The best known packings of variable-sized circles in a square with maximized sum of radii (complete up to N = 500)


Last update: 30-Sep-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   97-108   109-120   121-132   133-144   145-156   157-168   169-180   181-192   193-204   205-216   217-228   229-240   241-252   253-264   265-276   277-288   289-300   301-312   313-324   325-336   337-348   349-360   361-372   373-384   385-396   397-408   409-420   421-432   433-444   445-456   457-468   469-480   481-492   493-500  

Overview (contacts)

1-6   7-12   13-18   19-24   25-30   31-36   37-42   43-48   49-54   55-60   61-66   67-72   73-78   79-84   85-90   91-96   97-102   103-108   109-114   115-120   121-126   127-132   133-138   139-144   145-150   151-156   157-162   163-168   169-174   175-180   181-186   187-192   193-198   199-204   205-210   211-216   217-222   223-228   229-234   235-240   241-246   247-252   253-258   259-264   265-270   271-276   277-282   283-288   289-294   295-300   301-306   307-312   313-318   319-324   325-330   331-336   337-342   343-348   349-354   355-360   361-366   367-372   373-378   379-384   385-390   391-396   397-402   403-408   409-414   415-420   421-426   427-432   433-438   439-444   445-450   451-456   457-462   463-468   469-474   475-480   481-486   487-492   493-498   499-500  

Mean radii plots

2-50   50-100   100-150   150-200   200-250   250-300   300-350   350-400   400-450   450-500  

Standard deviation of radii plots

2-50   50-100   100-150   150-200   200-250   250-300   300-350   350-400   400-450   450-500  

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 D4
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 2.420202649747 0.858428348511 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 2.635983084919 0.862734562616 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 2.987285008592 0.867066089447 99 19 14 [8]
34 3.029799271186 0.869081132597 103 20 14 D1 [8]
35 3.074036363730 0.869011650679 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 3.248110798175 0.870172333104 117 20 19 [8]
40 3.292391572608 0.869253904522 120 20 20 [8]
41 3.336245021959 0.870463471234 123 20 21 [1]
42 3.380421747613 0.870873606318 126 20 22 [10]
43 3.422605518046 0.868635886924 129 20 23 [1]
44 3.466043568234 0.872329991512 132 21 23 [2]
45 3.503095055216 0.871496118603 135 22 23 [1]
46 3.539571078547 0.873734875767 138 23 23 [17]
47 3.578955041751 0.873515182162 141 24 23 [10]
48 3.616465058812 0.874916588165 144 24 24 [18]
49 3.655413576541 0.874887674201 148 24 25 D2 [1]
50 3.691712400814 0.874835126831 150 24 26 [11]
51 3.729658093785 0.875234133088 154 24 27 D1 [11]
52 3.766233110734 0.874576786203 156 24 28 [11]
53 3.805568432565 0.874359269184 159 24 29 [18]
54 3.842678286689 0.874546912423 162 23 31 [22]
55 3.882016352806 0.874956093680 165 24 31 [11]
56 3.918995278232 0.875487011034 168 24 32 [10]
57 3.954683754499 0.875557633128 171 25 32 [10]
58 3.990566331646 0.876764162060 174 25 33 [10]
59 4.023613974609 0.877500688540 177 27 32 [10]
60 4.057789066904 0.877871489691 180 27 33 [17]
61 4.091772479980 0.877821018367 183 27 34 [17]
62 4.123334225100 0.878497966628 186 27 35 [17]
63 4.155766324151 0.878567921428 189 28 35 [17]
64 4.190585830354 0.878610892803 193 28 36 D1 [1]
65 4.222869486756 0.878829689163 195 28 37 [17]
66 4.255807931448 0.878501255196 198 28 38 [17]
