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: 19-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 0.128823218364 43.9117600175 0.860244900068 45 10 20 12 [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 0.118726460389 51.9211470028 0.863534164854 53 12 18 20 [22]
39 0.117329565063 53.2261241661 0.864957542030 51 14 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 0.112024578661 58.5357124542 0.867359604322 63 12 20 23 [22]
44 0.110840298552 59.8451074870 0.868415896319 69 10 22 22 [22]
45 0.109641862875 61.1828708176 0.868621391232 71 10 21 24 [22]
46 0.108501855186 62.5088849540 0.869144653091 57 18 25 21 [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 0.103254484423 69.1633730821 0.870845407011 71 16 26 25 [22]
52 0.102351376646 70.4543777252 0.872133810591 61 22 27 25 [22]
53 0.101372436483 71.8154770848 0.871672623953 77 15 25 28 [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 0.098756414618 75.7754805346 0.873223392573 91 11 25 31 [22]
57 0.097933746199 77.0912451358 0.873801137708 83 16 28 29 [22]
58 0.097114862493 78.4202634935 0.874064109632 85 16 29 29 [22]
59 0.096347969078 79.7229648056 0.874895648704 81 19 30 29 [22]
60 0.095573052767 81.0476014751 0.875226842632 83 19 28 32 [22]
61 0.094836630063 82.3547786410 0.875918689790 91 16 28 33 [22]
62 0.094006210698 83.7605070512 0.874527600461 87 19 32 30 [22]
63 0.093315679997 85.0580945612 0.875405158176 93 17 30 33 [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 0.091306223905 88.9757352474 0.877395584211 91 21 28 38 [22]
67 0.090616999099 90.3291088129 0.877098041844 99 18 30 37 [22]
68 0.089974166482 91.6508768423 0.877414145776 107 15 31 37 [22]
69 0.089369034343 92.9474501320 0.878197199054 103 18 32 37 [22]
70 0.088724435073 94.2987155508 0.877939747033 113 14 29 41 [22]
71 0.088120189780 95.6211033392 0.878219793409 91 26 32 39 [22]
72 0.087530393990 96.9409708722 0.878537904317 107 19 31 41 [22]
73 0.086946061437 98.2678640539 0.878721847227 113 17 31 42 [22]
74 0.086381497309 99.5852761874 0.879068252506 101 24 34 40 [22]
75 0.085800909436 100.9342918943 0.878855047947 107 22 32 43 [22]
76 0.085243569853 102.2692726519 0.878888658160 113 20 33 43 [22]
77 0.084748694545 103.5409977048 0.879995626952 95 30 33 44 [22]
78 0.084170403658 104.9271535182 0.879155683815 115 21 31 47 [22]
79 0.083704768921 106.1850421652 0.880461291873 103 28 39 40 [22]
80 0.083184219646 107.5236619159 0.880413695240 131 15 32 48 [22]
81 0.082669536279 108.8671886301 0.880287900348 105 29 34 47 [22]
82 0.082202061772 110.1600731532 0.880974606800 121 22 36 46 [22]
83 0.081738176645 111.4587326637 0.881554271719 115 26 37 46 [22]
84 0.081237242354 112.8195778723 0.881148584603 131 19 34 50 [22]
85 0.080790988473 114.1160002061 0.881747367515 115 28 36 49 [22]
86 0.080326921581 115.4484488259 0.881782299476 131 21 36 50 [22]
87 0.079857527922 116.8002478394 0.881524268873 139 18 35 52 [22]
88 0.079444511837 118.0802965829 0.882343488873 121 28 36 52 [22]
89 0.079008149612 119.4051648898 0.882482633819 147 16 34 55 [22]
90 0.078579663342 120.7288575321 0.882635918741 143 19 35 55 [22]
91 0.078144177515 122.0742519467 0.882472038259 127 28 37 54 [22]
92 0.077720103069 123.4128966360 0.882408253002 125 30 36 56 [22]
93 0.077322819068 124.7193374128 0.882801554645 129 29 37 56 [22]
94 0.076920599168 126.0437362641 0.882935135155 135 27 37 57 [22]
95 0.076526853032 127.3643689588 0.883118167168 137 27 40 55 [22]
96 0.076118786897 128.7193263396 0.882826432828 133 30 36 60 [22]
97 0.075782659581 129.9618917606 0.884067933105 141 27 40 57 [22]
98 0.075389206157 131.3118341632 0.883839510677 141 28 41 57 [22]
99 0.075002445444 132.6606660904 0.883630554189 137 31 39 60 [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.

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    19-Jul-2026