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 | [1169.96467123275, 1169.96467123285] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1169.964671247 | 邱仲普 | 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。
从实时页面的 154 圆九位整数证书出发，将容器半径目标降至 1169.964671247，在每个圆心 ±32 个整数单位窗口内建立线性化修复模型并由 HiGHS 求解；独立整数检查确认全部 154 个包含约束与 11781 个成对分离约束。该解严格领先 1e-9；更低目标可扩大窗口或进行全局重启。
在讨论区查看与回复 →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": [
[
"2230.909506683",
"679.277349119"
],
[
"1529.830606335",
"1032.106215939"
],
[
"131.955640267",
"1697.144399717"
],
[
"2217.947766128",
"1212.941961957"
],
[
"1203.036900805",
"1212.590993797"
],
[
"608.313477241",
"1271.308591114"
],
[
"1105.130214541",
"389.276291541"
],
[
"1980.391231443",
"1686.486971928"
],
[
"1346.075463235",
"1067.630632887"
],
[
"1540.613172482",
"1023.142958462"
],
[
"983.541962768",
"1276.668963028"
],
[
"1749.460782459",
"1348.371423081"
],
[
"312.596073901",
"946.598245092"
],
[
"713.533013516",
"1923.220902654"
],
[
"335.560671147",
"687.353363796"
],
[
"1522.238308227",
"2268.123236749"
],
[
"517.643018941",
"981.966981637"
],
[
"705.962074191",
"2224.062687479"
],
[
"1045.241244252",
"25.946698123"
],
[
"2198.848177377",
"656.583694233"
],
[
"2314.542222827",
"1268.918999588"
],
[
"22.386804664",
"1196.725844041"
],
[
"1246.061307628",
"25.685039778"
],
[
"57.766953415",
"894.960803193"
],
[
"579.320269495",
"189.529617067"
],
[
"1860.901407376",
"258.411785181"
],
[
"1435.045871502",
"58.409621887"
],
[
"1687.138455759",
"152.024529619"
],
[
"58.613727548",
"1427.984356918"
],
[
"913.833129992",
"2280.659879262"
],
[
"272.233054453",
"1870.877606294"
],
[
"135.114397542",
"1643.302292842"
],
[
"1932.947675402",
"2012.903339854"
],
[
"2037.350024758",
"436.586737440"
],
[
"171.942508480",
"629.590021533"
],
[
"2246.160463493",
"1527.272742824"
],
[
"2122.259757193",
"1783.720396427"
],
[
"1003.273374698",
"1001.613496253"
],
[
"1038.016408421",
"932.794031961"
],
[
"1252.292369345",
"506.597007251"
],
[
"1218.447226649",
"2297.827015585"
],
[
"247.011555573",
"1403.261409909"
],
[
"1171.860154647",
"1888.110014652"
],
[
"735.730943555",
"1396.873591660"
],
[
"910.102930558",
"2077.605857002"
],
[
"601.281896624",
"606.718576327"
],
[
"537.387573330",
"2097.809414756"
],
[
"2029.202948236",
"1272.769486256"
],
[
"461.665477299",
"2038.799549710"
],
[
"1256.148333441",
"1229.303208887"
],
[
"2237.790605542",
"835.650704764"
],
[
"2284.614881986",
"1083.941919298"
],
[
"1278.061770902",
"879.567963126"
],
[
"1456.454191784",
"380.349937613"
],
[
"931.888189917",
"489.439096337"
],
[
"1700.790480802",
"2149.283261503"
],
[
"1925.427348370",
"1637.879986652"
],
[
"2193.150698375",
"734.933135135"
],
[
"2280.374667679",
"1196.124178244"
],
[
"982.084387529",
"76.028041305"
],
[
"855.786770914",
"106.612422847"
],
[
"1652.966660029",
"773.522366614"
],
[
"1693.240934017",
"655.119443937"
],
[
"1579.535435778",
"2197.190450521"
],
[
"1248.596210716",
"611.732359773"
],
[
"67.120127826",
"1120.840486425"
],
[
"1531.187645003",
"816.505364402"
],
[
"381.521039040",
"400.132533772"
],
[
"84.532204844",
"986.274082553"
],
[
"604.493436969",