67 4.289552804745 0.877204785993 201 27 40 [17]
68 4.324531544548 0.878098582388 204 27 41 [17]
69 4.357977925354 0.877206123511 207 27 42 D1 [22]
70 4.392230522115 0.879186208800 210 28 42 D1 [17]
71 4.426322769438 0.879358932062 213 28 43 [11]
72 4.458700040364 0.879338187765 216 28 44 [11]
73 4.489319191848 0.879490904751 219 28 45 [22]
74 4.518264420770 0.879682048636 222 29 45 [10]
75 4.550081805198 0.880777582990 225 30 45 [17]
76 4.580531017693 0.880521246375 228 29 47 [22]
77 4.608377826571 0.880656349010 231 30 47 [17]
78 4.636377420432 0.881673124349 234 32 46 [17]
79 4.666928346046 0.879284608132 237 28 51 [17]
80 4.695876897580 0.879316118103 240 28 52 [17]
81 4.726033735177 0.881670504106 244 32 49 D2 [1]
82 4.755873680213 0.879928160452 246 29 53 [17]
83 4.786393807284 0.879308643833 249 29 54 [17]
84 4.816567858213 0.880756242321 252 30 54 [17]
85 4.848035445600 0.881188270335 255 31 54 [17]
86 4.877360712054 0.881155084175 258 31 55 [17]
87 4.907558130324 0.881831503253 261 32 55 [17]
88 4.939497171364 0.881979228495 264 32 56 [10]
89 4.967596361079 0.882194573105 267 32 57 [17]
90 4.996999276706 0.883087421247 270 34 56 [1]
91 5.022578168631 0.883517421100 273 34 57 [22]
92 5.049048324631 0.882543691631 276 33 59 [17]
93 5.076445594003 0.882304841480 279 32 61 [17]
94 5.102335812493 0.882332455049 282 32 62 [17]
95 5.129349439048 0.883722275429 285 35 60 [22]
96 5.155358373110 0.883833852748 288 35 61 [22]
97 5.182315239435 0.882447334982 291 32 65 [22]
98 5.209107205724 0.882005024255 294 32 66 [22]
99 5.236421941919 0.882126924245 297 32 67 [17]
100 5.263989520982 0.881869913182 300 32 68 [22]
101 5.291263171788 0.881898004058 303 32 69 [22]
102 5.319420509692 0.882370370223 306 32 70 [22]
103 5.345851378686 0.882647956948 309 33 70 [22]
104 5.374249672894 0.883541255273 312 35 69 [22]
105 5.401448182925 0.883412287517 315 35 70 [21]
106 5.429079100881 0.884101982994 318 36 70 [14]
107 5.454618744051 0.884334024843 321 36 71 [17]
108 5.482068537918 0.884055064979 324 36 72 [17]
109 5.507925934200 0.884411360128 327 37 72 [14]
110 5.531919536401 0.885383679816 330 38 72 [20]
111 5.555132386885 0.884205597984 333 36 75 [22]
112 5.579487231138 0.883858339264 336 36 76 [17]
113 5.604269563353 0.884164579951 339 36 77 [22]
114 5.629424575175 0.884415622975 342 36 78 [22]
115 5.652307617702 0.884638217560 345 37 78 [22]
116 5.676573137511 0.885455079083 348 38 78 [17]
117 5.702235739011 0.884614163204 351 36 81 [2]
118 5.727559469522 0.884410805947 354 36 82 [22]
119 5.752646828336 0.884682616756 357 36 83 [22]
120 5.777303097445 0.884174155617 360 36 84 [23]
121 5.803819757799 0.884742186393 363 36 85 [17]
122 5.828135840589 0.884643675210 366 37 85 [17]
123 5.853393864742 0.884586718245 369 36 87 [17]
124 5.878043164365 0.884738506418 372 36 88 [17]
125 5.902296796096 0.885361855450 375 38 87 [17]
126 5.926814799439 0.885755127651 378 39 87 [22]
127 5.951639782676 0.885728890783 381 40 87 [22]
128 5.976280217625 0.886589707717 384 41 87 [22]
129 5.998868370219 0.886147621277 387 40 89 [22]
130 6.022424077079 0.885979259876 390 40 90 [22]
131 6.044773912185 0.886625430814 393 41 90 [22]
132 6.067101865258 0.886386759599 396 40 92 [22]
133 6.089190901004 0.886062274219 399 40 93 [22]
134 6.111583027658 0.886205607630 402 40 94 [22]
135 6.133616190771 0.886571264308 405 40 95 [17]
136 6.156404709652 0.886403927622 408 40 96 [17]
