The best known packings of unequal circles with inverse square root integer radii in a circle (complete up to N = 100)


Last update: 05-Jul-2026

Overview    Download    Results    History of updates    References

Overview

5-16   17-28   29-40   41-52   53-64   65-76   77-88   89-100  

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 circles in the container circle, the latter has always a *radius* of 1
ratio
= 1/radius, that is the radius of the circumcircle if r1=1
density
ratio of total area occupied by the circles to container area
contacts
number of contacts between circles and container and between the circles themselves, respectively
loose
number of circles that have still degrees of freedom for a movement inside the container (so called "rattlers")
boundary
number of circles that have contact to the container (rattlers too if possible)
symmetry group
of the packing (Schönfliess notation); if field is empty then the packing has symmetry element C1
reference
for the best known packing so far
records
the sequence of N 's that establish density records

N radius ratio density contacts loose boundary symmetry group reference
5 0.570922096619039327572248540682 1.7515524550931378665568811273 0.744257158931325262919831649210 8 1 8 [1]
6 0.552462703778895858181919898120 1.8100769394203586063526115504 0.747776845713385678253134806798 12 10 [1]
7 0.543855392719099794562810055793 1.8387240678084035628744535299 0.766911884377439960701417861841 12 1 10 [1]
8 0.538097011594676918415207784095 1.8584009545722078860518310212 0.786951170528927642877892446330 14 1 10 [31]
9 0.532251017145950947776483149073 1.8788127552338439170686917244 0.801421656550761748912120728233 14 2 11 [2]
10 0.522620172967119285734658120347 1.9134355153621579135747986081 0.799994503715671817039971557739 18 1 11 [31]
11 0.518352866238196249839914019637 1.9291877505322305145432013482 0.811409919523441134848081636471 20 1 13 [31]
12 0.512866950316335641876274515984 1.9498234374104265275455335062 0.816245289797483039072781824356 18 3 11 [31]
13 0.508844527279375491914800119383 1.9652368186932685571530009639 0.823408986603556154663566884407 22 2 13 [31]
14 0.504987063495310980797764145277 1.9802487475192231818475507113 0.829187198385919487092411030112 16 6 10 [16]
15 0.501829350259971028078346639563 1.9927092735447883259424864242 0.835638555906221923856147111261 18 6 11 [31]
16 0.498856178168462934962144041161 2.0045857779520204648299050634 0.841319719781977370987306980855 22 5 14 [16]
17 0.496214433802134310366198114497 2.0152577834903334777242732148 0.846916767441465729418659584290 30 2 14 [17]
18 0.493060522668214360455819502949 2.0281485822236686639861103870 0.849691107900951254052871420972 28 4 16 [17]
19 0.489716608581961948150612639469 2.0419973153363736119166355332 0.850827285608816047609664462513 34 2 15 [17]
20 0.487462145681 2.0514413455 0.854892535313 34 3 12 [20]
21 0.484888190971411805794536944239 2.0623311077067626961925510750 0.857084190404219922580654549281 32 5 15 [17]
22 0.483567606273408313065930244471 2.0679631700445246082006019953 0.863051022601780021218709439771 32 6 14 [17]
23 0.480821851142213573402595165555 2.0797723681327206147789405733 0.863329556915647433132889131557 36 5 15 [17]
24 0.478381196872302363133524238449 2.0903831641755288099475223776 0.864122627549923203618044223198 34 7 15 [17]
25 0.476751718388099055418281177941 2.0975278356227159485165655764 0.867337533092140648556940206836 40 5 13 [19]
26 0.474603330816 2.1070227179 0.868201571904 34 9 10 [20]
27 0.473222670592 2.1131701039 0.871451642173 44 5 10 [20]
28 0.471370007341 2.1214756655 0.872576891457 52 2 16 [20]
29 0.469908040244 2.1280759518 0.874786903436 36 11 11 [20]
30 0.468111875195 2.1362414692 0.875416447171 42 9 13 [20]
31 0.466511538255 2.1435697041 0.876461515337 50 6 16 [20]
32 0.464833270522 2.1513090035 0.876918932625 46 9 11 [20]
33 0.463676655479 2.1566753214 0.879075430314 60 3 19 [20]
34 0.462334598612 2.1629356812 0.880280897607 46 11 14 [20]
