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 | [878.80376460205, 878.80376460215] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 878.803764616 | 游侠 | 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": [
[
"494.834394681",
"598.707208042"
],
[
"1671.208967735",
"1248.805141033"
],
[
"1206.621086544",
"615.739451268"
],
[
"1138.175420842",
"1328.196166481"
],
[
"559.790581810",
"842.465763097"
],
[
"1394.898278934",
"1238.070755930"
],
[
"1553.384600794",
"999.652380519"
],
[
"1004.481193960",
"124.467126968"
],
[
"806.146715042",
"12.154190396"
],
[
"1503.192419159",
"274.761060868"
],
[
"931.295294823",
"1323.547323372"
],
[
"1017.607028032",
"868.904084809"
],
[
"922.205684245",
"944.054719368"
],
[
"106.030957919",
"1266.756027168"
],
[
"1540.152518208",
"764.535846685"
],
[
"248.890481992",
"771.185989331"
],
[
"782.013225024",
"22.679733365"
],
[
"1451.275744487",
"671.629263422"
],
[
"491.474706122",
"1646.246143975"
],
[
"982.529140028",
"26.334345434"
],
[
"529.424401395",
"95.564857061"
],
[
"1313.003493116",
"1617.279522612"
],
[
"913.133166796",
"1733.727834920"
],
[
"56.038313770",
"1110.065711076"
],
[
"670.030932643",
"1706.206649017"
],
[
"745.083733388",
"1112.041094874"
],
[
"61.960845898",
"638.063403353"
],
[
"1476.008891175",
"1484.537836224"
],
[
"150.992309880",
"440.191510377"
],
[
"30.106407275",
"877.134226514"
],
[
"1624.118843523",
"474.955889468"
],
[
"1628.558809577",
"1272.000312168"
],
[
"1712.755958682",
"1018.650321348"
],
[
"324.631206801",
"241.232319714"
],
[
"1710.318943429",
"736.041563689"
],
[
"1001.810930371",
"337.311106613"
],
[
"930.505940492",
"1102.872813792"
],
[
"475.984921626",
"1283.885430569"
],
[
"293.195484045",
"556.281521650"
],
[
"1225.407124101",
"462.504857762"
],
[
"1297.022644721",
"739.041702240"
],
[
"1261.086878791",
"134.589606869"
],
[
"410.481917211",
"1014.251679648"
],
[
"653.087546830",
"688.169133168"
],
[
"899.136762927",
"1490.385218848"
],
[
"1109.842100364",
"1678.768404949"
],
[
"961.669340044",
"1563.396785085"
],
[
"161.982674671",
"1298.806645927"
],
[
"993.556771797",
"1162.766820906"
],
[
"916.571750041",
"1382.972315329"
],
[
"244.415686975",
"1243.981483065"
],
[
"336.340854074",
"1502.434655249"
],
[
"1322.499758112",
"1236.605116875"
],
[
"1438.899006651",
"273.333228608"
],
[
"556.475644885",
"1486.528828428"
],
[
"1043.441530555",
"72.658970965"
],
[
"721.805689611",
"72.172963338"
],
[
"671.785454530",
"1159.386630933"
],
[
"609.615795915",
"104.551564178"
],
[
"1103.699583092",
"1167.226884880"
],
[
"672.773301201",
"1498.510708441"
],
[
"1304.191835735",
"1350.144114663"
],
[
"1155.950057347",
"111.548218397"
],
[
"1340.853713423",
"207.674105281"
],
[
"1073.745253897",
"1569.477862675"
],
[
"1005.249709987",
"1457.811572037"
],
[
"484.316042894",
"425.406440136"
],
[
"610.355240668",
"377.041983924"
],
[
"1078.689531712",
"811.525132353"
],
[
"1414.792195238",
"1154.724985332"
],
[
"992.526229840",
"1678.562555146"
],
[
"1207.941247363",
"236.488621874"
],
[
"367.431574244",
"1259.572874264"
],
[
"450.324130163",
"1560.063114936"
],
[
"1197.100946639",
"1264.700908539"
],
[
"656.363349081",
"1026.277057032"
],
[
"134.462706838",
"1176.873666915"
],
[
"1211.697550638",
"1607.136378639"
],
[
"1178.800716854",
"702.517431308"
],
[
"792.912648405",
"1424.495633218"
],
[
"418.223466695",
"557.829139747"
],
[
"1137.334785286",
"546.937535661"