137 6.179272030442 0.886424394359 411 40 97 [2]
138 6.201196207424 0.886680421605 414 40 98 [22]
139 6.224048511183 0.886488100077 417 40 99 [22]
140 6.248286289975 0.886525040430 420 40 100 [22]
141 6.270968968004 0.886591190326 423 40 101 [2]
142 6.293740807174 0.886344592838 426 40 102 [22]
143 6.319024034482 0.886521754998 429 40 103 [2]
144 6.341449107518 0.886127577247 432 40 104 [22]
145 6.364667389585 0.886684054251 435 40 105 [2]
146 6.386708980523 0.887087230866 438 42 104 [17]
147 6.410094597843 0.886759619574 441 41 106 [22]
148 6.432461590552 0.887321218416 444 44 104 [22]
149 6.455635125850 0.888302731351 447 45 104 D1 [22]
150 6.478189273051 0.888174096402 450 45 105 [22]
151 6.499346535645 0.888360380985 453 45 106 [22]
152 6.520366820754 0.888195923564 456 45 107 [22]
153 6.541408426141 0.888376491024 459 45 108 [22]
154 6.561927143305 0.888008715319 462 45 109 [22]
155 6.582091347104 0.887946674330 465 46 109 [22]
156 6.602458999560 0.888070256218 468 46 110 [22]
157 6.622760196297 0.888517839103 471 45 112 [22]
158 6.643473474163 0.888291881534 474 45 113 [22]
159 6.663891763480 0.887839630787 477 44 115 [17]
160 6.686377484769 0.888483808233 480 46 114 [15]
161 6.707499445479 0.888622253898 483 46 115 [22]
162 6.728706099717 0.888773453818 486 46 116 [22]
163 6.749584271556 0.887897402681 489 44 119 [17]
164 6.771350627575 0.887911575755 492 44 120 [17]
165 6.792830162827 0.888083577627 495 44 121 [22]
166 6.814618735087 0.888073050171 498 44 122 [22]
167 6.836877925708 0.887883393429 501 44 123 [22]
168 6.858106347377 0.888155696766 504 45 123 [2]
169 6.880734941065 0.888277688447 507 45 124 [2]
170 6.901615116068 0.888158634614 510 45 125 [22]
171 6.922613061509 0.888449071175 513 46 125 [22]
172 6.943340559805 0.888483807498 516 46 126 [22]
173 6.964232150231 0.888783788885 519 47 126 [22]
174 6.985262310499 0.889290941390 522 48 126 [22]
175 7.004707257451 0.889424973065 525 48 127 [22]
176 7.023425437597 0.889247810051 528 48 128 [22]
177 7.042544930878 0.888897884163 531 47 130 [22]
178 7.061720669271 0.888852964046 534 47 131 [22]
179 7.080903944143 0.889435355703 537 48 131 [22]
180 7.099941756488 0.889129905864 540 48 132 [22]
181 7.120170554574 0.889141306389 543 48 133 [22]
182 7.138696832806 0.889548778648 546 49 133 [17]
183 7.158659503357 0.889738851285 549 49 134 [17]
184 7.178399270760 0.889713608932 552 49 135 [17]
185 7.197625081950 0.889550616569 555 49 136 [22]
186 7.218383296615 0.889730118196 558 49 137 [22]
187 7.237822453752 0.889857545873 561 49 138 [22]
188 7.256806779802 0.889726172180 564 49 139 [22]
189 7.277259920003 0.888966700277 567 47 142 [22]
190 7.297283358464 0.888953067993 570 47 143 [22]
191 7.314938403871 0.890041248796 573 49 142 [17]
192 7.336598450128 0.889155924951 576 48 144 [22]
193 7.357305320712 0.889296590660 579 48 145 [22]
194 7.376248679483 0.889530934984 582 49 145 [22]
195 7.395054318099 0.889605073007 585 49 146 [22]
196 7.414752186446 0.889762709027 588 50 146 [22]
197 7.435507140839 0.889947388143 591 50 147 [22]
198 7.454135017058 0.889900936101 594 51 147 [22]
199 7.475311776105 0.890172495469 597 51 148 [22]
200 7.493815304616 0.890000628821 600 51 149 [22]
201 7.513577601745 0.890530839057 603 52 149 [22]
202 7.530737482199 0.890587402082 606 52 150 [22]
203 7.549453338312 0.890346714899 609 52 151 [22]
204 7.567040637207 0.890918267936 612 54 150 [22]