35 0.460853371077 2.1698875668 0.880717611985 48 11 16 [20]
36 0.459301870582 2.1772173467 0.880657528681 50 11 13 [20]
37 0.458026808993 2.1832783155 0.881444714186 64 5 22 [20]
38 0.457010575847 2.1881331699 0.883033976318 56 10 15 [20]
39 0.455540870731 2.1951927132 0.882684553380 68 5 19 [20]
40 0.454540624932 2.2000233756 0.883977710556 52 14 12 [20]
41 0.453527830785 2.2049363504 0.885059560770 70 6 21 [20]
42 0.452554740096 2.2096774410 0.886141990686 42 21 13 [20]
43 0.451235263946 2.2161388524 0.885717415048 50 18 12 [20]
44 0.450177826296 2.2213444146 0.886176958757 74 7 18 [20]
45 0.449194559799 2.2262068366 0.886793963445 52 19 13 [20]
46 0.448304812076 2.2306251752 0.887653455983 64 14 18 [20]
47 0.447539197551 2.2344411517 0.888885695194 76 9 21 [20]
48 0.446395997097 2.2401634569 0.888501776374 62 17 16 [20]
49 0.445518731290 2.2445745370 0.889063760923 84 7 20 [20]
50 0.445013975690 2.2471204381 0.891011100754 70 15 18 [20]
51 0.442351255091 2.2606469146 0.884217110341 50 26 26 [22]
52 0.441306051434 2.2660011046 0.883788738267 66 19 25 [22]
53 0.440913453926 2.2680187939 0.885884966879 66 20 23 [22]
54 0.440070479327 2.2723632849 0.886087123420 74 17 24 [22]
55 0.439172258550 2.2770108552 0.885980427125 52 29 27 [22]
56 0.438501246315 2.2804952287 0.886708743769 78 17 23 [22]
57 0.437644217853 2.2849610693 0.886606294905 80 17 25 [22]
58 0.437478798123 2.2858250601 0.889235974769 82 17 26 [22]
59 0.436270154044 2.2921577163 0.887555250769 71 23 32 [22]
60 0.435826627873 2.2944903685 0.888917281562 70 25 27 [22]
61 0.435143489225 2.2980925253 0.889236885801 87 17 27 [22]
62 0.434691277029 2.3004832460 0.890437296518 96 14 22 [22]
63 0.434177565650 2.3032051380 0.891326155003 92 17 25 [22]
64 0.433091431430 2.3089812622 0.889803020775 52 38 28 [22]
65 0.432770186737 2.3106952157 0.891364876301 81 24 28 [22]
66 0.432538590179 2.3119324442 0.893245796820 74 29 28 [22]
67 0.431529599899 2.3173381391 0.891862647600 84 25 32 [22]
68 0.430892538789 2.3207642509 0.891961720003 90 23 29 [22]
69 0.429944197261 2.3258832341 0.890718858943 84 27 32 [22]
70 0.429937993757 2.3259167939 0.893333822267 86 27 34 [22]
71 0.429441207768 2.3286074599 0.893868012406 116 13 24 [22]
72 0.428724889646 2.3324981221 0.893441360970 98 23 27 [22]
73 0.427884939043 2.3370768839 0.892451980576 106 20 27 [22]
74 0.427562001239 2.3388420793 0.893575762985 96 26 34 [22]
75 0.426821947925 2.3428973249 0.892914143304 104 23 30 [22]
76 0.426290730296 2.3458169013 0.893084007784 100 26 31 [22]
77 0.426312017359 2.3456997675 0.895533488182 100 27 28 [22]
78 0.425654650416 2.3493223885 0.895096662860 88 34 34 [22]
79 0.425058958854 2.3526148059 0.894880118160 104 27 30 [22]
80 0.424574131836 2.3553012890 0.895093151703 104 28 30 [22]
81 0.424354137937 2.3565223256 0.896388969417 116 22 35 [22]
82 0.423785402604 2.3596848637 0.896178002860 122 19 33 [22]
83 0.423770668620 2.3597669071 0.898279321589 132 15 28 [22]
84 0.422936436733 2.3644214902 0.896875577350 104 32 33 [22]
85 0.422357789191 2.3676608449 0.896521765651 106 32 31 [22]
86 0.422075189645 2.3692461072 0.897393923964 139 14 28 [22]
87 0.421734321571 2.3711610577 0.897989403911 124 24 36 [22]
88 0.421063973997 2.3749360234 0.897151675109 121 26 34 [22]
89 0.420795441527 2.3764515993 0.897997264760 130 24 36 [22]
90 0.420231194775 2.3796424740 0.897552779574 135 21 30 [22]
91 0.419785418699 2.3821694500 0.897586044988 124 29 32 [22]
92 0.419553943941 2.3834837318 0.898509759082 136 23 37 [22]
93 0.419576620531 2.3833549132 0.900499841371 148 19 36 [22]
94 0.418655348726 2.3885996036 0.898414294935 134 27 34 [22]
95 0.418402356839 2.3900438983 0.899171548549 134 26 34 [22]
96 0.417867723180 2.3931017988 0.898693987570 116 38 33 [22]
97 0.417423017002 2.3956513160 0.898578485344 136 29 34 [22]
98 0.417489163952 2.3952717492 0.900641837313 136 30 38 [22]
99 0.416650703753 2.4000919499 0.898781393107 134 32 38 [22]
100 0.416797448742 2.3992469316 0.901151809981 131 34 36 [22]



