The best known packings of variable-sized circles in a square with maximized sum of radii (complete up to N = 100)


Last update: 16-Aug-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 sum of radii 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
sum of radii
of all circles in the container square, the latter has always a side length of 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
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
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, see at bottom of the page
records
the sequence of N 's that establish density records

N sum of radii density contacts boundary core symmetry reference
1 0.500000000000 0.785398163397 4 1
2 0.585786437626 0.539012084451 5 2 D2 [1]
3 0.796287456147 0.678218187440 9 3 D1 [1]
4 1.006788474668 0.817424290428 13 4 D2 [1]
5 1.103553390593 0.819086418676 16 4 1 D4 [1]
6 1.202838910871 0.830664461672 19 6 D1 [1]
7 1.306546350758 0.832372446742 22 6 1 D1 [1]
8 1.423813915320 0.817424290433 24 8 D4 [1]
9 1.524365359370 0.839665382935 28 8 1 D1 [1]
10 1.591012852634 0.840405214793 31 8 2 D1 [1]
11 1.680058380778 0.845401576001 34 8 3 D1 [1]
12 1.765978318170 0.842428591982 37 8 4 D2 [1]
13 1.829542411692 0.843172230057 39 10 3 [1]
14 1.905667205293 0.847885561040 43 10 4 D1 [1]
15 1.980266508444 0.846184875220 45 12 3 [1]
16 2.053080418439 0.853389973899 49 12 4 D1 [1]
17 2.111185327830 0.850943221106 51 12 5 [1]
18 2.178530295972 0.856434344552 55 12 6 D1 [1]
19 2.236704571294 0.853822984295 57 12 7 [1]
20 2.301122834497 0.854890113387 60 12 8 [1]
21 2.362117319374 0.855805001847 64 12 9 D2 [1]
22 2.420202649748 0.858428348512 66 13 9 [1]
23 2.478013611963 0.858563010053 69 16 7 [1]
24 2.530311586971 0.860593386358 72 16 8 [2]
25 2.587275055266 0.863260424813 76 16 9 D2 [1]
26 2.635983084918 0.862734562615 78 16 10 [8]
27 2.685978684198 0.864374412342 82 16 11 D1 [11]
28 2.737739985536 0.862283425201 84 16 12 [1]
29 2.790344154631 0.862992740834 87 16 13 [1]
30 2.842668747462 0.863678798204 90 16 14 [1]
31 2.889969851933 0.863902357568 93 16 15 [1]
32 2.939572771205 0.865911045200 96 18 14 [7]
33 2.987285008591 0.867066089447 99 19 14 [8]
34 3.029799271188 0.869081132598 103 20 14 D1 [8]
35 3.074036363728 0.869011650678 105 19 16 [8]
36 3.121754486102 0.869881804498 109 20 16 D2 [1]
37 3.161498916179 0.870181953644 111 20 17 [1]
38 3.205945627974 0.870613578474 115 20 18 D1 [1]
39 3.248110798173 0.870172333103 117 20 19 [8]
40 3.292391572609 0.869253904522 120 20 20 [8]
41 3.336245021959 0.870463471234 123 20 21 [1]
42 3.380421747611 0.870873606317 126 20 22 [10]
43 3.422605518048 0.868635886925 129 20 23 [1]
44 3.466043568234 0.872329991512 132 21 23 [2]
45 3.503095055216 0.871496118603 135 22 23 [1]
46 3.539568936862 0.872596494882 138 24 22 [2]
47 3.578955041750 0.873515182161 141 24 23 [10]
48 3.615166622821 0.873096687255 144 24 24 [10]
49 3.655413576541 0.874887674201 148 24 25 D2 [1]
50 3.691712400780 0.874835126816 150 24 26 [11]
51 3.729658093746 0.875234133070 154 24 27 D1 [11]
52 3.766233110734 0.874576786203 156 24 28 [11]
53 3.804408802004 0.875187663456 159 24 29 [11]
54 3.842635794938 0.874323099376 162 24 30 [11]
55 3.882016352766 0.874956093662 165 24 31 [11]
56 3.918995278232 0.875487011034 168 24 32 [10]
57 3.954683754497 0.875557633127 171 25 32 [10]
58 3.990566331646 0.876764162060 174 25 33 [10]