205 7.585145048634 0.890351288527 615 51 154 [22]
206 7.602847648701 0.890346021303 618 51 155 [22]
207 7.620986073417 0.890454478852 621 52 155 [22]
208 7.638874318313 0.890431587708 624 52 156 [22]
209 7.656781531190 0.890728635164 627 53 156 [22]
210 7.674608739197 0.890469780080 630 52 158 [22]
211 7.693286838327 0.890495738672 633 52 159 [22]
212 7.712135369769 0.890618622157 636 52 160 [22]
213 7.730352006424 0.890001121929 639 50 163 [22]
214 7.749920828063 0.889858288263 642 50 164 [22]
215 7.768139491031 0.889645224706 645 50 165 [22]
216 7.787370320365 0.889767444469 648 50 166 [22]
217 7.805323305341 0.890065917003 651 51 166 [22]
218 7.824384394934 0.890330584948 654 51 167 [22]
219 7.843270863590 0.889911558361 657 51 168 [22]
220 7.862978781947 0.890354013349 660 51 169 [17]
221 7.881322193729 0.890745224994 663 52 169 [22]
222 7.900508311326 0.890490556288 666 52 170 [17]
223 7.918453776494 0.890387275584 669 52 171 [22]
224 7.936705752698 0.890995728309 672 53 171 [22]
225 7.955497269730 0.890868903498 675 53 172 [22]
226 7.973731098060 0.891279908735 678 54 172 [22]
227 7.991139399236 0.891140716658 681 54 173 [22]
228 8.008136439065 0.891265775166 684 55 173 [22]
229 8.025414113644 0.891293312050 687 55 174 [22]
230 8.042086155658 0.891357576565 690 55 175 [22]
231 8.059954604485 0.891742080399 693 56 175 [22]
232 8.075645300615 0.891683281816 696 56 176 [22]
233 8.093167942343 0.891431104949 699 56 177 [22]
234 8.109403544494 0.891437152840 702 56 178 [22]
235 8.124965979221 0.891208716234 705 55 180 [22]
236 8.142183830581 0.891324930085 708 55 181 [22]
237 8.159361330801 0.891552335870 711 56 181 [22]
238 8.176808422664 0.891358067544 714 56 182 [22]
239 8.193958609549 0.891754274055 717 56 183 [22]
240 8.211062781761 0.891467392617 720 56 184 [22]
241 8.228747504774 0.891495817542 723 56 185 [17]
242 8.245913651626 0.891031472007 726 54 188 [22]
243 8.264972382603 0.890992729875 729 54 189 [22]
244 8.282696623107 0.890934867764 732 54 190 [22]
245 8.300102521900 0.891134308508 735 55 190 [22]
246 8.317883144176 0.891154017068 738 55 191 [22]
247 8.336758433778 0.891392261360 741 55 192 [22]
248 8.354624968831 0.891379613726 744 55 193 [22]
249 8.372100365514 0.891603966753 747 56 193 [22]
250 8.389973532838 0.891690197961 750 56 194 [23]
251 8.406300148899 0.891675575304 753 56 195 [22]
252 8.424518228844 0.891853845776 756 57 195 [22]
253 8.441592362873 0.891818772736 759 57 196 [17]
254 8.458872827124 0.891833528053 762 57 197 [22]
255 8.477029407017 0.892448189915 765 59 196 [17]
256 8.492946191754 0.892428161009 768 59 197 [22]
257 8.508422799988 0.892092535661 771 58 199 [22]
258 8.524409881740 0.892111533798 774 58 200 [22]
259 8.541425245228 0.891996374678 777 58 201 [22]
260 8.557871248284 0.892166423174 780 58 202 [22]
261 8.573654744714 0.891941246366 783 58 203 [22]
262 8.589424076094 0.891958775250 786 58 204 [22]
263 8.604914868198 0.892133781749 789 58 205 [22]
264 8.621387757251 0.892301315311 792 58 206 [22]
265 8.636677283514 0.892197461791 795 58 207 [22]
266 8.654332484807 0.892341834975 798 58 208 [22]
267 8.670342483549 0.892326330876 801 58 209 [22]
268 8.686758922131 0.892373861884 804 58 210 [22]
269 8.702116357477 0.892547816472 807 59 210 [22]
270 8.717369002184 0.892822407873 810 60 210 [22]
271 8.733631185820 0.892346623950 813 60 211 [22]
272 8.750101147793 0.892525551155 816 59 213 [22]
