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 | [1367.29765586075, 1367.29765586085] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1367.297655877 | 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 171 圆九位整数证书出发，将容器半径目标降至 1367.297655877，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 171 个包含约束与 14535 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": [
[
"1569.335510068",
"2713.484512624"
],
[
"51.660277135",
"1716.779575458"
],
[
"1219.345011234",
"2721.508945099"
],
[
"1176.331389320",
"406.392233609"
],
[
"2503.937860594",
"1840.013755726"
],
[
"2092.034025486",
"1798.543796944"
],
[
"2721.257184721",
"1303.436211310"
],
[
"2331.943533784",
"1338.275791133"
],
[
"1915.318220073",
"2241.203970543"
],
[
"282.248170124",
"2158.659328174"
],
[
"2086.102062852",
"2275.066151305"
],
[
"1790.473517510",
"1165.920984770"
],
[
"30.492437009",
"1150.381000490"
],
[
"292.599495472",
"1715.619053897"
],
[
"1716.847888868",
"2233.545224701"
],
[
"1156.421126856",
"34.020315829"
],
[
"1572.483517297",
"1500.995117004"
],
[
"1560.284230563",
"397.960058780"
],
[
"598.110858771",
"1148.806075465"
],
[
"2054.413796950",
"208.789752144"
],
[
"1403.477951873",
"2711.814372727"
],
[
"463.585667264",
"1241.856064194"
],
[
"306.600753103",
"1552.228921937"
],
[
"1155.185862849",
"2691.381387718"
],
[
"35.119722952",
"1530.584513907"
],
[
"37.961883644",
"1188.679501273"
],
[
"971.981520240",
"2647.884705943"
],
[
"2227.673440364",
"340.931857632"
],
[
"979.171937040",
"86.648706215"
],
[
"2549.505978365",
"751.441087704"
],
[
"2663.135966378",
"1692.870713652"
],
[
"2395.713934792",
"515.898483064"
],
[
"2166.242738340",
"2435.711928166"
],
[
"1885.342864503",
"2595.616946410"
],
[
"715.302606496",
"2528.983598980"
],
[
"451.664113438",
"401.165095157"
],
[
"487.304937483",
"2364.651416885"
],
[
"2649.868846412",
"1018.389873488"
],
[
"115.188875780",
"923.973682776"
],
[
"258.513896153",
"637.696640620"
],
[
"112.641931747",
"1797.302871997"
],
[
"696.256435012",
"224.442653451"
],
[
"1848.726883972",
"1604.354512577"
],
[
"265.845688127",
"2100.718179438"
],
[
"2689.560003970",
"1357.634656972"
],
[
"713.850802517",
"1888.403590494"
],
[
"1201.498828617",
"691.123831386"
],
[
"1455.290529577",
"1035.331587006"
],
[
"671.514814095",
"2212.501899625"
],
[
"1837.912935306",
"1367.466924061"
],
[
"799.126073992",
"1572.267250674"
],
[
"2353.962321864",
"2236.829376701"
],
[
"515.114809413",
"1453.113949154"
],
[
"1513.833896434",
"62.526560200"
],
[
"1400.620319274",
"55.456130500"
],
[
"2401.713128594",
"1055.638853998"
],
[
"1602.289050132",
"2656.350813289"
],
[
"1096.824982128",
"1276.628582512"
],
[
"1701.068086928",
"580.191352572"
],
[
"2111.652658790",
"1345.401764526"
],
[
"1478.847957616",
"2668.765103697"
],
[
"2108.267424915",
"1050.880293126"
],
[
"2403.933812769",
"1351.557131097"
],
[
"2274.422029677",
"2150.365370973"
],
[
"1933.460515501",
"1675.755628800"
],
[
"1222.835761247",
"2396.246468873"
],
[
"1958.619363066",
"422.678490161"
],
[
"1811.360204989",
"146.239569201"
],
[
"1263.231300878",
"2263.807176975"
],
[
"1557.820218831",
"976.922145799"
],
[
"2071.157132528",
"502.658857623"
],
[
"1110.148125819",
"2476.979421023"
],
[
"1522.427698167",
"2314.125569740"
],
[
"1287.800385248",
"605.880077119"
],
[
"1445.983529856",
"2036.789315779"
],
[
"78.810153794",
"1282.153113837"
],
[
"78.785460416",
"1435.153111846"
],
[
"1075.492326837",
"2623.139244369"
],
[
"2413.897639141",
"2118.426626671"
],
[
