文献中也称packing circles with radii 1 to ngraduated circle packing
将半径分别为 1, 2, …, n 的 n 个圆互不重叠地放入一个圆内，使外围容器圆的半径尽可能小。
| 纪录 | 数值 / 区间 | 作者 / 持有人 | 来源 |
|---|---|---|---|
| 外部已知最好 | [484.16788596315, 484.16788596325] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 484.167885968 | zzzcy | 纪录详情 ↓ |
本站纪录尚未确认达到或超越此参考纪录。
当前展示：本站纪录
半径最大的几个圆决定了容器半径的下界与主体骨架，而半径较小的圆可以填入大圆之间及靠近边界的空隙。每当 n 增加 1 时，新增的最大圆往往会改变大圆之间的接触关系，使整体排布发生重排。
将半径为 1..n 的圆装入最小圆曾是 2005 年 Al Zimmermann 编程竞赛的赛题。本站已导入 Packomania 收录的 n = 1..200 公开已知最优构型；n ≥ 5 的最优性尚未给出数学证明，较大规模的子题合并为同一张数据表供算法挑战。
查看来源
查看构型、求解笔记与纪录历史。
纪录保持者暂未分享求解过程。
单位就是最小那个圆的半径：第 i 个圆的半径正好是 i。容器是你自己给出的圆，圆心在 (radius, radius)，所以坐标范围是 0 到 2·radius，radius 越小越好。坐标和半径用同一个单位，直接写成小数，例如 "7.5"，最多九位小数。
提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。
{
"centers": [
[
"364.194696311",
"605.096881133"
],
[
"936.872618764",
"321.255577125"
],
[
"402.712747652",
"207.314502398"
],
[
"907.172719113",
"556.200035717"
],
[
"302.736375323",
"807.524611276"
],
[
"89.590981079",
"575.058178581"
],
[
"182.259416437",
"344.406319894"
],
[
"647.561475514",
"575.815893571"
],
[
"275.896850205",
"274.042817839"
],
[
"746.080896084",
"878.212677741"
],
[
"760.229578693",
"641.311497123"
],
[
"758.503571224",
"100.674713805"
],
[
"313.704992484",
"923.225475696"
],
[
"552.872629522",
"19.215457054"
],
[
"21.555380905",
"561.815612525"
],
[
"884.139149570",
"727.261361293"
],
[
"300.660417202",
"420.752722523"
],
[
"88.590233695",
"237.587270065"
],
[
"606.311104067",
"933.013289652"
],
[
"300.159762490",
"732.864064120"
],
[
"860.058019261",
"213.651680415"
],
[
"31.853968722",
"389.319311953"
],
[
"934.157924470",
"383.258359619"
],
[
"937.653926648",
"562.301708246"
],
[
"624.289590151",
"282.548690475"
],
[
"459.820133057",
"539.119248961"
],
[
"369.101830221",
"226.247798618"
],
[
"427.158861327",
"407.137732497"
],
[
"347.812800440",
"49.904031766"
],
[
"405.359554631",
"36.889767169"
],
[
"625.340256676",
"746.834083968"
],
[
"797.242526613",
"530.696041770"
],
[
"451.413383003",
"934.145223024"
],
[
"272.580231971",
"779.290021260"
],
[
"330.956937302",
"687.295061238"
],
[
"128.617651036",
"757.002056211"
],
[
"211.901288702",
"129.442614450"
],
[
"451.901495553",
"863.146900887"
],
[
"55.609552608",
"604.634485508"
],
[
"523.205115741",
"557.513377651"
],
[
"135.507699659",
"600.598881726"
],
[
"773.459809329",
"818.566783184"
],
[
"831.369000598",
"756.345198773"
],
[
"700.384110951",
"100.764297186"
],
[
"313.846787846",
"609.146211050"
],
[
"286.567243285",
"93.193542376"
],
[
"94.504592290",
"681.336349163"
],
[
"220.999712782",
"294.586728780"
],
[
"195.338030878",
"809.665049990"
],
[
"374.029245255",
"904.133638176"
],
[
"550.021901251",
"266.168600461"
],
[
"279.842103010",
"864.982875274"
],
[
"448.091813617",
"243.715732019"
],
[
"481.473325264",
"345.705338703"
],
[
"488.403290264",
"55.020899848"
],
[
"536.077188958",
"906.703649431"
],
[
"521.908160890",
"161.891960685"
],
[
"159.186781166",
"208.475396859"
],
[
"600.593878974",
"75.251367092"
],
[
"707.915045689",
"298.351912248"
],
[
"500.169424378",
"459.175404703"
],
[
"717.334879272",
"582.182875494"
],
[
"282.398029651",
"202.113777677"
],
[
"615.013776090",
"508.654705084"
],
[
"639.914431199",
"192.851937555"
],
[
"225.231363257",
"542.300818345"
],
[
"217.413385228",
"409.530794546"
],
[
"394.769248763",
"134.780954213"
],
[
"773.772651156",
"186.689393442"
],
[
"602.502648317",
"375.198092754"
],
[
"377.508014792",
"783.183656106"
],
[
"723.627685072",
"716.035034967"
],
[
"104.089675386",
"327.325339790"
],
[
"595.837105044",
"645.395619945"
],
[
"523.309296596",
"775.568787467"
],
[
"217.473792664",
"684.088758942"
],
[
"889.943436762",
"473.280872736"
],
[
"657.887099138",
"850.861498846"
],
[
"738.380667427",
"436.338661761"
],
[
