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 | [1549.36880316195, 1549.36880316205] | 见来源署名 | packomania.com ↗ |
| 本站纪录 | 1549.368803184 | 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。
从此前的 186 个圆心出发，在 1e-9 整数格点上将每个坐标限制在 ±32 单位窗口，求解公共半径最小化整数 MILP；约束包括大圆边界和 186*185/2=17205 个成对分离条件。HiGHS 得到半径 1549.368803184 的可行解，并通过独立整数验证器、几何审计和装箱验证。后续可扩大局部窗口或以此解作为新基准继续搜索。
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": [
[
"2673.264533007",
"2599.841890767"
],
[
"2964.437391885",
"1970.805083581"
],
[
"485.954024983",
"1362.965173377"
],
[
"2106.471255921",
"2239.290915083"
],
[
"1192.209586372",
"363.965346401"
],
[
"1566.124617386",
"2345.787057639"
],
[
"375.813040761",
"1262.542589030"
],
[
"676.824199375",
"2465.135998643"
],
[
"2311.581299181",
"2887.872695460"
],
[
"2935.044308896",
"2219.715610431"
],
[
"1412.125120835",
"771.547725609"
],
[
"3012.857768327",
"2018.465933070"
],
[
"1614.569622830",
"14.466485136"
],
[
"2663.242644620",
"918.968683383"
],
[
"654.268355465",
"747.113165035"
],
[
"347.326564016",
"2495.283623689"
],
[
"22.169660975",
"1673.816274615"
],
[
"954.019769067",
"1350.423978260"
],
[
"2529.944707107",
"629.630072548"
],
[
"2357.673887452",
"1053.550463510"
],
[
"3015.232247214",
"1976.717159951"
],
[
"1755.942338602",
"3062.672990841"
],
[
"3058.363063231",
"1778.046870397"
],
[
"364.363380229",
"594.392230431"
],
[
"204.885683682",
"2260.992050634"
],
[
"1377.676081721",
"35.871938619"
],
[
"2250.562522759",
"2900.447571831"
],
[
"2517.607644335",
"2722.628094854"
],
[
"839.023644399",
"205.395651197"
],
[
"119.886397250",
"2063.756538635"
],
[
"2682.408299924",
"539.284626166"
],
[
"2044.569293305",
"2983.584396658"
],
[
"2231.017552709",
"195.080827799"
],
[
"2497.801277903",
"367.696784385"
],
[
"2882.620508545",
"2267.288415759"
],
[
"1999.288244226",
"104.450584066"
],
[
"2275.136052139",
"2304.329560366"
],
[
"672.707439687",
"2777.957563705"
],
[
"3059.492011680",
"1526.711406047"
],
[
"143.082528169",
"1001.294434782"
],
[
"600.106845632",
"377.212326497"
],
[
"1663.414795526",
"46.320493767"
],
[
"974.743565142",
"2941.831659782"
],
[
"2860.049748764",
"808.985902145"
],
[
"509.928146503",
"2450.832139261"
],
[
"1298.290951224",
"3031.623056218"
],
[
"59.981461910",
"1746.213862973"
],
[
"2318.475483082",
"1363.698713431"
],
[
"2996.862974459",
"1154.568638935"
],
[
"1686.577772984",
"3042.446331835"
],
[
"64.063250966",
"1351.944167182"
],
[
"1336.853451084",
"957.642723579"
],
[
"2180.469325362",
"1161.626826116"
],
[
"2173.038657434",
"2267.510216577"
],
[
"391.156790599",
"1062.727625445"
],
[
"1518.983863755",
"1848.029940972"
],
[
"1791.080742169",
"2649.125430621"
],
[
"1889.920839942",
"2373.663976798"
],
[
"1190.829528839",
"2329.099970942"
],
[
"2065.996654510",
"2761.213650067"
],
[
"2810.740495023",
"1594.776569178"
],
[
"1114.266802470",
"127.063559695"
],
[
"1286.483218781",
"551.550570386"
],
[
"425.245610497",
"2520.237875585"
],
[
"1317.029494843",
"2747.555201583"
],
[
"792.258597777",
"1538.591791034"
],
[
"1602.901396936",