"1087.202833037",
"110.890614696"
],
[
"983.669475198",
"937.874850590"
],
[
"1143.450306027",
"970.508079440"
],
[
"1390.158983694",
"1884.628955599"
],
[
"2153.192022493",
"1869.081385863"
],
[
"1292.603641368",
"446.952644839"
],
[
"1041.924460936",
"1886.958377543"
],
[
"1374.296865345",
"2374.600823917"
],
[
"1945.389707107",
"226.064957159"
],
[
"2577.714092457",
"1778.317055756"
],
[
"1408.002027306",
"2197.827625086"
],
[
"657.209610958",
"606.574353289"
],
[
"1259.431968753",
"96.569870464"
],
[
"1820.812368912",
"2196.209680322"
],
[
"1650.638889446",
"125.925565882"
],
[
"815.083983718",
"508.229641016"
],
[
"1544.622780890",
"1148.272938493"
],
[
"858.950827950",
"1882.711187858"
],
[
"658.996853836",
"1510.865084963"
],
[
"2292.722705188",
"1987.505031769"
],
[
"2099.879375311",
"333.195449644"
],
[
"1094.579676914",
"793.573335851"
],
[
"2490.258675340",
"1950.335798477"
],
[
"2213.968213922",
"2306.231907460"
],
[
"1705.964371204",
"1266.463002523"
],
[
"1966.056211328",
"1842.601427709"
],
[
"834.944738284",
"2250.041979081"
],
[
"2106.571855918",
"1683.858384944"
],
[
"1698.572577836",
"2355.766671638"
],
[
"1743.267135213",
"2568.114020436"
],
[
"1026.697441861",
"1123.965135358"
],
[
"1582.440326859",
"1910.394922897"
],
[
"956.090799694",
"2067.935899412"
],
[
"2254.030952165",
"480.186236273"
],
[
"1530.623200765",
"2500.945894490"
],
[
"1174.672907271",
"285.421403413"
],
[
"551.944228248",
"1895.038707358"
],
[
"1797.259191956",
"1986.794237029"
],
[
"1732.641118539",
"1046.071643451"
],
[
"725.546839271",
"2053.427890848"
],
[
"712.607086127",
"1722.170416169"
],
[
"2138.138286886",
"683.471479551"
],
[
"1612.917843289",
"2141.393020299"
],
[
"648.460111998",
"2382.949782311"
],
[
"600.283076069",
"1296.770030604"
],
[
"1217.122745827",
"1768.836912782"
],
[
"1717.764284963",
"1496.160111856"
],
[
"1754.935153706",
"1746.414635327"
],
[
"1066.912102840",
"2280.817647736"
],
[
"2519.116112030",
"912.660076467"
],
[
"1488.509383384",
"1367.195953776"
],
[
"1976.507717826",
"1484.541250752"
],
[
"1894.966456310",
"611.478260918"
],
[
"2394.134649403",
"682.391463755"
],
[
"622.540846044",
"384.261338038"
],
[
"166.545075216",
"1090.258393838"
],
[
"167.219281079",
"1628.927360298"
],
[
"2253.122013531",
"914.431958005"
],
[
"568.646332370",
"817.755663273"
],
[
"2252.846583861",
"1485.634864682"
],
[
"428.073839112",
"577.286163467"
],
[
"1292.164246311",
"2591.291492288"
],
[
"969.268935644",
"1669.552025820"
],
[
"2347.284867164",
"1751.365763927"
],
[
"1499.381851119",
"553.370237058"
],
[
"893.629604330",
"2494.085185293"
],
[
"447.098981641",
"1074.431025431"
],
[
"2257.393684045",
"1198.402928026"
],
[
"1675.549213457",
"786.241575888"
],
[
"848.040329353",
"752.120349170"
],
[
"452.562305255",
"1648.301532650"
],
[
"279.557524069",
"827.533849105"
],
[
"1219.163607939",
"2045.829394420"
],
[
"1954.724364042",
"1201.377895958"
],
[
"1442.547434270",
"260.840154738"
],
[
"1958.965077951",
"893.407091585"
],
[
"280.014121609",
"1901.204508466"
],
[
"1291.449018592",
"1158.542622555"
],
[
"1066.544748628",
"536.095098329"
],
[
"2563.540315905",
"1197.213217568"
],
[
"1739.139747757",
"364.338355085"
],
[
"1480.812026488",
"1658.094133037"
],
[
"2054.640642516",
"2094.477981900"
],
[
"897.564647056",
"263.172535546"
],
[
"464.410452086",
"2162.734608662"
],
[
"1966.625932218",
"2409.410372711"
],
[
"759.595436726",
"1054.448814562"
],
[
"2557.087799075",
"1523.149353995"
],
[
"1208.896647130",
"1472.883370744"
