The best known packings of unequal circles with inverse square root 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 0.343145750508 4.1213203436 0.554879118757 5 2
3 0.331288928562 5.2282182054 0.632128180991 7 3
4 0.327671056155 6.1036822216 0.702724017944 7 1 4
5 0.323308618074 6.9162028245 0.749814357380 11 5 [16]
6 0.308826772086 7.9315977894 0.734084140259 7 3 6 [16]
7 0.305710487848 8.6544342318 0.761288270075 15 6 1 [16]
8 0.302700786604 9.3439701841 0.782354522083 13 2 7 1 [16]
9 0.300483166159 9.9839203585 0.802450477978 17 1 8 1 [31]
10 0.297724173461 10.6215011815 0.815629140575 15 3 8 2 [31]
11 0.296004701494 11.2046355129 0.831259066015 17 3 9 2 [31]
12 0.290510026690 11.9242067291 0.822779355567 19 3 9 3 [22]
13 0.287825185416 12.5268790160 0.827661674868 9 9 12 1 [31]
14 0.285809048870 13.0914587959 0.834437676481 25 2 11 3 [31]
15 0.284073316049 13.6337456826 0.841234563731 23 4 12 3 [22]
16 0.281524203537 14.2083698302 0.841766635237 27 3 12 4 [31]
17 0.279747899942 14.7386472838 0.845639942957 27 4 13 4 [31]
18 0.277734129429 15.2759068388 0.846971867630 29 4 11 7 [31]
19 0.276785064192 15.7483170425 0.853860488608 22 9 15 4 [31]
20 0.274523761686 16.2905240972 0.851803633852 29 6 14 6 [22]
21 0.273047985610 16.7830415768 0.853823484651 23 10 15 6 [31]
22 0.271864307647 17.2527824650 0.856991139353 25 10 17 5 [22]
23 0.270582437618 17.7241049550 0.858929066994 17 15 18 5 [22]
24 0.269482775300 18.1791933830 0.861467843254 27 11 18 6 [22]
25 0.268056524780 18.6527822969 0.861402731786 31 10 18 7 [22]
26 0.266491178625 19.1339148256 0.859952675644 29 12 18 8 [22]
27 0.265946518334 19.5383359604 0.864670613208 55 15 12 [31]
28 0.264691484868 19.9912083487 0.864389792314 43 7 18 10 [22]
29 0.263398267874 20.4449514821 0.863479880282 47 6 18 11 [31]
30 0.262496754265 20.8658792387 0.864794926200 19 21 19 11 [22]
31 0.261757467776 21.2706992092 0.866874256860 45 9 16 15 [31]
32 0.261171349058 21.6595513631 0.869692998155 31 17 21 11 [22]
33 0.260491491748 22.0527841734 0.871630939896 47 10 18 15 [22]
34 0.259310744497 22.4863489793 0.869960200411 43 13 22 12 [22]
35 0.258910964916 22.8498618628 0.873296868959 41 15 20 15 [31]
36 0.258167442028 23.2407307167 0.874104668971 51 11 23 13 [22]
37 0.257581675662 23.6148884220 0.875776084436 41 17 23 14 [22]
38 0.256843040696 24.0007048128 0.876214410998 45 16 22 16 [22]
39 0.256441818121 24.3524946288 0.878776427475 63 8 21 18 [22]
40 0.254994165246 24.8027452481 0.873989598068 51 15 23 17 [22]
41 0.254297684088 25.1796403904 0.874176837617 43 20 24 17 [22]
42 0.253901570847 25.5246183660 0.876277637308 43 21 25 17 [22]
43 0.253124020063 25.9060302641 0.875599914373 51 18 24 19 [22]
44 0.252685116327 26.2510498328 0.877124921323 41 24 26 18 [22]
45 0.252284739398 26.5898125606 0.878790972879 51 20 23 22 [22]
46 0.251670477415 26.9492474954 0.878842530204 57 18 24 22 [22]
47 0.251280081230 27.2829209815 0.880338630560 59 18 23 24 [22]
48 0.250694239024 27.6360687715 0.880351883812 43 27 25 23 [22]
49 0.250389225905 27.9564744637 0.882230605441 53 23 26 23 [22]
50 0.249899978504 28.2955919172 0.882710158335 55 23 27 23 [22]
51 0.249298738562 28.6460672434 0.882296226172 65 19 29 22 [22]
52 0.248522779708 29.0158614812 0.880543824140 53 26 26 26 [22]
53 0.248493657035 29.2969646636 0.883997664826 65 21 28 25 [22]
54 0.247873677420 29.6460249625 0.883166618398 85 12 26 28 [22]
55 0.247287518630 29.9901852232 0.882487567681 65 23 28 27 [22]