273 8.766441467546 0.892588769608 819 59 214 [22]
274 8.782947126324 0.892652869758 822 60 214 [22]
275 8.799100032873 0.892242445483 825 58 217 [22]
276 8.815248638705 0.892233476465 828 58 218 [22]
277 8.831359412359 0.892129278397 831 58 219 [22]
278 8.847454899623 0.892156122316 834 58 220 [22]
279 8.865026989516 0.892576678964 837 60 219 [22]
280 8.881441244871 0.892474953500 840 60 220 [22]
281 8.897344530591 0.892539806163 843 60 221 [22]
282 8.914254825361 0.892451968757 846 60 222 [22]
283 8.931076985022 0.892495002398 849 60 223 [22]
284 8.946990762449 0.892445728354 852 60 224 [22]
285 8.963362670814 0.892574700813 855 60 225 [22]
286 8.979493799189 0.892563980464 858 60 226 [22]
287 8.996138387721 0.893079014853 861 63 224 [17]
288 9.013518286200 0.893346478215 864 64 224 [17]
289 9.027041535254 0.893594826034 867 65 224 [17]
290 9.042284795006 0.892551171348 870 60 230 [22]
291 9.058001856273 0.892731039916 873 60 231 [22]
292 9.072594144101 0.892711170314 876 60 232 [22]
293 9.088041361012 0.892498603645 879 60 233 [22]
294 9.103186323553 0.892621506768 882 60 234 [22]
295 9.116204790462 0.892844611237 885 61 234 [22]
296 9.131106084861 0.892725234763 888 61 235 [22]
297 9.147139780469 0.892766605333 891 60 237 [22]
298 9.161800772953 0.892871655795 894 61 237 [22]
299 9.176674773948 0.892847320726 897 61 238 [22]
300 9.192717162187 0.893008329243 900 62 238 [22]
301 9.206706957124 0.892681921573 903 61 240 [22]
302 9.222457409110 0.892994928676 906 62 240 [22]
303 9.237872384633 0.892846635683 909 61 242 [22]
304 9.253504120852 0.892925051635 912 61 243 [22]
305 9.269547427839 0.892898643255 915 61 244 [22]
306 9.284302999971 0.892910606833 918 61 245 [22]
307 9.305003208927 0.893014261753 921 61 246 [22]
308 9.319968125993 0.892994373658 924 61 247 [22]
309 9.335666509317 0.892987253481 927 62 247 [22]
310 9.352370720792 0.893120644099 930 62 248 [22]
311 9.367949471824 0.893000896761 933 62 249 [22]
312 9.383900680341 0.892867438679 936 62 250 [22]
313 9.399575833292 0.893354404002 939 63 250 [22]
314 9.414131191837 0.893044966391 942 62 252 [22]
315 9.430651133292 0.893113211381 945 63 252 [22]
316 9.446030411797 0.893085427348 948 63 253 [22]
317 9.461295946830 0.893293528703 951 63 254 [22]
318 9.475501126651 0.893283365442 954 64 254 [22]
319 9.491078399249 0.893504207311 957 65 254 [22]
320 9.506126610664 0.893531904655 960 65 255 [22]
321 9.521410018032 0.893722385972 963 66 255 [22]
322 9.535451604148 0.893865055553 966 67 255 [22]
323 9.549618944734 0.894018447945 969 67 256 [22]
324 9.563484194583 0.894039663192 972 67 257 [22]
325 9.577108334936 0.893893154751 975 68 257 [22]
326 9.590955525107 0.893879510645 978 67 259 [22]
327 9.605597654827 0.893829078220 981 67 260 [22]
328 9.619828359469 0.893830538637 984 67 261 [22]
329 9.634735015835 0.893934934696 987 67 262 [22]
330 9.649407813676 0.893924758003 990 67 263 [22]
331 9.661510288207 0.893795546687 993 66 265 [22]
332 9.676557878917 0.893736817326 996 66 266 [22]
333 9.690284870774 0.893806400907 999 66 267 [22]
334 9.709694515096 0.894076873367 1002 67 267 [22]
335 9.719504825163 0.893592308768 1005 65 270 [22]
336 9.733809322739 0.893693150249 1008 66 270 [22]
337 9.748019005603 0.893776988648 1011 66 271 [22]
338 9.762187506496 0.893882088693 1014 67 271 [22]
339 9.775655735345 0.893736435835 1017 66 273 [22]
340 9.798209662529 0.893327291834 1020 65 275 [22]