],
[
"313.220376025",
"1360.088313075"
],
[
"1361.851231591",
"838.371952582"
],
[
"886.664254665",
"1367.587192041"
]
],
"radius": "1367.297655877"
}{
"n": 171
}{
"centers": [
[
"1569.335510068",
"2713.484512624"
],
[
"51.660277135",
"1716.779575458"
],
[
"1219.345011234",
"2721.508945099"
],
[
"1176.331389320",
"406.392233609"
],
[
"2503.937860594",
"1840.013755726"
],
[
"2092.034025486",
"1798.543796944"
],
[
"2721.257184721",
"1303.436211310"
],
[
"2331.943533784",
"1338.275791133"
],
[
"1915.318220073",
"2241.203970543"
],
[
"282.248170124",
"2158.659328174"
],
[
"2086.102062852",
"2275.066151305"
],
[
"1790.473517510",
"1165.920984770"
],
[
"30.492437009",
"1150.381000490"
],
[
"292.599495472",
"1715.619053897"
],
[
"1716.847888868",
"2233.545224701"
],
[
"1156.421126856",
"34.020315829"
],
[
"1572.483517297",
"1500.995117004"
],
[
"1560.284230563",
"397.960058780"
],
[
"598.110858771",
"1148.806075465"
],
[
"2054.413796950",
"208.789752144"
],
[
"1403.477951873",
"2711.814372727"
],
[
"463.585667264",
"1241.856064194"
],
[
"306.600753103",
"1552.228921937"
],
[
"1155.185862849",
"2691.381387718"
],
[
"35.119722952",
"1530.584513907"
],
[
"37.961883644",
"1188.679501273"
],
[
"971.981520240",
"2647.884705943"
],
[
"2227.673440364",
"340.931857632"
],
[
"979.171937040",
"86.648706215"
],
[
"2549.505978365",
"751.441087704"
],
[
"2663.135966378",
"1692.870713652"
],
[
"2395.713934792",
"515.898483064"
],
[
"2166.242738340",
"2435.711928166"
],
[
"1885.342864503",
"2595.616946410"
],
[
"715.302606496",
"2528.983598980"
],
[
"451.664113438",
"401.165095157"
],
[
"487.304937483",
"2364.651416885"
],
[
"2649.868846412",
"1018.389873488"
],
[
"115.188875780",
"923.973682776"
],
[
"258.513896153",
"637.696640620"
],
[
"112.641931747",
"1797.302871997"
],
[
"696.256435012",
"224.442653451"
],
[
"1848.726883972",
"1604.354512577"
],
[
"265.845688127",
"2100.718179438"
],
[
"2689.560003970",
"1357.634656972"
],
[
"713.850802517",
"1888.403590494"
],
[
"1201.498828617",
"691.123831386"
],
[
"1455.290529577",
"1035.331587006"
],
[
"671.514814095",
"2212.501899625"
],
[
"1837.912935306",
"1367.466924061"
],
[
"799.126073992",
"1572.267250674"
],
[
"2353.962321864",
"2236.829376701"
],
[
"515.114809413",
"1453.113949154"
],
[
"1513.833896434",
"62.526560200"
],
[
"1400.620319274",
"55.456130500"
],
[
"2401.713128594",
"1055.638853998"
],
[
"1602.289050132",
"2656.350813289"
],
[
"1096.824982128",
"1276.628582512"
],
[
"1701.068086928",
"580.191352572"
],
[
"2111.652658790",
"1345.401764526"
],
[
"1478.847957616",
"2668.765103697"
],
[
"2108.267424915",
"1050.880293126"
],
[
"2403.933812769",
"1351.557131097"
],
[
"2274.422029677",
"2150.365370973"
],
[
"1933.460515501",
"1675.755628800"
],
[
"1222.835761247",
"2396.246468873"
],
[
"1958.619363066",
"422.678490161"
],
[
"1811.360204989",
"146.239569201"
],
[
"1263.231300878",
"2263.807176975"
],
[
"1557.820218831",
"976.922145799"
],
[
"2071.157132528",
"502.658857623"
],
[
"1110.148125819",
"2476.979421023"
],
[
"1522.427698167",
"2314.125569740"
],
[
"1287.800385248",
"605.880077119"
],
[
"1445.983529856",
"2036.789315779"
],
[
"78.810153794",
"1282.153113837"
],
[
"78.785460416",
"1435.153111846"
],
[
"1075.492326837",
"2623.139244369"
],
[
"2413.897639141",
"2118.426626671"
],
[
"1087.202833037",
"110.890614696"
],
[
"983.669475198",
"937.874850590"
],
[
"1143.450306027",
"970.508079440"
],
[
"1390.158983694",
"1884.628955599"
],
[
"2153.192022493",
"1869.081385863"
],
[
"1292.603641368",
