The best known packings of unequal circles with radii of i1/2, i=1,2,3,..., in a circle (complete up to N = 100)


Last update: 21-Jul-2026

Overview    Download    Results    History of updates    References

Overview

1-12   13-24   25-36   37-48   49-60   61-72   73-84   85-96   97-100  

Download

You may download ASCII files which contain all the values of radius, ratio etc. by using the links given in the table header below.
All coordinates of all packings are packed as ASCII files here.
All packings are stored as nice PDF files here.
All contact graphs of all packings are stored as nice PDF files here.
For industrial applications, for instance if a machine has to do an important job at every circle center,
it is useful to know a tour visiting each of the circle centers once which is of minimal length.
This problem is known as the "Traveling Salesman Problem" (TSP). Thus (very near) optimal tours are provided for every packing.
All optimal TSP tours of all packings are stored as nice PDF files here.

Results

The table below summarizes the current status of the search.
Please use the links in the following table to view a picture for a certain configuration.
Furthermore, note that for certain values of N several distinct optimal configurations exist; however, only one is shown here.
Proven optimal packings are indicated by a radius in bold face type.

Legend:
N
the number of circles; colors correspond to active researchers in the past, see "References" at the bottom of the page
radius
of the largest circle in the container square, the latter has always a side length of 1
ratio
is the side length of the circumscribed square if r1=1
density
ratio of total area occupied by the circles to container area, also known as packing fraction ϕ
contacts
number of contacts between circles and container and mutually between the circles
loose
number of circles that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of circles that are near to the container boundary (including rattlers if any)
core
number of circles that are at the container core
reference
for the best known packing so far, see at bottom of the page
records
the sequence of N 's that establish density records

