P92 · 装箱与覆盖 · 经典问题 · 难

单位圆内圆的半径之和 · n = 44

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 = 44

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	[6.113925, 6.113935]	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

1234567891011121314151617181920212223242526272829303132333435363738394041424344

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": [
    [
      "1.183909821",
      "1.776324589",
      "0.202188812"
    ],
    [
      "0.933764218",
      "0.713391747",
      "0.184626460"
    ],
    [
      "0.690485212",
      "1.194504066",
      "0.164964770"
    ],
    [
      "0.563775072",
      "0.279072524",
      "0.157368043"
    ],
    [
      "1.677949912",
      "1.500652004",
      "0.157225705"
    ],
    [
      "1.054204120",
      "0.157803051",
      "0.156060554"
    ],
    [
      "0.619263727",
      "0.586117931",
      "0.154650982"
    ],
    [
      "1.149287882",
      "0.452603373",
      "0.153694431"
    ],
    [
      "0.329429931",
      "0.482694705",
      "0.153082656"
    ],
    [
      "0.851804076",
      "0.388270870",
      "0.150666022"
    ],
    [
      "1.352568329",
      "0.226752286",
      "0.150166808"
    ],
    [
      "1.265963385",
      "0.730857546",
      "0.148031533"
    ],
    [
      "0.641226733",
      "1.773641287",
      "0.147216851"
    ],
    [
      "1.373460595",
      "1.482073718",
      "0.147829858"
    ],
    [
      "0.412508118",
      "1.055030389",
      "0.146040328"
    ],
    [
      "0.149450782",
      "0.927373508",
      "0.146355705"
    ],
    [
      "0.650949944",
      "0.885578008",
      "0.146480805"
    ],
    [
      "1.821549016",
      "1.233766077",
      "0.145839964"
    ],
    [
      "1.532055360",
      "1.237483703",
      "0.143677559"
    ],
    [
      "0.171847278",
      "1.216802959",
      "0.143938989"
    ],
    [
      "0.400368245",
      "1.614177673",
      "0.141645495"
    ],
    [
      "1.713820840",
      "0.522585830",
      "0.141242477"
    ],
    [
      "0.388742107",
      "0.769847209",
      "0.140131419"
    ],
    [
      "1.552946007",
      "0.753037693",
      "0.139806935"
    ],
    [
      "1.398675013",
      "0.986805265",
      "0.140276654"
    ],
    [
      "1.831221898",
      "0.777664395",
      "0.139556529"
    ],
    [
      "1.249534131",
      "1.223168568",
      "0.139206105"
    ],
    [
      "1.120021872",
      "0.977474873",
      "0.138532650"
    ],
    [
      "1.089498449",
      "1.449505472",
      "0.137993846"
    ],
    [
      "1.436073702",
      "0.501803460",
      "0.137281092"
    ],
    [
      "0.425030595",
      "1.337210513",
      "0.136417517"
    ],
    [
      "0.648339134",
      "1.491503708",
      "0.135010361"
    ],
    [
      "0.881945012",
      "1.626611635",
      "0.134852303"
    ],
    [
      "1.671684159",
      "0.998636401",
      "0.132988728"
    ],
    [
      "0.983056301",
      "1.205924039",
      "0.127829112"
    ],
    [
      "1.503335258",
      "1.718982554",
      "0.122342723"
    ],
    [
      "0.882682160",
      "1.876824595",
      "0.115361742"
    ],
    [
      "0.789141377",
      "0.135293928",
      "0.109956208"
    ],
    [
      "0.876660034",
      "0.998043732",
      "0.105696876"
    ],
    [
      "0.163202714",
      "0.678006660",
      "0.103390046"
    ],
    [
      "1.586098222",
      "0.317489924",
      "0.100371671"
    ],
    [
      "0.221277894",
      "1.454165203",
      "0.098515585"
    ],
    [
      "0.860151936",
      "1.395054793",
      "0.097727809"
    ],
    [
      "1.902300641",
      "1.004013536",
      "0.097690432"
    ]
  ]
}