"2251.450155308"
],
[
"986.631401765",
"1944.993521233"
],
[
"370.497443948",
"1371.485006979"
],
[
"462.280852936",
"801.376513887"
],
[
"2473.465877118",
"2137.335383288"
],
[
"427.324222045",
"1891.658914233"
],
[
"2671.946299457",
"2508.265692908"
],
[
"1154.205969668",
"257.066878591"
],
[
"605.503133985",
"824.626719022"
],
[
"2441.890513851",
"1375.729099165"
],
[
"2203.343631227",
"2000.194039276"
],
[
"345.868968595",
"702.897124316"
],
[
"1215.723519401",
"1008.971306866"
],
[
"540.438562390",
"1790.124949431"
],
[
"751.509217169",
"747.853706321"
],
[
"2620.312350270",
"2180.290165602"
],
[
"2557.471822769",
"2614.246360361"
],
[
"383.971603160",
"1740.816178279"
],
[
"2124.581614284",
"760.461350743"
],
[
"2124.355124085",
"2895.042978052"
],
[
"2144.459307181",
"588.281063565"
],
[
"1651.578621104",
"2701.185509364"
],
[
"558.953647585",
"2167.473745589"
],
[
"214.726512143",
"2139.696897270"
],
[
"823.072447790",
"591.448130897"
],
[
"2971.125781608",
"1869.572726672"
],
[
"1039.264520139",
"2317.608750894"
],
[
"1053.608314156",
"948.575444190"
],
[
"2032.691061196",
"2318.657941453"
],
[
"2374.047214643",
"1972.099794140"
],
[
"509.566700215",
"967.408007759"
],
[
"2091.590661076",
"2134.865022291"
],
[
"1275.255295322",
"125.138677801"
],
[
"1638.505599156",
"1318.398803424"
],
[
"2200.577330208",
"2420.473779384"
],
[
"1585.512027478",
"362.244893786"
],
[
"1255.081458357",
"1400.872932606"
],
[
"395.551190337",
"2064.767188028"
],
[
"2092.011177620",
"210.809692396"
],
[
"977.224488756",
"224.240739384"
],
[
"2542.960983971",
"503.805633084"
],
[
"2295.980430666",
"2160.571482322"
],
[
"591.342633856",
"1972.140815521"
],
[
"630.338730880",
"2353.229418135"
],
[
"1507.899278931",
"1681.398215363"
],
[
"1957.375889604",
"656.110223003"
],
[
"2582.361269590",
"1989.018851818"
],
[
"1348.492169480",
"1847.047754662"
],
[
"2748.444860899",
"2336.532129031"
],
[
"2010.365073714",
"1125.463886426"
],
[
"2977.361040920",
"1659.702284937"
],
[
"1469.173466399",
"1455.696395600"
],
[
"2231.949307263",
"1805.395392928"
],
[
"1284.693692436",
"1621.898079579"
],
[
"954.546521329",
"757.756783670"
],
[
"2351.855353049",
"1197.008028981"
],
[
"1962.374848397",
"891.331577078"
],
[
"2010.873447147",
"424.953729664"
],
[
"1455.489561688",
"2614.961070027"
],
[
"1111.013810565",
"1796.114320596"
],
[
"2476.102995551",
"1773.823175282"
],
[
"1501.481250086",
"884.714551751"
],
[
"2012.764971953",
"1921.990620977"
],
[
"1769.391587959",
"503.709756017"
],
[
"1704.193868933",
"1539.862234298"
],
[
"2139.776519676",
"1342.095924762"
],
[
"287.696737758",
"905.719686079"
],
[
"1873.590358118",
"1334.730886158"
],
[
"1736.496482077",
"766.660154402"
],
[
"1509.978119432",
"136.549017138"
],
[
"2607.052502201",
"1241.228104354"
],
[
"779.132479136",
"366.703210670"
],
[
"855.594779007",
"2459.345923839"
],
[
"1343.574554672",
"356.744052612"
],
[
"615.736971894",
"2603.804334349"
],
[
"793.275842418",
"1330.594278510"
],
[
"2216.777373059",
"969.019127222"
],
[
"2315.319287666",
"1555.672770095"