341 9.805007478330 0.893995317317 1023 67 274 [22]
342 9.828802384035 0.893485600480 1026 65 277 [22]
343 9.833219378303 0.893870245292 1029 67 276 [22]
344 9.857945183870 0.893854895871 1032 66 278 [22]
345 9.862417622270 0.894026911830 1035 67 278 [22]
346 9.887015627054 0.893875176061 1038 67 279 [22]
347 9.889662892792 0.894152949640 1041 68 279 [22]
348 9.916834535046 0.893931471118 1044 67 281 [22]
349 9.931453646678 0.894038496668 1047 67 282 [22]
350 9.940090471554 0.893648535064 1050 65 285 [22]
351 9.960345542076 0.894122985598 1053 68 283 [22]
352 9.974894607704 0.894383187842 1056 69 283 [22]
353 9.988919587821 0.894302467230 1059 70 283 [22]
354 10.004674669830 0.894555804349 1062 70 284 [22]
355 10.011883344382 0.894158341580 1065 68 287 [22]
356 10.026760098873 0.894187199711 1068 68 288 [22]
357 10.040229808529 0.894117771201 1071 68 289 [22]
358 10.053590103715 0.894129696599 1074 68 290 [22]
359 10.067804687286 0.894158824981 1077 68 291 [22]
360 10.082022880434 0.894134142377 1080 68 292 [22]
361 10.096748872686 0.894183286242 1083 69 292 [22]
362 10.111182220237 0.894235791165 1086 69 293 [22]
363 10.125277523602 0.894686122405 1089 70 293 [22]
364 10.139103802287 0.894675578340 1092 70 294 [22]
365 10.153462171620 0.894734744725 1095 71 294 [20]
366 10.166756765071 0.894725522768 1098 72 294 [22]
367 10.181369332522 0.894633349977 1101 71 296 [22]
368 10.195502127629 0.894695476906 1104 71 297 [22]
369 10.209460915964 0.894712874069 1107 71 298 [22]
370 10.222766514638 0.894673800402 1110 72 298 [22]
371 10.238015724752 0.894497888815 1113 71 300 [22]
372 10.251261391169 0.894546177418 1116 72 300 [22]
373 10.265674630415 0.894584646477 1119 72 301 [22]
374 10.280099023022 0.894638506835 1122 72 302 [22]
375 10.294533787995 0.894724390054 1125 72 303 [22]
376 10.308942922882 0.894816235244 1128 72 304 [22]
377 10.322603841964 0.894862193080 1131 72 305 [22]
378 10.336829496116 0.894871214155 1134 72 306 [22]
379 10.350749285851 0.894676761765 1137 71 308 [22]
380 10.357587145033 0.894532286623 1140 70 310 [22]
381 10.378660854799 0.894872909030 1143 72 309 [22]
382 10.392829822423 0.894786062130 1146 72 310 [22]
383 10.408503510571 0.894626024326 1149 71 312 [22]
384 10.423034367204 0.894566825705 1152 72 312 [22]
385 10.424399434790 0.894611417346 1155 70 315 [22]
386 10.437431673991 0.894759447310 1158 71 315 [22]
387 10.458345608697 0.894528546468 1161 70 317 [22]
388 10.472818517163 0.894571439014 1164 70 318 [22]
389 10.486548146241 0.894320600899 1167 70 319 [22]
390 10.499937476376 0.894609277736 1170 71 319 [22]
391 10.512915238537 0.894592722134 1173 71 320 [22]
392 10.526288015021 0.894628564698 1176 71 321 [22]
393 10.537703970268 0.894708430547 1179 72 321 [22]
394 10.551354965462 0.894626255856 1182 71 323 [22]
395 10.564434141115 0.894789027779 1185 72 323 [22]
396 10.576994618622 0.894819945453 1188 72 324 [22]
397 10.589897201759 0.894573102675 1191 71 326 [22]
398 10.603648603343 0.894906130861 1194 72 326 [22]
399 10.616641862208 0.894951162956 1197 72 327 [22]
400 10.629555406253 0.894729029993 1200 72 328 [22]
401 10.643383237208 0.894885988555 1203 72 329 [22]
402 10.656528036072 0.894982204285 1206 73 329 [22]
403 10.670267535586 0.894894301577 1209 73 330 [22]
404 10.685799819731 0.894512002492 1212 70 334 [22]
405 10.699279861070 0.894500012124 1215 71 334 [22]