59 4.023613974606 0.877500688539 177 27 32 [10]
60 4.057375010904 0.877801322733 180 27 33 [1]
61 4.091602684217 0.877899009046 183 27 34 [10]
62 4.122816770019 0.877868629969 186 27 35 [11]
63 4.153848570061 0.877430080627 189 28 35 [11]
64 4.190585830354 0.878610892803 193 28 36 D1 [1]
65 4.219413302014 0.878356876349 195 28 37 [10]
66 4.254094763412 0.878429592692 198 28 38 [11]
67 4.287777805457 0.878098780362 201 27 40 [11]
68 4.322886593217 0.878117826656 204 27 41 [11]
69 4.356995844873 0.878741895709 207 28 41 [11]
70 4.392179819755 0.879139148529 210 28 42 [2]
71 4.426322769390 0.879358932043 213 28 43 [11]
72 4.458700040315 0.879338187746 216 28 44 [11]
73 4.488004575020 0.878835121728 219 28 45 [10]
74 4.518264420768 0.879682048635 222 29 45 [10]
75 4.548223289937 0.880207829133 225 29 46 [10]
76 4.578201562751 0.880750187906 228 30 46 [10]
77 4.605135681249 0.880791449546 231 31 46 [11]
78 4.635843558758 0.880645110599 234 32 46 [10]
79 4.664674436598 0.881288148502 237 32 47 [10]
80 4.695590667896 0.881087216471 240 32 48 [11]
81 4.726033735177 0.881670504106 244 32 49 D2 [1]
82 4.752137878070 0.881034104933 246 31 51 [11]
83 4.783919133241 0.879724778343 249 30 53 [11]
84 4.813516974275 0.880809583417 252 31 53 [11]
85 4.845134378577 0.881276607867 255 32 53 [11]
86 4.876312684220 0.881234110832 258 32 54 [10]
87 4.906503570705 0.881730994014 261 32 55 [11]
88 4.939497171360 0.881979228494 264 32 56 [10]
89 4.966708260631 0.881433957967 267 33 56 [2]
90 4.996999276706 0.883087421247 270 34 56 [1]
91 5.021462693130 0.882997368509 273 34 57 [10]
92 5.048426925392 0.883223324673 276 34 58 [10]
93 5.071029091028 0.883182351521 279 34 59 [10]
94 5.098748716187 0.883467205785 282 36 58 [10]
95 5.125567045105 0.883049384536 285 34 61 [10]
96 5.152146722729 0.883229405231 288 35 61 [10]
97 5.180639279289 0.883126820718 291 35 62 [10]
98 5.204165507569 0.882936834557 294 35 63 [10]
99 5.233333680412 0.884184249236 297 36 63 [13]
100 5.261573443983 0.884045506369 301 36 64 D1 [1]
104 5.312596980368 0.883417941269 312 36 68 [12]
110 5.451091711899 0.883522496596 330 35 75 [12]
113 5.466919623437 0.879395103092 340 28 85 D4 [2]
120 5.703544322766 0.885034678325 360 38 82 [12]
125 5.820095292518 0.884827546846 375 39 86 [12]
130 5.818452084708 0.882860215084 390 38 92 [12]
135 5.988451165267 0.883737776945 405 39 96 [12]
140 6.103360445068 0.883560677918 420 40 100 [12]
145 6.203179763987 0.882166715266 436 32 113 D4 [2]
150 6.246254196217 0.880386787598 450 40 110 [12]
155 6.412434005100 0.885757529914 465 41 114 [12]
160 6.535002405365 0.886094857714 480 42 118 [12]
170 6.746840077835 0.886604253785 510 43 127 [12]
181 6.939889809379 0.884421938014 544 36 145 D4 [2]
221 7.676927093459 0.886293611742 664 40 181 D4 [2]
265 8.414210400025 0.887872509444 796 44 221 D4 [2]
313 9.151683700015 0.889222778983 940 48 265 D4 [2]
365 9.889307019483 0.890391048793 1096 52 313 D4 [2]
421 10.627051029404 0.891412034803 1264 56 365 D4 [2]
481 11.364893695176 0.892312122358 1444 60 421 D4 [2]
545 12.102818122811 0.893111728021 1636 64 481 D4 [2]
613 12.840811129581 0.893826900219 1840 68 545 D4 [2]
685 13.578862268967 0.894470430463 2056 72 613 D4 [2]
761 14.316963149341 0.895052641377 2284 76 685 D4 [2]
841 15.055106947710 0.895581956188 2524 80 761 D4 [2]
925 15.793288056089 0.896065317340 2776 84 841 D4 [2]
1013 16.978328452483 0.899371591427 3039 117 896 [13]