56 0.247069169467 30.2883390498 0.884354337544 55 29 30 26 [22]
57 0.246780784316 30.5932832502 0.885647647935 65 25 30 27 [22]
58 0.246013305376 30.9567529049 0.883425779546 91 13 26 32 [22]
59 0.245541918305 31.2824213515 0.883253877817 85 17 33 26 [22]
60 0.245364167056 31.5692661457 0.885127792661 81 20 29 31 [22]
61 0.245166465838 31.8569248416 0.886797571787 83 20 30 31 [22]
62 0.244515529683 32.2024858062 0.885124290134 79 23 29 33 [22]
63 0.244807950303 32.4223699572 0.890231177749 77 25 30 33 [22]
64 0.244040177229 32.7814874208 0.887579430619 85 22 34 30 [22]
65 0.243648204387 33.0897482647 0.887599706947 65 33 32 33 [22]
66 0.243141818626 33.4127565983 0.886728066347 69 32 35 31 [22]
67 0.243083563137 33.6729997958 0.889073885856 81 27 32 35 [22]
68 0.242446950307 34.0124354660 0.887138836948 89 24 30 38 [22]
69 0.242107817977 34.3096060768 0.887327545790 73 33 34 35 [22]
70 0.241949386955 34.5799605886 0.888793870113 69 36 34 36 [22]
71 0.241505778650 34.8900544752 0.888118447919 85 29 33 38 [22]
72 0.241223911371 35.1759546804 0.888585541359 111 17 29 43 [22]
73 0.241168318196 35.4275545364 0.890679054909 93 27 37 36 [22]
74 0.240778155169 35.7271832281 0.890260728784 99 25 34 40 [22]
75 0.240287333600 36.0412424079 0.889053393749 105 23 31 44 [22]
76 0.240331826600 36.2740050306 0.891770253891 105 24 36 40 [22]
77 0.239791220929 36.5941853643 0.890108833476 97 29 35 42 [22]
78 0.239564555123 36.8658913745 0.890738389217 79 39 39 39 [22]
79 0.239115568871 37.1711238180 0.889676444006 99 30 32 47 [22]
80 0.239257557248 37.3834457430 0.892981322650 95 33 35 45 [22]
81 0.238965740181 37.6623025258 0.893019160822 111 26 33 48 [22]
82 0.238476917311 37.9717468687 0.891548275237 75 45 38 44 [22]
83 0.238163362504 38.2528760232 0.891352309917 101 33 38 45 [22]
84 0.238027809692 38.5045402963 0.892456932102 91 39 35 49 [22]
85 0.237682729972 38.7892904898 0.891959116549 93 39 38 47 [22]
86 0.237374803933 39.0674087639 0.891707843840 101 36 38 48 [22]
87 0.237224597830 39.3187685358 0.892611813869 107 34 36 51 [22]
88 0.236858303691 39.6052465693 0.891860242621 91 43 34 54 [22]
89 0.236781217395 39.8426076014 0.893258858579 89 45 38 51 [22]
90 0.236763849978 40.0687561948 0.895084588620 111 35 43 47 [22]
91 0.236294449684 40.3707832618 0.893466569543 81 51 45 46 [22]
92 0.236186236171 40.6105927344 0.894553307985 123 31 41 51 [22]
93 0.235989523512 40.8647410169 0.894945108925 121 33 40 53 [22]
94 0.235712673997 41.1321103377 0.894703440308 91 49 43 51 [22]
95 0.235626340636 41.3654700849 0.895884168142 121 35 43 52 [22]
96 0.235299256761 41.6404161492 0.895210497173 133 30 39 57 [22]
97 0.235257043373 41.8642420248 0.896681836633 117 39 40 57 [22]
98 0.235268086386 42.0775086357 0.898540412549 113 42 43 55 [22]
99 0.234438733327 42.4412563141 0.893960724117 123 38 45 54 [22]
100 0.234332037117 42.6744892548 0.894872298029 65 68 43 57 [22]


Updates

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

08-May-2013: First complete presentation from N=1 to N=72.
19-Jul-2026: André Müller has published his records on github in December 2025 [22]. For N=12, 15, 20, 22–26, 28, 30, 32–34, 36–72 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= 15, 24, 28, 30, 32–40, 42–96, 98–100, could be significantly 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.
[22]   , github.com/muellan/packing, December 2025.
[31]   , program csqs, 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