],
[
"889.153003250",
"298.940183868"
],
[
"595.871860416",
"1621.543666488"
],
[
"436.529262747",
"1401.662667861"
],
[
"266.602206059",
"1382.536889284"
],
[
"301.329379578",
"682.018706857"
],
[
"1130.325827398",
"376.810359585"
],
[
"517.561408246",
"937.063710378"
],
[
"1046.822687858",
"219.612313517"
],
[
"1011.834586222",
"666.162082494"
],
[
"1034.442349207",
"1302.531857277"
],
[
"1313.757226877",
"363.090274628"
],
[
"160.363058595",
"562.955856780"
],
[
"764.870023857",
"428.947492080"
],
[
"1351.758600974",
"1502.536858116"
],
[
"1427.440707965",
"784.154103864"
],
[
"1169.370358132",
"1436.301930195"
],
[
"830.618950698",
"1010.152997189"
],
[
"909.600924917",
"827.498000681"
],
[
"295.377730082",
"1100.779282793"
],
[
"1426.924930908",
"983.153435458"
],
[
"1502.195054707",
"417.008592889"
],
[
"603.044017440",
"548.886524179"
],
[
"1251.048535619",
"1092.975378567"
],
[
"711.200152568",
"235.245285286"
],
[
"961.636419248",
"474.631030094"
],
[
"883.269701628",
"108.012937639"
],
[
"1243.358286980",
"879.113600595"
],
[
"117.657279565",
"987.040804393"
],
[
"504.325253008",
"1136.625240227"
],
[
"120.483189645",
"765.058791082"
],
[
"493.919371637",
"736.452010555"
],
[
"695.980341299",
"840.235592925"
],
[
"622.876416832",
"1329.035275197"
],
[
"318.930518266",
"884.254212934"
],
[
"805.730616549",
"636.972519338"
],
[
"1599.842102208",
"1121.550719231"
],
[
"1048.224227083",
"997.040267085"
],
[
"835.567175493",
"1229.097088392"
],
[
"1486.274560963",
"1331.844355789"
],
[
"296.573187562",
"395.316956855"
],
[
"482.200217864",
"235.418221329"
],
[
"803.705447378",
"1629.862328128"
],
[
"1342.004740128",
"579.252431344"
],
[
"1588.933708009",
"628.945861968"
],
[
"1630.607463774",
"878.490042590"
]
],
"radius": "878.803764616"
}{
"n": 127
}{
"centers": [
[
"494.834394681",
"598.707208042"
],
[
"1671.208967735",
"1248.805141033"
],
[
"1206.621086544",
"615.739451268"
],
[
"1138.175420842",
"1328.196166481"
],
[
"559.790581810",
"842.465763097"
],
[
"1394.898278934",
"1238.070755930"
],
[
"1553.384600794",
"999.652380519"
],
[
"1004.481193960",
"124.467126968"
],
[
"806.146715042",
"12.154190396"
],
[
"1503.192419159",
"274.761060868"
],
[
"931.295294823",
"1323.547323372"
],
[
"1017.607028032",
"868.904084809"
],
[
"922.205684245",
"944.054719368"
],
[
"106.030957919",
"1266.756027168"
],
[
"1540.152518208",
"764.535846685"
],
[
"248.890481992",
"771.185989331"
],
[
"782.013225024",
"22.679733365"
],
[
"1451.275744487",
"671.629263422"
],
[
"491.474706122",
"1646.246143975"
],
[
"982.529140028",
"26.334345434"
],
[
"529.424401395",
"95.564857061"
],
[
"1313.003493116",
"1617.279522612"
],
[
"913.133166796",
"1733.727834920"
],
[
"56.038313770",
"1110.065711076"
],
[
"670.030932643",
"1706.206649017"
],
[
"745.083733388",
"1112.041094874"
],
[
"61.960845898",
"638.063403353"
],
[
"1476.008891175",
"1484.537836224"
],
[
"150.992309880",
"440.191510377"
],
[
"30.106407275",
"877.134226514"
],
[
"1624.118843523",
"474.955889468"
],
[
"1628.558809577",
"1272.000312168"
],
[
"1712.755958682",
"1018.650321348"
],
[
"324.631206801",
"241.232319714"
],
[
"1710.318943429",
"736.041563689"
],
[
"1001.810930371",
"337.311106613"
],
[
"930.505940492",
"1102.872813792"
],
[
"475.984921626",
"1283.885430569"
],
[
"293.195484045",
"556.281521650"
],
[
"1225.407124101",
"462.504857762"
],
[
"1297.022644721",
"739.041702240"
],
[
"1261.086878791",