Updates

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

about 2011: First complete presentation from N=1 to N=100 due to results by David W. Cantrell [1]. He posted his packings at the sci.math forum, but I can't found or remember the URL. Shortly later, my program found many improvements. Unfortunately, this web page disappeared 2012 for some mysterious reason. In the meantime, Erich Friedman's Packing Center [3] shows the best known results for N=1–40, but only with poor accuracy (3 decimal places) and without providing coordinates of the packings.
23-Jul-2026: Revival of the old web page as of 2012. In the last time, new approaches driven by LLM yield better results, see [4]–[7].
24-Jul-2026: The case N=26 attracted some attention in the last year. Now this record, credited to Yiping Wang [5], is present here, it has D1 symmetry. The former candidate lacks this property.
25-Jul-2026: It is not easy to reproduce an announced new record if no coordinates of the circles are available [7]. Now the case N=32 is shown here.
27-Jul-2026: What a surprise! Yiping Wang's record for N=26 could be beaten again [2]. The value of 'sum of radii' is raised from 2.635977394754 to 2.635983084918. The new candidate configuration has no symmetry.
01-Aug-2026: Command back! Everett Dutton pointed me yesterday to the fact that Haowei Lin [8] published his results two weeks ago. He refers to Jonathan Viquerat [9]. So the credits for N=26, 33 should go to him. Moreover, the database is updated by his new records for N=34, 35, 39, 40.
01-Aug-2026: Everett Dutton from Gurobi Optimization LLC [10] sent me significant improvements for the cases N=42, 47, 48, 53–59, 61, 69, 73–75, 77, 78, 80, 83, 86–88, 91, 95, 96, 99. Great work, Everett!
01-Aug-2026: Some of these new records could be immediately reclaimed [2]: N=58, 62, 69, 86, 88, 96, 99.
02-Aug-2026: Nearly all of my previous records are blown away by Everett Dutton [10]: N=58, 62, 86, 88, 96, 99.
03-Aug-2026: Today an old record by David Cantrell was beaten by Jason Liang: N=27. The packing has a remarkable D1 symmetry.
04-Aug-2026: An incredible amount of new candidates by Everett Dutton [10]: N=48, 51, 52, 54–56, 61, 63, 65, 68, 71, 74–80, 82–85, 87, 89, 91–95, 97, 98.
06-Aug-2026: A few candidates more by Everett Dutton [10]: N=50, 66, 67.
06-Aug-2026: For the cases N=66, 83, 89, I could do better [2].
08-Aug-2026: An own class of packings is presented today [2]. These examples are far from being records. But their full D4 symmetry makes them quite pretty to look at: N=113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013. The formula for generating the number of circles is 2 n2 - 2 n + 1. Be inspired and enjoy!
10-Aug-2026: It is a good time to fillup the table with additional entries. Today Rasim Abiyev contributes the case N=104 [12].
11-Aug-2026: Jason Liang [11] earned with 21 new records an own color in the table: N=50–55, 62, 63, 66–69, 71, 72, 77, 80, 82–85, 87.
12-Aug-2026: Rasim Abiyev [12] sent me two more new packings: N=110, 120.
13-Aug-2026: Next entries for N=125, 130, 135, 140, 150, 155, 160, 170 by Rasim Abiyev [12].
16-Aug-2026: Welcome to packomania! Byron Tasseff sent me two improvements: N=99, 1013 [13]. It is nice to see how the 1013 circles had been rearranged.

References

[1]   , sci.math forum 2011/12.
[2]   , program csqv, 2011–2026.
[3]   E. Friedman, Circles in Squares.
[4]   A. Novikov et al., AlphaEvolve: A coding agent for scientific and algorithmic discovery, Google DeepMind, Jun 2025.
[5]   Y. Wang, Seems a new circle packing result (2.635977) when reproducing your example #156, Jul 2025.
[6]   Breaking the world record, a mysterious student defeated Google's AlphaEvolve's optimal solution to a difficult problem with a margin of only 0.00006442, 36kr Europe, Jul 2025.
[7]   Sebastian Pokutta, Not every discovery needs an LLM, Apr 2026.
[8]   Haowei Lin, packing_records, Mid July 2026.
[9]   Jonathan Viquerat, Policy-based optimization, 2025.
[10]   , private communication, July/August 2026.
[11]   , circle-packing-sota , August 2026.
[12]   , Circle Packing Explorations, August 2026.
[13]   Byron Tasseff, private communication, August 2026.

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©  E. Specht    17-Aug-2026