],
[
"2747.569444124",
"1790.851601573"
],
[
"2505.809690389",
"976.844301717"
],
[
"374.360491753",
"2314.871075034"
],
[
"528.104619533",
"1213.188049726"
],
[
"1160.577683216",
"2893.564895232"
],
[
"1513.012194502",
"2053.943367359"
],
[
"2609.130933002",
"1529.220604477"
],
[
"1500.928253345",
"602.719282089"
],
[
"1402.432339777",
"2342.071108312"
],
[
"1985.142356166",
"2563.075863566"
],
[
"2676.869991608",
"728.999721981"
],
[
"199.338146124",
"1894.178673520"
],
[
"2836.938986427",
"2079.325317722"
],
[
"1691.230260779",
"2458.387372387"
],
[
"1023.050178477",
"1525.028726304"
],
[
"2362.897325696",
"703.579945032"
],
[
"1236.286934422",
"769.853875538"
],
[
"1736.325096127",
"1831.095103739"
],
[
"875.874233642",
"2761.108514366"
],
[
"1072.702630936",
"1205.867898544"
],
[
"2298.242179610",
"385.076079926"
],
[
"214.595020454",
"1195.645778261"
],
[
"1732.892381331",
"1068.638647773"
],
[
"2466.641369723",
"2377.483547179"
],
[
"562.494459015",
"584.227848951"
],
[
"1062.467619657",
"487.472404428"
],
[
"2822.189053185",
"1020.818793227"
],
[
"1821.206757697",
"2154.168248133"
],
[
"1408.683589344",
"1171.052803059"
],
[
"1813.150386391",
"201.997251296"
],
[
"551.280327742",
"1535.355531870"
],
[
"2001.291265344",
"1617.206510405"
],
[
"804.350280714",
"1781.290763435"
],
[
"1187.050314877",
"2090.451627132"
],
[
"1155.707808279",
"2564.494246039"
],
[
"1493.414925830",
"2917.593961094"
],
[
"785.039202090",
"1007.699314541"
],
[
"829.463576467",
"2139.411305121"
],
[
"1854.564248593",
"2880.985674049"
],
[
"2902.590345998",
"1367.645318849"
],
[
"2300.014012983",
"2688.681754092"
],
[
"190.677665784",
"1552.288345954"
]
],
"radius": "1549.368803184"
}{
"n": 186
}{
"centers": [
[
"2673.264533007",
"2599.841890767"
],
[
"2964.437391885",
"1970.805083581"
],
[
"485.954024983",
"1362.965173377"
],
[
"2106.471255921",
"2239.290915083"
],
[
"1192.209586372",
"363.965346401"
],
[
"1566.124617386",
"2345.787057639"
],
[
"375.813040761",
"1262.542589030"
],
[
"676.824199375",
"2465.135998643"
],
[
"2311.581299181",
"2887.872695460"
],
[
"2935.044308896",
"2219.715610431"
],
[
"1412.125120835",
"771.547725609"
],
[
"3012.857768327",
"2018.465933070"
],
[
"1614.569622830",
"14.466485136"
],
[
"2663.242644620",
"918.968683383"
],
[
"654.268355465",
"747.113165035"
],
[
"347.326564016",
"2495.283623689"
],
[
"22.169660975",
"1673.816274615"
],
[
"954.019769067",
"1350.423978260"
],
[
"2529.944707107",
"629.630072548"
],
[
"2357.673887452",
"1053.550463510"
],
[
"3015.232247214",
"1976.717159951"
],
[
"1755.942338602",
"3062.672990841"
],
[
"3058.363063231",
"1778.046870397"
],
[
"364.363380229",
"594.392230431"
],
[
"204.885683682",
"2260.992050634"
],
[
"1377.676081721",
"35.871938619"
],
[
"2250.562522759",
"2900.447571831"
],
[
"2517.607644335",
"2722.628094854"
],
[
"839.023644399",
"205.395651197"
],
[
"119.886397250",
"2063.756538635"
],
[
"2682.408299924",
"539.284626166"
],
[
"2044.569293305",
"2983.584396658"
],
[
"2231.017552709",
"195.080827799"
],
[
"2497.801277903",