N radius ratio density contacts loose boundary core reference
1 0.500000000000 2.0000000000 0.785398163397 4 1
2 0.343145750508 4.1213203436 0.554879118757 5 2 [31]
3 0.322481442452 5.3710092413 0.653415336689 5 1 3 [31]
4 0.301525245061 6.6329437842 0.714064166361 7 1 4 [31]
5 0.278593202069 8.0262833439 0.731496339965 11 5 [31]
6 0.262702420604 9.3241993627 0.758832749948 7 3 6 [31]
7 0.247311830027 10.6980378204 0.768598701156 15 6 1 [31]
8 0.233612631837 12.1073381285 0.771533931405 13 2 7 1 [31]
9 0.223488239380 13.4235251409 0.784565533604 11 4 8 1 [31]
10 0.216258275147 14.6226897353 0.808086835497 15 3 8 2 [31]
11 0.208545252245 15.9036216584 0.819788340641 19 2 8 3 [31]
12 0.200793434608 17.2520661440 0.823307831000 21 2 10 2 [31]
13 0.193952032257 18.5899123279 0.827249630454 25 1 8 5 [31]
14 0.186952222795 20.0139764633 0.823516684330 27 1 9 5 [22]
15 0.181846154442 21.2981316987 0.831090087258 19 6 12 3 [31]
16 0.177280605606 22.5630998175 0.839249810172 25 4 11 5 [31]
17 0.172714405736 23.8723898452 0.843430899234 31 2 12 5 [31]
18 0.168213670493 25.2217353957 0.844492997566 31 3 11 7 [18]
19 0.164123782850 26.5586063631 0.846238752428 29 5 14 5 [18]
20 0.160535665987 27.8575849641 0.850123927380 31 5 14 6 [18]
21 0.156677935603 29.2483793414 0.848317134573 35 4 15 6 [18]
22 0.153568099022 30.5429043511 0.852019882828 33 6 15 7 [18]
23 0.150331382514 31.9017323137 0.851982004310 35 6 15 8 [18]
24 0.147236046964 33.2729626106 0.851310879214 29 10 16 8 [18]
25 0.144597276172 34.5787979717 0.853912653644 37 7 14 11 [18]
26 0.141997358913 35.9092560076 0.855153691356 37 8 17 9 [18]
27 0.139543617726 37.2367615759 0.856441817569 39 8 17 10 [22]
28 0.137123984193 38.5891837468 0.856534226329 37 10 16 12 [22]
29 0.135063851743 39.8712515426 0.859645498270 43 8 16 13 [18]
30 0.132892984077 41.2153102972 0.859974706234 43 9 17 13 [18]
31 0.130678950163 42.6064362767 0.858383041531 49 7 16 15 [18]
32 0.128826446081 43.9106598186 0.860288008158 43 11 19 13 [22]
33 0.126954355314 45.2490395647 0.860783637452 47 10 18 15 [22]
34 0.125221545938 46.5650847157 0.862076970699 53 8 18 16 [22]
35 0.123462519671 47.9180223995 0.861970990074 49 11 20 15 [22]
36 0.121854995828 49.2388511379 0.862994999768 45 14 23 13 [22]
37 0.120288128534 50.5682697407 0.863672337524 49 13 21 16 [22]
38 0.118729833206 51.9196720531 0.863583228628 61 8 18 20 [22]
39 0.117349671146 53.2170046786 0.865254012881 53 13 21 18 [22]
40 0.115984378101 54.5293721782 0.866368652757 59 11 21 19 [22]
41 0.114654460387 55.8471446800 0.867263515842 53 15 24 17 [22]
42 0.113386181596 57.1563536863 0.868377561613 59 13 22 20 [22]
43 0.112032716632 58.5314604647 0.867485626730 73 7 19 24 [22]
44 0.110865167086 59.8316834320 0.868805621960 63 13 21 23 [22]
45 0.109641862875 61.1828708176 0.868621391232 71 10 21 24 [22]
46 0.108522839502 62.4967980406 0.869480871659 69 12 24 22 [22]
47 0.107314672847 63.8836649129 0.868319044414 61 17 25 22 [22]
48 0.106326514434 65.1596948056 0.870159991817 75 11 22 26 [22]
49 0.105300319210 66.4765316242 0.870861832054 69 15 28 21 [22]
50 0.104262480065 67.8198696931 0.870855613154 65 18 23 27 [22]
51 0.103256197000 69.1622259588 0.870874294901 75 14 23 28 [22]
52 0.102351376646 70.4543777252 0.872133810591 61 22 27 25 [22]
53 0.101373473888 71.8147421617 0.871690464737 83 12 24 29 [22]
54 0.100507702826 73.1134930134 0.872732723862 73 18 26 28 [22]
55 0.099648146646 74.4238476754 0.873466705578 77 17 30 25 [22]
56 0.098757594246 75.7745754206 0.873244253699 89 12 25 31 [22]
57 0.097933746199 77.0912451358 0.873801137708 83 16 28 29 [22]
58 0.097115721144 78.4195701391 0.874079565949 73 22 29 29 [22]
59 0.096347969078 79.7229648056 0.874895648704 81 19 30 29 [22]
60 0.095573844170 81.0469303574 0.875241337509 89 16 29 31 [22]
61 0.094850948233 82.3423468229 0.876183197284 91 16 28 33 [22]
62 0.094008324778 83.7586234258 0.874566934941 89 18 31 31 [22]
63 0.093318125738 85.0558653042 0.875451046337 99 14 28 35 [22]
64 0.092665908446 86.3316416379 0.876744820775 89 20 30 34 [22]
65 0.091954926593 87.6761914448 0.876624928819 75 28 34 31 [22]
66 0.091306443070 88.9755216769 0.877399796292 103 15 28 38 [22]
67 0.090630686197 90.3154672589 0.877363021570 99 18 29 38 [22]
68 0.089986310222 91.6385084676 0.877651009501 115 11 31 37 [22]
69 0.089369034343 92.9474501320 0.878197199054 103 18 32 37 [22]
70 0.088733422571 94.2891643637 0.878117620951 105 18 31 39 [22]
71 0.088137544552 95.6022750128 0.878565748296 101 21 30 41 [22]
72 0.087530393990 96.9409708722 0.878537904317 107 19 31 41 [22]
73 0.086950814254 98.2624926357 0.878817918708 113 17 28 45 [22]
74 0.086384783638 99.5814876741 0.879135140946 101 24 31 43 [22]
75 0.085808282001 100.9256197178 0.879006088160 117 17 32 43 [22]
76 0.085254537427 102.2561162164 0.879114831151 113 20 33 43 [22]
77 0.084755085378 103.5331903482 0.880128351517 117 19 32 45 [22]
78 0.084171855459 104.9253437291 0.879186012047 109 24 33 45 [22]
79 0.083723529649 106.1612482727 0.880856011216 119 20 37 42 [22]
80 0.083184219646 107.5236619159 0.880413695240 131 15 32 48 [22]
81 0.082680397840 108.8528869614 0.880519229316 115 24 31 50 [22]
82 0.082206602866 110.1539879090 0.881071944968 123 21 35 47 [22]
83 0.081741591471 111.4540763792 0.881627931711 113 27 36 47 [22]
84 0.081248092437 112.8045116513 0.881383973519 137 16 35 49 [22]
85 0.080798424002 114.1054986054 0.881909676694 135 18 33 52 [22]
86 0.080335210303 115.4365372355 0.881964286422 123 25 35 51 [22]
87 0.079859433599 116.7974606472 0.881566341823 129 23 35 52 [22]
88 0.079445937062 118.0781782750 0.882375147423 117 30 36 52 [22]
89 0.079008821835 119.4041489670 0.882497650684 141 19 36 53 [22]
90 0.078579663342 120.7288575321 0.882635918741 143 19 35 55 [22]
91 0.078151265632 122.0631801289 0.882632135887 125 29 35 56 [22]
92 0.077731114552 123.3954138175 0.882658312153 125 30 36 56 [22]
93 0.077326266380 124.7137772515 0.882880272949 127 30 37 56 [22]
94 0.076934760361 126.0205357032 0.883260264323 151 19 35 59 [22]
95 0.076527902678 127.3626220469 0.883142393118 147 22 39 56 [22]
96 0.076123003321 128.7121966244 0.882924239804 137 28 37 59 [22]
97 0.075784508372 129.9587212917 0.884111069018 143 26 40 57 [22]
98 0.075392548005 131.3060136395 0.883917869964 141 28 38 60 [22]
99 0.075008684922 132.6496309257 0.883777579313 147 26 38 61 [22]
100 0.074663023429 133.9351065730 0.884407456787 137 32 41 59 [22]


Updates

Please note that the results are taken from a running search. For updates look at the list below.

