The best known packings of unequal circles with inverse square root integer radii in a square (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
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 0.284114639367 13.6317627097 0.841479325418 21 5 14 1 [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 0.269705849671 18.1641573275 0.862894657532 25 12 19 5 [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 0.264809731599 19.9822815807 0.865162269737 47 5 15 13 [22]
29 0.263398267874 20.4449514821 0.863479880282 47 6 18 11 [31]
30 0.262524126666 20.8637036321 0.864975292222 17 22 19 11 [22]
31 0.261757467776 21.2706992092 0.866874256860 45 9 16 15 [31]
32 0.261228810526 21.6547869972 0.870075730291 31 17 21 11 [22]
33 0.260593768869 22.0441289577 0.872315533500 49 9 18 15 [22]
34 0.259312730247 22.4861767847 0.869973524425 45 12 22 12 [22]
35 0.258943495821 22.8469912494 0.873516333762 29 21 21 14 [31]
36 0.258185807930 23.2390775004 0.874229040126 51 11 23 13 [22]
37 0.257660682023 23.6076474010 0.876313409096 33 21 24 13 [22]
38 0.256847053369 24.0003298543 0.876241789506 43 17 22 16 [22]
39 0.256463968132 24.3503913781 0.878928241625 51 14 23 16 [22]
40 0.255014357832 24.8007813133 0.874128023265 57 12 21 19 [22]
41 0.254297684088 25.1796403904 0.874176837617 43 20 24 17 [22]
42 0.254220694993 25.4925772215 0.878481775623 37 24 26 16 [22]
43 0.253125459669 25.9058829280 0.875609874101 43 22 25 18 [22]
44 0.252782732709 26.2409125403 0.877802747550 35 27 26 18 [22]
45 0.252305143013 26.5876622743 0.878933123675 69 11 21 24 [22]
46 0.251815928095 26.9336814173 0.879858661967 49 22 23 23 [22]
47 0.251321040577 27.2784745148 0.880625649207 65 15 23 24 [22]
48 0.250772418927 27.6274530505 0.880901051254 43 27 23 25 [22]
49 0.250430853582 27.9518274201 0.882523974803 63 18 26 23 [22]
50 0.249964723605 28.2882628792 0.883167609846 57 22 28 22 [22]
51 0.249398746534 28.6345802767 0.883004247039 65 19 29 22 [22]
52 0.248644269706 29.0016840503 0.881404939689 59 23 27 25 [22]
53 0.248606997120 29.2836081592 0.884804246535 79 14 28 25 [22]
54 0.247909258446 29.6417700348 0.883420184894 65 22 27 27 [22]
55 0.247513005005 29.9628639188 0.884097674387 73 19 27 28 [22]
56 0.247165750977 30.2765036983 0.885045876451 61 26 29 27 [22]
57 0.246986226470 30.5678358796 0.887122844610 73 21 29 28 [22]
58 0.246238176085 30.9284824431 0.885041524458 91 13 24 34 [22]
59 0.245788559369 31.2510304287 0.885029184363 85 17 33 26 [22]
60 0.245558630285 31.5442657561 0.886531363685 93 14 27 33 [22]
61 0.245348030962 31.8333497329 0.888111545388 95 14 27 34 [22]
62 0.244690600221 32.1794456628 0.886392223185 77 24 29 33 [22]
63 0.244949050745 32.4036933765 0.891257682089 89 19 33 30 [22]
64 0.244184947594 32.7620522020 0.888632808900 79 25 33 31 [22]
65 0.243920212226 33.0528482027 0.889582638423 37 47 31 34 [22]
66 0.243323691336 33.3877821761 0.888055126858 91 21 35 31 [22]
67 0.243215964414 33.6546689753 0.890042660336 97 19 32 35 [22]
68 0.242847773209 33.9562975697 0.890074567812 85 26 28 40 [22]
69 0.242356152668 34.2744501078 0.889148777535 53 43 34 35 [22]
70 0.242189352076 34.5456982052 0.890557753843 85 28 34 36 [22]
71 0.241672643319 34.8659643783 0.889346135208 107 18 30 41 [22]
72 0.241513883458 35.1337208974 0.890723139324 75 35 33 39 [22]
73 0.241457019651 35.3851950863 0.892812786672 103 22 35 38 [22]
74 0.240958576930 35.7004319027 0.891595422852 119 15 32 42 [22]
75 0.240535247836 36.0040955152 0.890888885379 113 19 31 44 [22]
76 0.240587379230 36.2354746745 0.893667758694 94 29 40 36 [22]
77 0.240150039239 36.5395084473 0.892774705134 111 22 35 42 [22]
78 0.239800027258 36.8296908358 0.892490294050 65 46 41 37 [22]
79 0.239345824914 37.1353643646 0.891390694650 119 20 31 48 [22]
80 0.239523949749 37.3418688167 0.894970943893 115 23 35 45 [22]
81 0.239064659157 37.6467188070 0.893758637742 115 24 32 49 [22]
82 0.238760554715 37.9266380451 0.893670298816 95 35 39 43 [22]
83 0.238411968274 38.2129875656 0.893214149457 95 36 40 43 [22]
84 0.238272233886 38.4650416058 0.894290751997 95 37 38 46 [22]
85 0.237966788834 38.7429880551 0.894092382976 89 41 37 48 [22]
86 0.237774949951 39.0016631164 0.894716706447 121 26 35 51 [22]
87 0.237472158181 39.2777794439 0.894475790850 107 34 36 51 [22]
88 0.237143812066 39.5575639858 0.894011630454 125 26 35 53 [22]
89 0.237064978534 39.7949169481 0.895401123369 118 31 36 53 [22]
90 0.237109585458 40.0103309285 0.897700599865 101 40 41 49 [22]
91 0.236758866209 40.2915935817 0.896982085080 113 35 41 50 [22]
92 0.236494919875 40.5575859799 0.896893109654 133 26 41 51 [22]
93 0.236287726257 40.8131683932 0.897208291631 117 35 40 53 [22]
94 0.236028055454 41.0771494777 0.897099252486 119 35 40 54 [22]
95 0.235980603090 41.3033707735 0.898580103672 135 28 40 55 [22]
96 0.235565642816 41.5933276771 0.897238608820 145 24 36 60 [22]
97 0.235257043373 41.8642420248 0.896681836633 117 39 40 57 [22]
98 0.235649418967 42.0094179736 0.901455558626 137 30 39 59 [22]
99 0.234950281835 42.3488505455 0.897866249375 119 40 42 57 [22]
100 0.234490894926 42.6455790668 0.896086008625 67 67 45 55 [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    21-Jul-2026