "1.768208882",
      "0.159088690"
    ],
    [
      "1.489605382",
      "0.736876245",
      "0.158533056"
    ],
    [
      "0.422647165",
      "0.387257695",
      "0.158103670"
    ],
    [
      "0.277248339",
      "1.433631698",
      "0.157143895"
    ],
    [
      "0.236895430",
      "0.641798987",
      "0.157007383"
    ],
    [
      "0.168164326",
      "1.139520661",
      "0.156544841"
    ],
    [
      "1.028149568",
      "0.450306795",
      "0.156113241"
    ],
    [
      "0.852071290",
      "0.707857757",
      "0.155874030"
    ],
    [
      "0.775757174",
      "1.244473381",
      "0.155205417"
    ],
    [
      "0.872813023",
      "1.539100923",
      "0.154996494"
    ],
    [
      "0.665670910",
      "0.955469081",
      "0.154055751"
    ],
    [
      "1.801574629",
      "0.728083375",
      "0.153560080"
    ],
    [
      "1.386935537",
      "1.393984318",
      "0.153195253"
    ],
    [
      "0.545238580",
      "0.673102977",
      "0.152920734"
    ],
    [
      "0.472409650",
      "1.193332798",
      "0.152422734"
    ],
    [
      "1.333679883",
      "0.470543231",
      "0.150086508"
    ],
    [
      "1.169695772",
      "1.603862642",
      "0.148867697"
    ],
    [
      "0.724992789",
      "0.432296080",
      "0.147578079"
    ],
    [
      "0.578494750",
      "1.472969092",
      "0.146660038"
    ],
    [
      "0.369010603",
      "0.913930297",
      "0.145498597"
    ],
    [
      "1.834698740",
      "1.200048846",
      "0.141663511"
    ],
    [
      "1.657341553",
      "0.982574749",
      "0.138961780"
    ],
    [
      "1.471752739",
      "0.247725452",
      "0.112042994"
    ],
    [
      "1.241030215",
      "1.854725180",
      "0.111939924"
    ],
    [
      "0.614433125",
      "0.198754422",
      "0.110811442"
    ],
    [
      "0.409236394",
      "1.665783207",
      "0.109905107"
    ],
    [
      "1.584168442",
      "1.220201450",
      "0.109676015"
    ],
    [
      "0.117072852",
      "0.879105777",
      "0.108834604"
    ],
    [
      "1.897477354",
      "0.964785309",
      "0.101832044"
    ]
  ]
}

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": 39
}

起步布局答案

{
  "circles": [
    [
      "1.332203895",
      "1.051695385",
      "0.193441828"
    ],
    [
      "0.979540466",
      "1.000116757",
      "0.162973447"
    ],
    [
      "1.693499471",
      "1.469485008",
      "0.162528992"
    ],
    [
      "1.168895297",
      "0.735789731",
      "0.162178868"
    ],
    [
      "1.205237813",
      "0.186839014",
      "0.161338358"
    ],
    [
      "1.464830008",
      "1.698414551",
      "0.161042419"
    ],
    [
      "1.085638703",
      "1.305916306",
      "0.160708803"
    ],
    [
      "0.969298163",
      "1.839203506",
      "0.160235075"
    ],
    [
      "0.883986983",
      "0.168490174",
      "0.160436058"
    ],
    [
      "1.642892224",
      "0.458375979",
      "0.159365125"
    ],
    [
      "0.657966457",
      "1.768208882",
      "0.159088690"
    ],
    [
      "1.489605382",
      "0.736876245",
      "0.158533056"
    ],
    [
      "0.422647165",
      "0.387257695",
      "0.158103670"
    ],
    [
      "0.277248339",
      "1.433631698",
      "0.157143895"
    ],
    [
      "0.236895430",
      "0.641798987",
      "0.157007383"
    ],
    [
      "0.168164326",
      "1.139520661",
      "0.156544841"
    ],
    [
      "1.028149568",
      "0.450306795",
      "0.156113241"
    ],
    [
      "0.852071290",
      "0.707857757",
      "0.155874030"
    ],
    [
      "0.775757174",
      "1.244473381",
      "0.155205417"
    ],
    [
      "0.872813023",
      "1.539100923",
      "0.154996494"
    ],
    [
      "0.665670910",
      "0.955469081",
      "0.154055751"
    ],
    [
      "1.801574629",
      "0.728083375",
      "0.153560080"
    ],
    [
      "1.386935537",
      "1.393984318",
      "0.153195253"
    ],
    [
      "0.545238580",
      "0.673102977",
      "0.152920734"
    ],
    [
      "0.472409650",
      "1.193332798",
      "0.152422734"
    ],
    [
      "1.333679883",
      "0.470543231",
      "0.150086508"
    ],
    [
      "1.169695772",
      "1.603862642",
      "0.148867697"
    ],
    [
      "0.724992789",
      "0.432296080",
      "0.147578079"
    ],
    [
      "0.578494750",
      "1.472969092",
      "0.146660038"
    ],
    [
      "0.369010603",
      "0.913930297",
      "0.145498597"
    ],
    [
      "1.834698740",
      "1.200048846",
      "0.141663511"
    ],
    [
      "1.657341553",
      "0.982574749",
      "0.138961780"
    ],
    [
      "1.471752739",
      "0.247725452",
      "0.112042994"
    ],
    [
      "1.241030215",
      "1.854725180",
      "0.111939924"
    ],
    [
      "0.614433125",
      "0.198754422",
      "0.110811442"
    ],
    [
      "0.409236394",
      "1.665783207",
      "0.109905107"
    ],
    [
      "1.584168442",
      "1.220201450",
      "0.109676015"
    ],
    [
      "0.117072852",
      "0.879105777",
      "0.108834604"
    ],
    [
      "1.897477354",
      "0.964785309",
      "0.101832044"
    ]
  ]
}

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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