"367.696784385"
],
[
"2882.620508545",
"2267.288415759"
],
[
"1999.288244226",
"104.450584066"
],
[
"2275.136052139",
"2304.329560366"
],
[
"672.707439687",
"2777.957563705"
],
[
"3059.492011680",
"1526.711406047"
],
[
"143.082528169",
"1001.294434782"
],
[
"600.106845632",
"377.212326497"
],
[
"1663.414795526",
"46.320493767"
],
[
"974.743565142",
"2941.831659782"
],
[
"2860.049748764",
"808.985902145"
],
[
"509.928146503",
"2450.832139261"
],
[
"1298.290951224",
"3031.623056218"
],
[
"59.981461910",
"1746.213862973"
],
[
"2318.475483082",
"1363.698713431"
],
[
"2996.862974459",
"1154.568638935"
],
[
"1686.577772984",
"3042.446331835"
],
[
"64.063250966",
"1351.944167182"
],
[
"1336.853451084",
"957.642723579"
],
[
"2180.469325362",
"1161.626826116"
],
[
"2173.038657434",
"2267.510216577"
],
[
"391.156790599",
"1062.727625445"
],
[
"1518.983863755",
"1848.029940972"
],
[
"1791.080742169",
"2649.125430621"
],
[
"1889.920839942",
"2373.663976798"
],
[
"1190.829528839",
"2329.099970942"
],
[
"2065.996654510",
"2761.213650067"
],
[
"2810.740495023",
"1594.776569178"
],
[
"1114.266802470",
"127.063559695"
],
[
"1286.483218781",
"551.550570386"
],
[
"425.245610497",
"2520.237875585"
],
[
"1317.029494843",
"2747.555201583"
],
[
"792.258597777",
"1538.591791034"
],
[
"1602.901396936",
"2251.450155308"
],
[
"986.631401765",
"1944.993521233"
],
[
"370.497443948",
"1371.485006979"
],
[
"462.280852936",
"801.376513887"
],
[
"2473.465877118",
"2137.335383288"
],
[
"427.324222045",
"1891.658914233"
],
[
"2671.946299457",
"2508.265692908"
],
[
"1154.205969668",
"257.066878591"
],
[
"605.503133985",
"824.626719022"
],
[
"2441.890513851",
"1375.729099165"
],
[
"2203.343631227",
"2000.194039276"
],
[
"345.868968595",
"702.897124316"
],
[
"1215.723519401",
"1008.971306866"
],
[
"540.438562390",
"1790.124949431"
],
[
"751.509217169",
"747.853706321"
],
[
"2620.312350270",
"2180.290165602"
],
[
"2557.471822769",
"2614.246360361"
],
[
"383.971603160",
"1740.816178279"
],
[
"2124.581614284",
"760.461350743"
],
[
"2124.355124085",
"2895.042978052"
],
[
"2144.459307181",
"588.281063565"
],
[
"1651.578621104",
"2701.185509364"
],
[
"558.953647585",
"2167.473745589"
],
[
"214.726512143",
"2139.696897270"
],
[
"823.072447790",
"591.448130897"
],
[
"2971.125781608",
"1869.572726672"
],
[
"1039.264520139",
"2317.608750894"
],
[
"1053.608314156",
"948.575444190"
],
[
"2032.691061196",
"2318.657941453"
],
[
"2374.047214643",
"1972.099794140"
],
[
"509.566700215",
"967.408007759"
],
[
"2091.590661076",
"2134.865022291"
],
[
"1275.255295322",
"125.138677801"
],
[
"1638.505599156",
"1318.398803424"
],
[
"2200.577330208",
"2420.473779384"
],
[
"1585.512027478",
"362.244893786"
],
[
"1255.081458357",
"1400.872932606"
],
[
"395.551190337",
"2064.767188028"
],
[
"2092.011177620",
"210.809692396"
],
[
"977.224488756",
"224.240739384"
],
[
"2542.960983971",
"503.805633084"
],
[
"2295.980430666",
"2160.571482322"
],
[
"591.342633856",
"1972.140815521"
],
[
"630.338730880",
"2353.229418135"
],
[
"1507.899278931",
"1681.398215363"
],
[