"446.952644839"
],
[
"1041.924460936",
"1886.958377543"
],
[
"1374.296865345",
"2374.600823917"
],
[
"1945.389707107",
"226.064957159"
],
[
"2577.714092457",
"1778.317055756"
],
[
"1408.002027306",
"2197.827625086"
],
[
"657.209610958",
"606.574353289"
],
[
"1259.431968753",
"96.569870464"
],
[
"1820.812368912",
"2196.209680322"
],
[
"1650.638889446",
"125.925565882"
],
[
"815.083983718",
"508.229641016"
],
[
"1544.622780890",
"1148.272938493"
],
[
"858.950827950",
"1882.711187858"
],
[
"658.996853836",
"1510.865084963"
],
[
"2292.722705188",
"1987.505031769"
],
[
"2099.879375311",
"333.195449644"
],
[
"1094.579676914",
"793.573335851"
],
[
"2490.258675340",
"1950.335798477"
],
[
"2213.968213922",
"2306.231907460"
],
[
"1705.964371204",
"1266.463002523"
],
[
"1966.056211328",
"1842.601427709"
],
[
"834.944738284",
"2250.041979081"
],
[
"2106.571855918",
"1683.858384944"
],
[
"1698.572577836",
"2355.766671638"
],
[
"1743.267135213",
"2568.114020436"
],
[
"1026.697441861",
"1123.965135358"
],
[
"1582.440326859",
"1910.394922897"
],
[
"956.090799694",
"2067.935899412"
],
[
"2254.030952165",
"480.186236273"
],
[
"1530.623200765",
"2500.945894490"
],
[
"1174.672907271",
"285.421403413"
],
[
"551.944228248",
"1895.038707358"
],
[
"1797.259191956",
"1986.794237029"
],
[
"1732.641118539",
"1046.071643451"
],
[
"725.546839271",
"2053.427890848"
],
[
"712.607086127",
"1722.170416169"
],
[
"2138.138286886",
"683.471479551"
],
[
"1612.917843289",
"2141.393020299"
],
[
"648.460111998",
"2382.949782311"
],
[
"600.283076069",
"1296.770030604"
],
[
"1217.122745827",
"1768.836912782"
],
[
"1717.764284963",
"1496.160111856"
],
[
"1754.935153706",
"1746.414635327"
],
[
"1066.912102840",
"2280.817647736"
],
[
"2519.116112030",
"912.660076467"
],
[
"1488.509383384",
"1367.195953776"
],
[
"1976.507717826",
"1484.541250752"
],
[
"1894.966456310",
"611.478260918"
],
[
"2394.134649403",
"682.391463755"
],
[
"622.540846044",
"384.261338038"
],
[
"166.545075216",
"1090.258393838"
],
[
"167.219281079",
"1628.927360298"
],
[
"2253.122013531",
"914.431958005"
],
[
"568.646332370",
"817.755663273"
],
[
"2252.846583861",
"1485.634864682"
],
[
"428.073839112",
"577.286163467"
],
[
"1292.164246311",
"2591.291492288"
],
[
"969.268935644",
"1669.552025820"
],
[
"2347.284867164",
"1751.365763927"
],
[
"1499.381851119",
"553.370237058"
],
[
"893.629604330",
"2494.085185293"
],
[
"447.098981641",
"1074.431025431"
],
[
"2257.393684045",
"1198.402928026"
],
[
"1675.549213457",
"786.241575888"
],
[
"848.040329353",
"752.120349170"
],
[
"452.562305255",
"1648.301532650"
],
[
"279.557524069",
"827.533849105"
],
[
"1219.163607939",
"2045.829394420"
],
[
"1954.724364042",
"1201.377895958"
],
[
"1442.547434270",
"260.840154738"
],
[
"1958.965077951",
"893.407091585"
],
[
"280.014121609",
"1901.204508466"
],
[
"1291.449018592",
"1158.542622555"
],
[
"1066.544748628",
"536.095098329"
],
[
"2563.540315905",
"1197.213217568"
],
[
"1739.139747757",
"364.338355085"
],
[
"1480.812026488",
"1658.094133037"
],
[
"2054.640642516",
"2094.477981900"
],
[
"897.564647056",
"263.172535546"
],
[
"464.410452086",
"2162.734608662"
],
[
"1966.625932218",
"2409.410372711"
],
[
"759.595436726",
"1054.448814562"
],
[
"2557.087799075",
"1523.149353995"
],
[
"1208.896647130",
"1472.883370744"
],
[
"313.220376025",
"1360.088313075"
],
[
"1361.851231591",
"838.371952582"
],
[
"886.664254665",
"1367.587192041"
]
],
"radius": "1367.297655877"
}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。