14-May-2013: First complete presentation from N=1 to N=72.
19-Apr-2015: Improvements for N=18–45, 47, 50, 55, and 60 by Kun He, Menglong Huang, and Chenkai Yang [17]. All these packings could be slightly improved by simple exchange heuristics [31], nevertheless the credits should go to the authors.
24-Sep-2015: The next round on the record spiral up for N=18–45, 50, 55, and 60 by Zhizhong Zeng, Xinguo Yu, Kun He, and Zhanghua Fu [18].
29-Sep-2015: A new record for an astonishing small instance of N=14 by Kun He, Mohammed Dosh, Shenghao Zou [19].
19-Jul-2026: André Müller has published his records on github in December 2025 [22]. For N=14, 27, 28, 32–72 he found new records, and this repository could be extended up to N=100.
21-Jul-2026: Almost all of the new candidates, namely N=32, 38, 39, 43, 44, 46, 51, 53, 56, 58, 60–63, 66–68, 70, 71, 73–79, 81–89, 91–99, could be slightly improved by simple swapping of circles [31]. But the credits should remain at the author.

References

[1]   Ignacio Castillo, Frank J. Kampas, János D. Pintér, Solving circle packing problems by global optimization: Numerical results and industrial applications, European Journal of Operational Research 191 (2008), 786–802.
[2]   I. Al-Mudahka, Mhand Hifi, Rym M'Hallah Packing circles in the smallest circle: an adaptive hybrid algorithm, J. Operational Research Society 62 (2010), 1917–1930.
[3]   Wenqi Huang, Ruchu Xu, Two personification strategies for solving circles packing problem, Science in China (Series E) 42 (1999), 595–602.
[4]   Huaiqing Wang, Wenqi Huang, Quan Zhang, Dongmin Xu, An improved algorithm for the packing of unequal circles within a larger containing circle, European Journal of Operational Research 141 (2002), 440–453.
[5]   Wenqi Huang and Yan Kang, A Short Note on a Simple Search Heuristic for the Diskspacking Problem, Annals of Operations Research 131 (2004), 101–108.
[6]   De-fu Zhang, Xin Li, A Personified Annealing Algorithm for Circles Packing Problem, Acta Automatica Sinica 31 (2005), 590–595.
[7]   De-fu Zhang, An-sheng Deng, An effective hybrid algorithm for the problem of packing circles into a larger containing circle, Computers & Operations Research 32 (2005), 1941–1951.
[8]   Hakim Akeb, Yu Li, A hybrid heuristic for packing unequal circles into a circular container, Service Systems and Service Management, 2006 Int. Conf. on (2006), 922–927.
[9]   Wen Qi Huang, Yu Li, Chu Min Li, Ru Chu Xu, New heuristics for packing unequal circles into a circular container, Computers & Operations Research 33 (2006), 2125–2142.
[10]   Zhipeng Lü, Wenqi Huang, PERM for solving circle packing problem, Computers & Operations Research 35 (2008), 1742–1755.
[11]   Ignacio Castillo, Frank J. Kampas, János D. Pintér, Solving circle packing problems by global optimization: Numerical results and industrial applications, European Journal of Operational Research 191 (2008), 786–802.
[12]   Mhand Hifi, Rym M'Hallah, Adaptive and restarting techniques-based algorithms for circular packing problems, Comput. Optim. Appl. 39 (2008), 17–35.
[13]   Bernardetta Addis, Marco Locatelli, Fabio Schoen, Efficiently packing unequal disks in a circle, Operations Research Letters 36 (2008), 37–42.
[14]   A. Grosso, A. R. M. J. U. Jamali, M. Locatelli, F. Schoen, Solving the problem of packing equal and unequal circles in a circular container, J. Glob. Optim. 47 (2010) 1, 63–81.
[15]   Jingfa Liu, Shengjun Xue, Zhaoxia Liu, Danhua Xu, An improved energy landscape paving algorithm for the problem of packing circles into a larger containing circle, Computers & Industrial Engineering 57 (2009), 1144–1149.
[16]   , Packing unequal circles using formulation space search, Computers & Operations Research 40 (2013), 1276–1288.
[17]   , Computers & Operations Research 58 (2015), 67–74.
[18]   Zhizhong Zeng, Xinguo Yu, Kun He, Zhanghua Fu, private communication, September 2015.
[19]   Kun He, Mohammed Dosh, Shenghao Zou, private communication, September 2015.
[22]   , github.com/muellan/packing, December 2025.
[31]   , program csqr, 2005–2026.
[32]   Eckard Specht, A precise algorithm to detect voids in polydisperse circle packings, Proc. R. Soc. A 471 (2015), 20150421.

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©  E. Specht    21-Jul-2026