The best known packings of unequal circles with integer radii in a square (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 0.390524291751 5.1213203436 0.598902316059 5 2
3 0.351471862576 8.5355339059 0.603693534587 5 1 3
4 0.334735107215 11.9497474683 0.660014797289 5 2 4
5 0.320445439141 15.6032802757 0.709709703127 7 2 4 1 [16]
6 0.308913652863 19.4229032754 0.757814603624 7 3 6 [16]
7 0.293884516036 23.8188799275 0.775238331435 11 2 6 1 [16]
8 0.274864248385 29.1052766848 0.756548153681 11 3 7 1 [16]
9 0.266632649665 33.7543058260 0.785844052832 7 6 6 3 [16]
10 0.259198794035 38.5804264146 0.812599508588 15 3 9 1 [16]
11 0.247163327190 44.5049843156 0.802570301381 15 4 9 2 [31]
12 0.239133461623 50.1811830036 0.810926398507 15 5 10 2 [31]
13 0.232178672615 55.9913615390 0.820713765587 15 6 9 4 [31]
14 0.226354370430 61.8499213131 0.833561431951 19 5 9 5 [31]
15 0.218890022409 68.5275639104 0.829546950560 17 7 13 2 [31]
16 0.213306762250 75.0093425601 0.835315900179 25 4 12 4 [17]
17 0.208583778822 81.5020232924 0.844211594720 21 7 13 4 [18]
18 0.203610493383 88.4040881240 0.847777243445 23 7 12 6 [17]
19 0.198431041375 95.7511479474 0.846367244411 25 7 11 8 [18]
20 0.193953211391 103.1176532553 0.847941181266 19 11 14 6 [31]
21 0.189941602299 110.5602971954 0.850962963235 23 10 15 6 [17]
22 0.185867536223 118.3638651864 0.850987881603 33 6 13 9 [18]
23 0.182534535157 126.0035531370 0.855598420798 25 11 16 7 [18]
24 0.179007777510 134.0723868749 0.856381487321 31 9 13 11 [18]
25 0.175783323972 142.2205442197 0.858138689425 33 9 13 12 [18]
26 0.172502694180 150.7222836351 0.857544513724 27 13 18 8 [18]
27 0.169771311851 159.0374704985 0.860764184295 35 10 18 9 [18]
28 0.166845093699 167.8203378909 0.860479097197 37 10 17 11 [22]
29 0.164300639059 176.5057042147 0.862685296594 35 12 18 11 [18]
30 0.161583478754 185.6625456478 0.861713316159 25 18 18 12 [18]
31 0.159189016456 194.7370534113 0.862886461779 45 9 17 14 [18]
32 0.156794336236 204.0890045400 0.862852739685 37 14 19 13 [18]
33 0.154634301277 213.4067262400 0.864271266631 39 14 19 14 [18]
34 0.152583663931 222.8285723649 0.865870450598 43 13 19 15 [18]
35 0.150413388907 232.6920512484 0.865096231238 41 15 22 13 [22]
36 0.148608142987 242.2478289305 0.867572785241 53 10 20 16 [22]
37 0.146590093103 252.4045057670 0.866664484203 49 13 21 16 [22]
38 0.144841113776 262.3564470703 0.868068644476 59 9 24 14 [18]
39 0.143010409756 272.7074208546 0.867673632307 41 19 23 16 [18]
40 0.141457873682 282.7696964392 0.869884831612 43 19 23 17 [22]
41 0.139857251975 293.1560531972 0.870787386573 53 15 18 23 [22]
42 0.138193104447 303.9225449639 0.870180725722 55 15 25 17 [22]
43 0.136689294198 314.5820618381 0.870905629582 55 16 25 18 [22]
44 0.135434330638 324.8806989537 0.874190328780 47 21 29 15 [22]
45 0.133844527455 336.2109819180 0.872542460153 55 18 23 22 [22]
46 0.132424778658 347.3670144377 0.872489297886 41 26 27 19 [22]
47 0.131031049417 358.6936089508 0.872195909534 57 19 24 23 [22]
48 0.129895370219 369.5281819442 0.874807591891 59 19 25 23 [22]
49 0.128553772010 381.1634558353 0.874132745763 57 21 28 21 [22]
50 0.127322454703 392.7037074225 0.874440318598 65 18 26 24 [22]
51 0.126241881493 403.9863743858 0.876346651798 61 21 29 22 [22]
52 0.125093493192 415.6890872029 0.876859250102 63 21 27 25 [22]
53 0.123859542458 427.9040512197 0.875707830142 59 24 27 26 [22]
54 0.122770267713 439.8459089968 0.876154015383 67 21 26 28 [22]
55 0.121739531481 451.7842259692 0.877021400075 63 24 28 27 [22]
56 0.120684699926 464.0190515810 0.877138804006 73 20 30 26 [22]
57 0.119655992307 476.3656119600 0.877240161152 59 28 32 25 [22]
58 0.118694816504 488.6481289437 0.877954499467 57 30 34 24 [22]
59 0.117768032812 500.9848478507 0.878819457462 67 26 31 28 [22]
60 0.116753995159 513.9010439710 0.878023375335 63 29 36 24 [22]