"1612.210055429"
],
[
"464.253184103",
"1597.592549564"
],
[
"1929.128582942",
"1202.984410411"
],
[
"1540.088516982",
"1111.955740693"
],
[
"899.453028473",
"1583.073023919"
],
[
"1187.554170224",
"2005.788069855"
],
[
"2244.082493251",
"962.528882668"
],
[
"1523.797469018",
"1261.783732062"
],
[
"984.274073527",
"1709.867342134"
],
[
"1731.187420097",
"1874.220562502"
],
[
"1818.170956903",
"585.229066305"
],
[
"1584.168623548",
"1811.089956126"
],
[
"661.910539998",
"2131.499632514"
],
[
"734.168502176",
"1066.600970428"
],
[
"1328.673121840",
"95.777889724"
],
[
"1543.173847104",
"664.794954030"
],
[
"1288.760679747",
"1639.472819092"
],
[
"748.388058695",
"1527.260697759"
],
[
"1745.706843928",
"254.115936086"
],
[
"345.486452371",
"1490.380314090"
],
[
"421.832428854",
"932.982859784"
],
[
"498.842921980",
"507.552201925"
],
[
"1383.804510064",
"1167.110761031"
],
[
"1149.734307033",
"1753.051299733"
],
[
"827.869285721",
"2190.097368974"
],
[
"786.421505347",
"2005.698124121"
],
[
"392.803988408",
"1911.194317263"
],
[
"1804.830635133",
"2034.949291711"
],
[
"730.814499412",
"1254.958727824"
],
[
"515.545210325",
"1759.670086713"
],
[
"156.495382610",
"1513.045417867"
],
[
"346.784345248",
"1300.384747102"
],
[
"1142.036717762",
"102.365229300"
],
[
"704.896214108",
"713.793658461"
],
[
"532.643927869",
"315.504051675"
],
[
"902.962177972",
"650.280872648"
],
[
"1400.269006308",
"532.738894439"
],
[
"591.572346430",
"1952.427069412"
],
[
"1367.479081346",
"745.210247698"
],
[
"621.148775886",
"909.637771916"
],
[
"719.184757560",
"210.630228153"
],
[
"1110.255640456",
"1297.393752532"
],
[
"303.524009694",
"562.865152542"
],
[
"1097.509866676",
"517.607521717"
],
[
"1887.480284723",
"1021.154089082"
],
[
"1423.348490272",
"963.922981796"
],
[
"1490.852290910",
"1636.234013318"
],
[
"1889.650055616",
"401.382270946"
],
[
"900.218792127",
"1389.520259301"
],
[
"126.596268391",
"1296.096016001"
],
[
"1588.908054333",
"2013.352090482"
],
[
"2069.101228230",
"872.027187147"
],
[
"1660.258358370",
"957.387356496"
],
[
"1622.071044215",
"1435.974157798"
],
[
"1838.822699808",
"788.181044653"
],
[
"477.139237457",
"724.317896800"
],
[
"927.084011992",
"1147.003744465"
],
[
"2108.666019826",
"1116.850852806"
],
[
"2110.190572565",
"1619.036637213"
],
[
"1098.286128145",
"1537.095089148"
],
[
"260.441330696",
"1086.128162979"
],
[
"1723.326718338",
"1203.029339315"
],
[
"1737.570692181",
"1663.317139480"
],
[
"997.876189858",
"1920.428455428"
],
[
"288.796507288",
"1706.054258105"
],
[
"2185.963963028",
"1367.188680505"
],
[
"556.360271973",
"1411.912281031"
],
[
"866.729002502",
"891.022743549"
],
[
"1524.521839642",
"200.820565843"
],
[
"524.235171671",
"1138.795130467"
],
[
"2036.400450345",
"613.083835956"
],
[
"1117.402242098",
"770.827373626"
],
[
"1636.685755245",
"457.373203790"
],
[
"1950.170887212",
"1837.748091645"
],
[
"1178.998380441",
"1049.848451356"
],
[
"1066.869005286",
"2189.731243048"
],
[
"732.224760511",
"466.297899796"
],
[
"1265.100930368",
"318.828158173"
],
[
"1362.097215423",
"1866.994246602"
],
[
"763.525776746",
"1762.774709004"
],
[
"215.736334041",
"809.720016792"
],
[