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": 44
}

起步布局答案

{
  "circles": [
    [
      "1.183909821",
      "1.776324589",
      "0.202188812"
    ],
    [
      "0.933764218",
      "0.713391747",
      "0.184626460"
    ],
    [
      "0.690485212",
      "1.194504066",
      "0.164964770"
    ],
    [
      "0.563775072",
      "0.279072524",
      "0.157368043"
    ],
    [
      "1.677949912",
      "1.500652004",
      "0.157225705"
    ],
    [
      "1.054204120",
      "0.157803051",
      "0.156060554"
    ],
    [
      "0.619263727",
      "0.586117931",
      "0.154650982"
    ],
    [
      "1.149287882",
      "0.452603373",
      "0.153694431"
    ],
    [
      "0.329429931",
      "0.482694705",
      "0.153082656"
    ],
    [
      "0.851804076",
      "0.388270870",
      "0.150666022"
    ],
    [
      "1.352568329",
      "0.226752286",
      "0.150166808"
    ],
    [
      "1.265963385",
      "0.730857546",
      "0.148031533"
    ],
    [
      "0.641226733",
      "1.773641287",
      "0.147216851"
    ],
    [
      "1.373460595",
      "1.482073718",
      "0.147829858"
    ],
    [
      "0.412508118",
      "1.055030389",
      "0.146040328"
    ],
    [
      "0.149450782",
      "0.927373508",
      "0.146355705"
    ],
    [
      "0.650949944",
      "0.885578008",
      "0.146480805"
    ],
    [
      "1.821549016",
      "1.233766077",
      "0.145839964"
    ],
    [
      "1.532055360",
      "1.237483703",
      "0.143677559"
    ],
    [
      "0.171847278",
      "1.216802959",
      "0.143938989"
    ],
    [
      "0.400368245",
      "1.614177673",
      "0.141645495"
    ],
    [
      "1.713820840",
      "0.522585830",
      "0.141242477"
    ],
    [
      "0.388742107",
      "0.769847209",
      "0.140131419"
    ],
    [
      "1.552946007",
      "0.753037693",
      "0.139806935"
    ],
    [
      "1.398675013",
      "0.986805265",
      "0.140276654"
    ],
    [
      "1.831221898",
      "0.777664395",
      "0.139556529"
    ],
    [
      "1.249534131",
      "1.223168568",
      "0.139206105"
    ],
    [
      "1.120021872",
      "0.977474873",
      "0.138532650"
    ],
    [
      "1.089498449",
      "1.449505472",
      "0.137993846"
    ],
    [
      "1.436073702",
      "0.501803460",
      "0.137281092"
    ],
    [
      "0.425030595",
      "1.337210513",
      "0.136417517"
    ],
    [
      "0.648339134",
      "1.491503708",
      "0.135010361"
    ],
    [
      "0.881945012",
      "1.626611635",
      "0.134852303"
    ],
    [
      "1.671684159",
      "0.998636401",
      "0.132988728"
    ],
    [
      "0.983056301",
      "1.205924039",
      "0.127829112"
    ],
    [
      "1.503335258",
      "1.718982554",
      "0.122342723"
    ],
    [
      "0.882682160",
      "1.876824595",
      "0.115361742"
    ],
    [
      "0.789141377",
      "0.135293928",
      "0.109956208"
    ],
    [
      "0.876660034",
      "0.998043732",
      "0.105696876"
    ],
    [
      "0.163202714",
      "0.678006660",
      "0.103390046"
    ],
    [
      "1.586098222",
      "0.317489924",
      "0.100371671"
    ],
    [
      "0.221277894",
      "1.454165203",
      "0.098515585"
    ],
    [
      "0.860151936",
      "1.395054793",
      "0.097727809"
    ],
    [
      "1.902300641",
      "1.004013536",
      "0.097690432"
    ]
  ]
}

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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