61 0.115863832229 526.4800829256 0.878741962774 73 25 32 29 [22]
62 0.115086111776 538.7270370266 0.880852782017 83 21 30 32 [22]
63 0.114115651990 552.0715072944 0.879695154081 77 25 35 28 [22]
64 0.113222798803 565.2571803260 0.879406128100 69 30 33 31 [22]
65 0.112473970480 577.9114911886 0.881058103654 77 27 31 34 [22]
66 0.111667186723 591.0420235061 0.881520223712 77 28 29 37 [22]
67 0.110854792047 604.3942599396 0.881607840762 69 33 34 33 [22]
68 0.110122636411 617.4933893356 0.882698864273 77 30 33 35 [22]
69 0.109251745384 631.5688573897 0.881290603561 69 35 35 34 [22]
70 0.108566653629 644.7651987065 0.882614274603 77 32 38 32 [22]
71 0.107854963149 658.2914492485 0.883261045273 81 31 32 39 [22]
72 0.107058300258 672.5307596561 0.882262207109 81 32 36 36 [22]
73 0.106439758359 685.8339508230 0.883959925328 85 31 37 36 [22]
74 0.105703658062 700.0703793675 0.883475414873 79 35 38 36 [22]
75 0.105075138390 713.7749342916 0.884561106469 91 30 35 40 [22]
76 0.104402531282 727.9516987449 0.884686179688 87 33 37 39 [22]
77 0.103728337429 742.3236688114 0.884563534753 87 34 38 39 [22]
78 0.103070380604 756.7644510762 0.884501396272 85 36 40 38 [22]
79 0.102434311386 771.2259586762 0.884605319870 95 32 36 43 [22]
80 0.101767643043 786.1044788686 0.883972923996 81 40 40 40 [22]
81 0.101231297677 800.1477987415 0.885410500895 99 32 34 47 [22]
82 0.100588603495 815.2016943358 0.884798472062 85 40 40 42 [22]
83 0.100046706407 829.6125178009 0.885771860042 97 35 38 45 [22]
84 0.099453668005 844.6143986944 0.885659074914 75 47 45 39 [22]
85 0.098876546890 859.6578528836 0.885647375777 101 35 42 43 [22]
86 0.098325456314 874.6463349772 0.885926041842 111 31 39 47 [22]
87 0.097778873619 889.7627552860 0.886115127109 97 39 39 48 [22]
88 0.097258714489 904.8032401246 0.886617447047 111 33 42 46 [22]
89 0.096743709185 919.9564576319 0.887053113973 109 35 40 49 [22]
90 0.096205514209 935.4973126019 0.886902768274 95 43 43 47 [22]
91 0.095751207987 950.3796548693 0.888146626310 119 32 37 54 [22]
92 0.095186486490 966.5237513485 0.887188829300 89 48 43 49 [22]
93 0.094752116916 981.5084140278 0.888511344160 113 37 42 51 [22]
94 0.094209863733 997.7723804633 0.887664658916 95 47 45 49 [22]
95 0.093766588001 1013.1540671927 0.888537667610 111 40 39 56 [22]
96 0.093239099434 1029.6109741810 0.887672105875 105 44 44 52 [22]
97 0.092781554056 1045.4664290431 0.887995694587 113 41 43 54 [22]
98 0.092362992326 1061.0310204557 0.888934898480 109 44 41 57 [22]
99 0.091941339424 1076.7735234251 0.889688868522 129 35 40 59 [22]
100 0.091466448537 1093.2970679358 0.889282378614 109 46 43 57 [22]


Updates

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

06-May-2013: First complete presentation from N=1 to N=72.
21-May-2013: First improvements for N=23, 40, 42, 49, 51, 54, 58, 61, 63, 64, 65, 67, 68, 70 and 72 by Eckard Specht [31].
19-Apr-2015: Improvements for N=16, 18, 19, 21–46, 49, 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.
23-Apr-2015: Further improvements for N=29, 30, 33, 35, 39, 41, 42, 44, 45, 49, 55, and 56 by Eckard Specht [31].
24-Sep-2015: The next round on the record spiral up for N=17, 19, 22–45, 50, 55, and 60 by Zhizhong Zeng, Xinguo Yu, Kun He, and Zhanghua Fu [18].
14-Oct-2015: Some small improvements (some can easily improved further) for N=52, 61–63, 66–68 by Kun He, Mohammed Dosh, Shenghao Zou [19].
19-Oct-2015: By some exchange heuristics and relocations (see [32]), some packings could be improved for N=32, 39, 42, 52, 61–68 by Eckard Specht [19].
28-Oct-2015: Some tiny improvements for N=47, 69, and 72 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=28, 35–37, 40–72 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, October 2015.
[22]   , github.com/muellan/packing, December 2025.
[31]   , program csqn, 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