"968.829408260",
"286.778428249"
],
[
"1348.396231016",
"1409.009428581"
],
[
"1364.092242354",
"2168.228979283"
],
[
"1899.147986948",
"1427.332441213"
]
],
"radius": "1169.964671247"
}{
"n": 154
}{
"centers": [
[
"2230.909506683",
"679.277349119"
],
[
"1529.830606335",
"1032.106215939"
],
[
"131.955640267",
"1697.144399717"
],
[
"2217.947766128",
"1212.941961957"
],
[
"1203.036900805",
"1212.590993797"
],
[
"608.313477241",
"1271.308591114"
],
[
"1105.130214541",
"389.276291541"
],
[
"1980.391231443",
"1686.486971928"
],
[
"1346.075463235",
"1067.630632887"
],
[
"1540.613172482",
"1023.142958462"
],
[
"983.541962768",
"1276.668963028"
],
[
"1749.460782459",
"1348.371423081"
],
[
"312.596073901",
"946.598245092"
],
[
"713.533013516",
"1923.220902654"
],
[
"335.560671147",
"687.353363796"
],
[
"1522.238308227",
"2268.123236749"
],
[
"517.643018941",
"981.966981637"
],
[
"705.962074191",
"2224.062687479"
],
[
"1045.241244252",
"25.946698123"
],
[
"2198.848177377",
"656.583694233"
],
[
"2314.542222827",
"1268.918999588"
],
[
"22.386804664",
"1196.725844041"
],
[
"1246.061307628",
"25.685039778"
],
[
"57.766953415",
"894.960803193"
],
[
"579.320269495",
"189.529617067"
],
[
"1860.901407376",
"258.411785181"
],
[
"1435.045871502",
"58.409621887"
],
[
"1687.138455759",
"152.024529619"
],
[
"58.613727548",
"1427.984356918"
],
[
"913.833129992",
"2280.659879262"
],
[
"272.233054453",
"1870.877606294"
],
[
"135.114397542",
"1643.302292842"
],
[
"1932.947675402",
"2012.903339854"
],
[
"2037.350024758",
"436.586737440"
],
[
"171.942508480",
"629.590021533"
],
[
"2246.160463493",
"1527.272742824"
],
[
"2122.259757193",
"1783.720396427"
],
[
"1003.273374698",
"1001.613496253"
],
[
"1038.016408421",
"932.794031961"
],
[
"1252.292369345",
"506.597007251"
],
[
"1218.447226649",
"2297.827015585"
],
[
"247.011555573",
"1403.261409909"
],
[
"1171.860154647",
"1888.110014652"
],
[
"735.730943555",
"1396.873591660"
],
[
"910.102930558",
"2077.605857002"
],
[
"601.281896624",
"606.718576327"
],
[
"537.387573330",
"2097.809414756"
],
[
"2029.202948236",
"1272.769486256"
],
[
"461.665477299",
"2038.799549710"
],
[
"1256.148333441",
"1229.303208887"
],
[
"2237.790605542",
"835.650704764"
],
[
"2284.614881986",
"1083.941919298"
],
[
"1278.061770902",
"879.567963126"
],
[
"1456.454191784",
"380.349937613"
],
[
"931.888189917",
"489.439096337"
],
[
"1700.790480802",
"2149.283261503"
],
[
"1925.427348370",
"1637.879986652"
],
[
"2193.150698375",
"734.933135135"
],
[
"2280.374667679",
"1196.124178244"
],
[
"982.084387529",
"76.028041305"
],
[
"855.786770914",
"106.612422847"
],
[
"1652.966660029",
"773.522366614"
],
[
"1693.240934017",
"655.119443937"
],
[
"1579.535435778",
"2197.190450521"
],
[
"1248.596210716",
"611.732359773"
],
[
"67.120127826",
"1120.840486425"
],
[
"1531.187645003",
"816.505364402"
],
[
"381.521039040",
"400.132533772"
],
[
"84.532204844",
"986.274082553"
],
[
"604.493436969",
"1612.210055429"
],
[
"464.253184103",
"1597.592549564"
],
[
"1929.128582942",
"1202.984410411"
],
[
"1540.088516982",
"1111.955740693"
],
[
"899.453028473",
"1583.073023919"
],
[
"1187.554170224",
"2005.788069855"
],
[
"2244.082493251",