"363.666948975",
"494.503498481"
],
[
"438.569583379",
"643.987805443"
],
[
"346.087918102",
"332.790559278"
],
[
"849.517808512",
"318.472059742"
],
[
"854.747706919",
"631.515570516"
],
[
"85.000000013",
"484.167885982"
]
],
"radius": "484.167885968"
}{
"n": 85
}{
"centers": [
[
"364.194696311",
"605.096881133"
],
[
"936.872618764",
"321.255577125"
],
[
"402.712747652",
"207.314502398"
],
[
"907.172719113",
"556.200035717"
],
[
"302.736375323",
"807.524611276"
],
[
"89.590981079",
"575.058178581"
],
[
"182.259416437",
"344.406319894"
],
[
"647.561475514",
"575.815893571"
],
[
"275.896850205",
"274.042817839"
],
[
"746.080896084",
"878.212677741"
],
[
"760.229578693",
"641.311497123"
],
[
"758.503571224",
"100.674713805"
],
[
"313.704992484",
"923.225475696"
],
[
"552.872629522",
"19.215457054"
],
[
"21.555380905",
"561.815612525"
],
[
"884.139149570",
"727.261361293"
],
[
"300.660417202",
"420.752722523"
],
[
"88.590233695",
"237.587270065"
],
[
"606.311104067",
"933.013289652"
],
[
"300.159762490",
"732.864064120"
],
[
"860.058019261",
"213.651680415"
],
[
"31.853968722",
"389.319311953"
],
[
"934.157924470",
"383.258359619"
],
[
"937.653926648",
"562.301708246"
],
[
"624.289590151",
"282.548690475"
],
[
"459.820133057",
"539.119248961"
],
[
"369.101830221",
"226.247798618"
],
[
"427.158861327",
"407.137732497"
],
[
"347.812800440",
"49.904031766"
],
[
"405.359554631",
"36.889767169"
],
[
"625.340256676",
"746.834083968"
],
[
"797.242526613",
"530.696041770"
],
[
"451.413383003",
"934.145223024"
],
[
"272.580231971",
"779.290021260"
],
[
"330.956937302",
"687.295061238"
],
[
"128.617651036",
"757.002056211"
],
[
"211.901288702",
"129.442614450"
],
[
"451.901495553",
"863.146900887"
],
[
"55.609552608",
"604.634485508"
],
[
"523.205115741",
"557.513377651"
],
[
"135.507699659",
"600.598881726"
],
[
"773.459809329",
"818.566783184"
],
[
"831.369000598",
"756.345198773"
],
[
"700.384110951",
"100.764297186"
],
[
"313.846787846",
"609.146211050"
],
[
"286.567243285",
"93.193542376"
],
[
"94.504592290",
"681.336349163"
],
[
"220.999712782",
"294.586728780"
],
[
"195.338030878",
"809.665049990"
],
[
"374.029245255",
"904.133638176"
],
[
"550.021901251",
"266.168600461"
],
[
"279.842103010",
"864.982875274"
],
[
"448.091813617",
"243.715732019"
],
[
"481.473325264",
"345.705338703"
],
[
"488.403290264",
"55.020899848"
],
[
"536.077188958",
"906.703649431"
],
[
"521.908160890",
"161.891960685"
],
[
"159.186781166",
"208.475396859"
],
[
"600.593878974",
"75.251367092"
],
[
"707.915045689",
"298.351912248"
],
[
"500.169424378",
"459.175404703"
],
[
"717.334879272",
"582.182875494"
],
[
"282.398029651",
"202.113777677"
],
[
"615.013776090",
"508.654705084"
],
[
"639.914431199",
"192.851937555"
],
[
"225.231363257",
"542.300818345"
],
[
"217.413385228",
"409.530794546"
],
[
"394.769248763",
"134.780954213"
],
[
"773.772651156",
"186.689393442"
],
[
"602.502648317",
"375.198092754"
],
[
"377.508014792",
"783.183656106"
],
[
"723.627685072",
"716.035034967"
],
[
"104.089675386",
"327.325339790"
],
[
"595.837105044",
"645.395619945"
],
[
"523.309296596",
"775.568787467"
],
[
"217.473792664",
"684.088758942"
],
[
"889.943436762",
"473.280872736"
],
[
"657.887099138",
"850.861498846"
],
[
"738.380667427",
"436.338661761"
],
[
"363.666948975",
"494.503498481"
],
[
"438.569583379",
"643.987805443"
],
[
"346.087918102",
"332.790559278"
],
[
"849.517808512",
"318.472059742"
],
[
"854.747706919",
"631.515570516"
],
[
"85.000000013",
"484.167885982"
]
],
"radius": "484.167885968"
}提交容器圆半径 radius 与各圆圆心坐标 centers（按半径 1, 2, …, n 的顺序排列）。容器圆的圆心固定在 (radius, radius)。所有数值均以十进制字符串表示（如 "7.5"）。 · 验证器 v1.0.0
欢迎交流解题思路、分享构造方法或探讨卡壳之处。所有登录用户均可发帖；发言使用与排行榜一致的公开署名，并附带唯一注册编号（#UID）以防冒名。新发言经审核后公开。
按选手与子题归档，默认展示最新一篇。历史版本与回复完整保留，点击即可展开查看或回复。求解笔记可在提交破纪录答案时一并发布。