"1957.375889604",
"656.110223003"
],
[
"2582.361269590",
"1989.018851818"
],
[
"1348.492169480",
"1847.047754662"
],
[
"2748.444860899",
"2336.532129031"
],
[
"2010.365073714",
"1125.463886426"
],
[
"2977.361040920",
"1659.702284937"
],
[
"1469.173466399",
"1455.696395600"
],
[
"2231.949307263",
"1805.395392928"
],
[
"1284.693692436",
"1621.898079579"
],
[
"954.546521329",
"757.756783670"
],
[
"2351.855353049",
"1197.008028981"
],
[
"1962.374848397",
"891.331577078"
],
[
"2010.873447147",
"424.953729664"
],
[
"1455.489561688",
"2614.961070027"
],
[
"1111.013810565",
"1796.114320596"
],
[
"2476.102995551",
"1773.823175282"
],
[
"1501.481250086",
"884.714551751"
],
[
"2012.764971953",
"1921.990620977"
],
[
"1769.391587959",
"503.709756017"
],
[
"1704.193868933",
"1539.862234298"
],
[
"2139.776519676",
"1342.095924762"
],
[
"287.696737758",
"905.719686079"
],
[
"1873.590358118",
"1334.730886158"
],
[
"1736.496482077",
"766.660154402"
],
[
"1509.978119432",
"136.549017138"
],
[
"2607.052502201",
"1241.228104354"
],
[
"779.132479136",
"366.703210670"
],
[
"855.594779007",
"2459.345923839"
],
[
"1343.574554672",
"356.744052612"
],
[
"615.736971894",
"2603.804334349"
],
[
"793.275842418",
"1330.594278510"
],
[
"2216.777373059",
"969.019127222"
],
[
"2315.319287666",
"1555.672770095"
],
[
"2747.569444124",
"1790.851601573"
],
[
"2505.809690389",
"976.844301717"
],
[
"374.360491753",
"2314.871075034"
],
[
"528.104619533",
"1213.188049726"
],
[
"1160.577683216",
"2893.564895232"
],
[
"1513.012194502",
"2053.943367359"
],
[
"2609.130933002",
"1529.220604477"
],
[
"1500.928253345",
"602.719282089"
],
[
"1402.432339777",
"2342.071108312"
],
[
"1985.142356166",
"2563.075863566"
],
[
"2676.869991608",
"728.999721981"
],
[
"199.338146124",
"1894.178673520"
],
[
"2836.938986427",
"2079.325317722"
],
[
"1691.230260779",
"2458.387372387"
],
[
"1023.050178477",
"1525.028726304"
],
[
"2362.897325696",
"703.579945032"
],
[
"1236.286934422",
"769.853875538"
],
[
"1736.325096127",
"1831.095103739"
],
[
"875.874233642",
"2761.108514366"
],
[
"1072.702630936",
"1205.867898544"
],
[
"2298.242179610",
"385.076079926"
],
[
"214.595020454",
"1195.645778261"
],
[
"1732.892381331",
"1068.638647773"
],
[
"2466.641369723",
"2377.483547179"
],
[
"562.494459015",
"584.227848951"
],
[
"1062.467619657",
"487.472404428"
],
[
"2822.189053185",
"1020.818793227"
],
[
"1821.206757697",
"2154.168248133"
],
[
"1408.683589344",
"1171.052803059"
],
[
"1813.150386391",
"201.997251296"
],
[
"551.280327742",
"1535.355531870"
],
[
"2001.291265344",
"1617.206510405"
],
[
"804.350280714",
"1781.290763435"
],
[
"1187.050314877",
"2090.451627132"
],
[
"1155.707808279",
"2564.494246039"
],
[
"1493.414925830",
"2917.593961094"
],
[
"785.039202090",
"1007.699314541"
],
[
"829.463576467",
"2139.411305121"
],
[
"1854.564248593",
"2880.985674049"
],
[
"2902.590345998",
"1367.645318849"
],
[
"2300.014012983",
"2688.681754092"
],
[
"190.677665784",
"1552.288345954"
]
],
"radius": "1549.368803184"
}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。