Updates

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

31-Mar-2011: First complete presentation from N=1 to N=50.
04-Apr-2011: First improvements for N=27, 28, 30, 32 and 33 by Eckard Specht [31].
05-Apr-2011: Extension up to N=100 by Eckard Specht [31]. All these packings are still far away from the optimum.
06-Apr-2011: Improvements for N=26, 42, 43, 46, 52, 53, 54, 55, 56, 57, 61, 62, 63, 64, 69, 76, 77, 80, 86, 98 and 99 by Eckard Specht [31].
07-Apr-2011: Further improvements for N=44, 71, 77, 78, 79, 81, 84, 89, 91 and 95 by Eckard Specht [31].
09-Apr-2011: More improvements for N=18, 42, 43, 44, 46, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 67, 68, 69, 71, 72, 73, 75, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95 and 96 by Eckard Specht [31].
12-Apr-2011: Improvements for N=26, 32, 41, 43, 53, 55, 57, 64, 65, 66, 80, 81, 82 and 83 by Eckard Specht [31].
14-Apr-2011: Improvements for N=71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85 and 86 by Eckard Specht [31].
17-Feb-2012: Zhizhong Zeng sent me new record packings for N=17, 18, 19, 21, 22 and 23 [17].
17-Feb-2012: Zhanghua Fu discovered two new record packings for N=14 and 16 [16].
17-Feb-2012: There were some improvements on my hard disk for N=83–97, now they can be seen here [31].
20-Feb-2012: For N=26 there is a better packing in [18] as Fig. 2, but ...
20-Feb-2012: More record packings for N=24, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 47 and 50 by Eckard Specht [31].
20-Feb-2012: Two new record packings for N=24 and 25 by Zhizhong Zeng and Wenqi Huang [17].
20-Feb-2012: ... we can do better [31].
21-Feb-2012: Further improvements for N=28, 29, 30, 31, 32, 34, 35, 37, 38, 42, 43, 44, 45, 46, 48, 49, 50, 52, 53, 55, 56 and 60 by Eckard Specht [31].
22-Feb-2012: Even more improvements for N=28, 29 and 30 by Eckard Specht [31].
23-Feb-2012: Further records for N=23 and 25 by Zhizhong Zeng and Wenqi Huang [17].
27-Feb-2012: One more improvement for N=21 by Zhizhong Zeng and Wenqi Huang [17].
04-Mar-2012: Further improvements for N=26, 27, 28, 29 and 30 by Zhizhong Zeng and Wenqi Huang [17].
04-Mar-2012: The packing for N=27 was only suboptimal and could be improved by Eckard Specht [31].
07-Mar-2012: Again, two improvements for N=26 and 27 by Zhizhong Zeng and Wenqi Huang [17].
08-Mar-2012: Four better packings were found for N=31, 32, 34 and 37 by Eckard Specht [31].
11-Mar-2012: Two new records for N=28 and 29 by Zhizhong Zeng and Wenqi Huang [17].
12-Mar-2012: Only small improvements for N=44, 45, 46, 50 and 52 by Eckard Specht [31].
13-Mar-2012: Again, two new records for N=25 and 30 by Zhizhong Zeng and Wenqi Huang [17].
14-Mar-2012: Due to a mistake, all packings before have had a precision of only 16 decimal places. Now they are provided in full accuracy of 30 digits. Many thanks to David Cantrell who discovered this inaccuracy!
22-Mar-2012: Only small improvements for N=42, 44, 45, 46, 47 and 50 by Eckard Specht [31].
22-Apr-2014: Five new candidates of optimal packings for N=31, 32, 33, 34 and 35 by Zhizhong Zeng, Kun He, Xinguo Yu, Wenqi Huang, and Zhanghua Fu [19].
17-Jul-2014: Two new candidates for N=25 and 30 by Zhizhong Zeng, Kun He, Xinguo Yu, Wenqi Huang, and Zhanghua Fu (their work is partly inspired by Tao Ye et al.) [19].
02-Jun-2024: Significant improvements for N= 20, 26–50 by Jianrong Zhou, Jiyao He, and Kun He [20].
05-Jul-2026: André Müller has published his records on github in December 2025 [22]. I could improve most of them by further swapping operations.

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, J\'{a}nos D. Pint\'{e}r, 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]   , Feasible Approaches for Solving the Arbitrary Sized Circle Packing Problem, Phd thesis.
[17]   , private communication, February–March 2012.
[18]   Wenqi Huang, Zhizhong Zeng, Ruchu Xu, Zhanghua Fu, Using iterated local search for efficiently packing unequal disks in a larger circle, Advanced Materials Research 430–432 (2012), 1477–1481.
[19]   Zhizhong Zeng, Kun He, Xinguo Yu, Wenqi Huang, Zhanghua Fu, private communication, April-July 2014.
[20]   , Solution-Hashing Search Based on Layout-Graph Transformation for Unequal Circle Packing, submitted to European Journal of Operational Research, under review, May 2024.
[21]   André Müller, Johannes Josef Schneider, Elmar Schömer, Packing a multidisperse system of hard discs in a circular environment, Phys. Rev. E 79, 021102 (2009).
[22]   , github.com/muellan/packing, December 2025.
[31]   , program ccis, 2005–2024.
[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    05-Jul-2026