"962.528882668"
],
[
"1523.797469018",
"1261.783732062"
],
[
"984.274073527",
"1709.867342134"
],
[
"1731.187420097",
"1874.220562502"
],
[
"1818.170956903",
"585.229066305"
],
[
"1584.168623548",
"1811.089956126"
],
[
"661.910539998",
"2131.499632514"
],
[
"734.168502176",
"1066.600970428"
],
[
"1328.673121840",
"95.777889724"
],
[
"1543.173847104",
"664.794954030"
],
[
"1288.760679747",
"1639.472819092"
],
[
"748.388058695",
"1527.260697759"
],
[
"1745.706843928",
"254.115936086"
],
[
"345.486452371",
"1490.380314090"
],
[
"421.832428854",
"932.982859784"
],
[
"498.842921980",
"507.552201925"
],
[
"1383.804510064",
"1167.110761031"
],
[
"1149.734307033",
"1753.051299733"
],
[
"827.869285721",
"2190.097368974"
],
[
"786.421505347",
"2005.698124121"
],
[
"392.803988408",
"1911.194317263"
],
[
"1804.830635133",
"2034.949291711"
],
[
"730.814499412",
"1254.958727824"
],
[
"515.545210325",
"1759.670086713"
],
[
"156.495382610",
"1513.045417867"
],
[
"346.784345248",
"1300.384747102"
],
[
"1142.036717762",
"102.365229300"
],
[
"704.896214108",
"713.793658461"
],
[
"532.643927869",
"315.504051675"
],
[
"902.962177972",
"650.280872648"
],
[
"1400.269006308",
"532.738894439"
],
[
"591.572346430",
"1952.427069412"
],
[
"1367.479081346",
"745.210247698"
],
[
"621.148775886",
"909.637771916"
],
[
"719.184757560",
"210.630228153"
],
[
"1110.255640456",
"1297.393752532"
],
[
"303.524009694",
"562.865152542"
],
[
"1097.509866676",
"517.607521717"
],
[
"1887.480284723",
"1021.154089082"
],
[
"1423.348490272",
"963.922981796"
],
[
"1490.852290910",
"1636.234013318"
],
[
"1889.650055616",
"401.382270946"
],
[
"900.218792127",
"1389.520259301"
],
[
"126.596268391",
"1296.096016001"
],
[
"1588.908054333",
"2013.352090482"
],
[
"2069.101228230",
"872.027187147"
],
[
"1660.258358370",
"957.387356496"
],
[
"1622.071044215",
"1435.974157798"
],
[
"1838.822699808",
"788.181044653"
],
[
"477.139237457",
"724.317896800"
],
[
"927.084011992",
"1147.003744465"
],
[
"2108.666019826",
"1116.850852806"
],
[
"2110.190572565",
"1619.036637213"
],
[
"1098.286128145",
"1537.095089148"
],
[
"260.441330696",
"1086.128162979"
],
[
"1723.326718338",
"1203.029339315"
],
[
"1737.570692181",
"1663.317139480"
],
[
"997.876189858",
"1920.428455428"
],
[
"288.796507288",
"1706.054258105"
],
[
"2185.963963028",
"1367.188680505"
],
[
"556.360271973",
"1411.912281031"
],
[
"866.729002502",
"891.022743549"
],
[
"1524.521839642",
"200.820565843"
],
[
"524.235171671",
"1138.795130467"
],
[
"2036.400450345",
"613.083835956"
],
[
"1117.402242098",
"770.827373626"
],
[
"1636.685755245",
"457.373203790"
],
[
"1950.170887212",
"1837.748091645"
],
[
"1178.998380441",
"1049.848451356"
],
[
"1066.869005286",
"2189.731243048"
],
[
"732.224760511",
"466.297899796"
],
[
"1265.100930368",
"318.828158173"
],
[
"1362.097215423",
"1866.994246602"
],
[
"763.525776746",
"1762.774709004"
],
[
"215.736334041",
"809.720016792"
],
[
"968.829408260",
"286.778428249"
],
[
"1348.396231016",
"1409.009428581"
],
[
"1364.092242354",
"2168.228979283"
],
[
"1899.147986948",
"1427.332441213"
]
],
"radius": "1169.964671247"
}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。