"134.589606869"
],
[
"410.481917211",
"1014.251679648"
],
[
"653.087546830",
"688.169133168"
],
[
"899.136762927",
"1490.385218848"
],
[
"1109.842100364",
"1678.768404949"
],
[
"961.669340044",
"1563.396785085"
],
[
"161.982674671",
"1298.806645927"
],
[
"993.556771797",
"1162.766820906"
],
[
"916.571750041",
"1382.972315329"
],
[
"244.415686975",
"1243.981483065"
],
[
"336.340854074",
"1502.434655249"
],
[
"1322.499758112",
"1236.605116875"
],
[
"1438.899006651",
"273.333228608"
],
[
"556.475644885",
"1486.528828428"
],
[
"1043.441530555",
"72.658970965"
],
[
"721.805689611",
"72.172963338"
],
[
"671.785454530",
"1159.386630933"
],
[
"609.615795915",
"104.551564178"
],
[
"1103.699583092",
"1167.226884880"
],
[
"672.773301201",
"1498.510708441"
],
[
"1304.191835735",
"1350.144114663"
],
[
"1155.950057347",
"111.548218397"
],
[
"1340.853713423",
"207.674105281"
],
[
"1073.745253897",
"1569.477862675"
],
[
"1005.249709987",
"1457.811572037"
],
[
"484.316042894",
"425.406440136"
],
[
"610.355240668",
"377.041983924"
],
[
"1078.689531712",
"811.525132353"
],
[
"1414.792195238",
"1154.724985332"
],
[
"992.526229840",
"1678.562555146"
],
[
"1207.941247363",
"236.488621874"
],
[
"367.431574244",
"1259.572874264"
],
[
"450.324130163",
"1560.063114936"
],
[
"1197.100946639",
"1264.700908539"
],
[
"656.363349081",
"1026.277057032"
],
[
"134.462706838",
"1176.873666915"
],
[
"1211.697550638",
"1607.136378639"
],
[
"1178.800716854",
"702.517431308"
],
[
"792.912648405",
"1424.495633218"
],
[
"418.223466695",
"557.829139747"
],
[
"1137.334785286",
"546.937535661"
],
[
"889.153003250",
"298.940183868"
],
[
"595.871860416",
"1621.543666488"
],
[
"436.529262747",
"1401.662667861"
],
[
"266.602206059",
"1382.536889284"
],
[
"301.329379578",
"682.018706857"
],
[
"1130.325827398",
"376.810359585"
],
[
"517.561408246",
"937.063710378"
],
[
"1046.822687858",
"219.612313517"
],
[
"1011.834586222",
"666.162082494"
],
[
"1034.442349207",
"1302.531857277"
],
[
"1313.757226877",
"363.090274628"
],
[
"160.363058595",
"562.955856780"
],
[
"764.870023857",
"428.947492080"
],
[
"1351.758600974",
"1502.536858116"
],
[
"1427.440707965",
"784.154103864"
],
[
"1169.370358132",
"1436.301930195"
],
[
"830.618950698",
"1010.152997189"
],
[
"909.600924917",
"827.498000681"
],
[
"295.377730082",
"1100.779282793"
],
[
"1426.924930908",
"983.153435458"
],
[
"1502.195054707",
"417.008592889"
],
[
"603.044017440",
"548.886524179"
],
[
"1251.048535619",
"1092.975378567"
],
[
"711.200152568",
"235.245285286"
],
[
"961.636419248",
"474.631030094"
],
[
"883.269701628",
"108.012937639"
],
[
"1243.358286980",
"879.113600595"
],
[
"117.657279565",
"987.040804393"
],
[
"504.325253008",
"1136.625240227"
],
[
"120.483189645",
"765.058791082"
],
[
"493.919371637",
"736.452010555"
],
[
"695.980341299",
"840.235592925"
],
[
"622.876416832",
"1329.035275197"
],
[
"318.930518266",
"884.254212934"
],
[
"805.730616549",
"636.972519338"
],
[
"1599.842102208",
"1121.550719231"
],
[
"1048.224227083",
"997.040267085"
],
[
"835.567175493",
"1229.097088392"
],
[
"1486.274560963",
"1331.844355789"
],
[
"296.573187562",
"395.316956855"
],
[
"482.200217864",
"235.418221329"
],
[
"803.705447378",
"1629.862328128"
],
[
"1342.004740128",
"579.252431344"
],
[
"1588.933708009",
"628.945861968"
],
[
"1630.607463774",
"878.490042590"
]
],
"radius": "878.803764616"
}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。