406 10.713481777078 0.894693021652 1218 71 335 [22]
407 10.727160300452 0.894623535554 1221 71 336 [22]
408 10.740743788955 0.894548974027 1224 71 337 [22]
409 10.754304303636 0.894742179238 1227 72 337 [22]
410 10.767959002931 0.894854484432 1230 72 338 [22]
411 10.781584594375 0.894645472932 1233 71 340 [22]
412 10.794987195226 0.894707137627 1236 71 341 [22]
413 10.809772222592 0.894639012071 1239 71 342 [22]
414 10.823111893800 0.894550058619 1242 71 343 [22]
415 10.836753113503 0.894439043978 1245 71 344 [22]
416 10.849897808353 0.894804418518 1248 72 344 [22]
417 10.864260725292 0.894841127886 1251 72 345 [22]
418 10.877439514354 0.894879964280 1254 73 345 [22]
419 10.891328953838 0.894858883339 1257 73 346 [22]
420 10.904688508438 0.895123135737 1260 75 345 [22]
421 10.918177618119 0.895046485065 1263 75 346 [22]
422 10.931891132307 0.895304677755 1266 75 347 [22]
423 10.945132887737 0.895224150598 1269 75 348 [22]
424 10.959562982145 0.895263563789 1272 75 349 [22]
425 10.972363540154 0.895254005920 1275 76 349 [22]
426 10.986233805390 0.895529654612 1278 77 349 [22]
427 10.998558129138 0.895572307935 1281 77 350 [22]
428 11.010629267826 0.895440281031 1284 77 351 [22]
429 11.022760186301 0.895416340016 1287 77 352 [22]
430 11.034031191924 0.895685244058 1290 78 352 [22]
431 11.046926788894 0.895506252591 1293 77 354 [22]
432 11.058264374314 0.895575417197 1296 78 354 [22]
433 11.069573219852 0.895209641462 1299 75 358 [22]
434 11.082737805283 0.895268871649 1302 75 359 [22]
435 11.095050841805 0.895351202945 1305 76 359 [22]
436 11.107812569878 0.895186011410 1308 75 361 [22]
437 11.120861607382 0.895322795365 1311 76 361 [22]
438 11.133663422819 0.895328787983 1314 76 362 [22]
439 11.145396609040 0.895367532420 1317 76 363 [22]
440 11.157976272856 0.895530241460 1320 77 363 [22]
441 11.169522435560 0.895557342077 1323 77 364 [22]
442 11.181832072224 0.895335025419 1326 76 366 [22]
443 11.195207710099 0.895365852703 1329 76 367 [22]
444 11.208657532459 0.895401729089 1332 76 368 [22]
445 11.221259916155 0.895317190593 1335 76 369 [22]
446 11.233447127648 0.895341580400 1338 76 370 [22]
447 11.246331389248 0.895382420429 1341 76 371 [22]
448 11.258946090227 0.895309879903 1344 75 373 [22]
449 11.271682773930 0.895255288431 1347 75 374 [22]
450 11.285552024486 0.895264492038 1350 75 375 [22]
451 11.298856614389 0.895376418081 1353 76 375 [22]
452 11.312054836109 0.895373903619 1356 76 376 [22]
453 11.325229372791 0.895436746250 1359 76 377 [22]
454 11.338237877359 0.895437904981 1362 76 378 [22]
455 11.350906862541 0.895378429863 1365 76 379 [22]
456 11.363864959547 0.895441887366 1368 76 380 [22]
457 11.376014719491 0.895419725147 1371 76 381 [22]
458 11.388906282292 0.895484097889 1374 76 382 [22]
459 11.400750295659 0.895467362508 1377 76 383 [22]
460 11.412700567346 0.895570934362 1380 77 383 [22]
461 11.425580426419 0.895486160135 1383 77 384 [22]
462 11.437566177571 0.895479427637 1386 78 384 [22]
463 11.450452426588 0.895506253761 1389 78 385 [22]
464 11.462580065449 0.895507709220 1392 77 387 [22]
465 11.475191505336 0.895540681728 1395 78 387 [22]
466 11.487639049093 0.895609476835 1398 79 387 [22]
467 11.500055528292 0.895777337457 1401 79 388 [22]
468 11.513306429273 0.895806157968 1404 79 389 [22]
469 11.522965080147 0.895668649128 1407 78 391 [22]
470 11.535717024814 0.895769092373 1410 79 391 [22]
471 11.548024053592 0.895772083138 1413 79 392 [22]
472 11.559922693009 0.895923413236 1416 80 392 [22]
473 11.572268460196 0.895633867813 1419 79 394 [22]
474 11.584603098149 0.895694895929 1422 79 395 [22]
475 11.596644766915 0.895877637362 1425 79 396 [22]
476 11.609006039390 0.895737058925 1428 79 397 [22]
477 11.621121796626 0.895806009569 1431 80 397 [22]
478 11.633559016748 0.895895579079 1434 80 398 [22]
479 11.646544636677 0.895908030556 1437 80 399 [22]
480 11.657967569371 0.895845358416 1440 80 400 [22]
481 11.670331156637 0.895910485186 1443 80 401 [22]
482 11.682781965040 0.895791439645 1446 80 402 [22]
483 11.694407066264 0.895902389389 1449 80 403 [22]
484 11.706739384534 0.895839348291 1452 80 404 [22]
485 11.719289534218 0.895817732597 1455 80 405 [22]
486 11.731171627011 0.895888802813 1458 80 406 [22]
487 11.742424558708 0.896033575659 1461 81 406 [22]
488 11.753818598945 0.896042207003 1464 81 407 [22]
489 11.767847428021 0.895774728945 1467 80 409 [22]
490 11.780098018438 0.895848805548 1470 80 410 [22]
491 11.792921656649 0.895883823517 1473 80 411 [22]
492 11.805589909537 0.895939047055 1476 80 412 [22]
494 11.824004090539 0.895812293341 1482 81 413 [22]
495 11.836128159654 0.896122252766 1485 82 413 [22]
496 11.847856118786 0.896023119462 1488 81 415 [22]
497 11.860375524037 0.895969434858 1491 81 416 [22]
498 11.872093599977 0.895991933316 1494 81 417 [22]
499 11.883589880191 0.896097214384 1497 82 417 [22]
500 11.901744072026 0.895890855534 1500 80 420 [22]
529 12.229389491201 0.896765372299 1588 88 441 D2 [2]
545 12.428407647582 0.896649140534 1635 87 458 [17]
576 12.765618381692 0.897174783722 1729 92 484 D1 [2]
613 13.178055048167 0.896338526932 1839 85 528 [17]
625 13.301701615661 0.897564931830 1876 96 529 D2 [2]
676 13.843506930660 0.897073766047 2028 91 585 [17]
685 13.915210345965 0.898264537385 2055 104 581 [20]
729 14.380692110222 0.897189892109 2187 94 635 [17]
761 14.691043947468 0.897509574913 2283 95 666 [17]
784 14.925744508409 0.898035293924 2352 102 682 [17]
841 15.467205852816 0.898187318407 2523 106 735 [17]
900 16.003323612102 0.898918443361 2700 114 786 [17]
925 16.213671971894 0.897985950832 2775 103 822 [17]
961 16.546634626446 0.899464575925 2883 119 842 [17]
1013 16.978328452492 0.899371591428 3039 117 896 [13]
1024 17.055056471349 0.899581660637 3073 124 900 D1 [2]
1089 17.591180823252 0.899804753318 3268 128 961 D2 [2]
1156 18.127461042421 0.900008212218 3469 132 1024 D1 [2]
1225 18.663589972222 0.900206372823 3676 136 1089 D2 [2]
1296 19.199876694862 0.900387807590 3889 140 1156 D1 [2]
1369 19.736013462864 0.900565026790 4108 144 1225 D2 [2]
1444 20.272302808822 0.900727801036 4333 148 1296 D1 [2]
1521 20.808443169665 0.900887199006 4564 152 1369 D2 [2]
1600 21.344736790743 0.901034072457 4801 156 1444 D1 [2]
1681 21.880883155746 0.901178230098 5044 160 1521 D2 [2]
1764 22.417178320883 0.901311405346 5293 164 1600 D1 [2]
1849 22.953327615747 0.901442386974 5548 168 1681 D2 [2]
1936 23.489625632919 0.901563709725 5809 172 1764 D1 [2]
2025 24.025779688154 0.901683257556 6076 176 1849 D2 [2]
2116 24.562078565552 0.901794229890 6349 180 1936 D1 [2]
2209 25.098235079600 0.901903763528 6628 184 2025 D2 [2]
2304 25.634535868132 0.902005666155 6913 188 2116 D1 [2]
2401 26.170696261929 0.902106403773 7204 192 2209 D2 [2]
2500 26.706997466009 0.902200295922 7501 196 2304 D1 [2]
2601 27.243159969444 0.902293245564 7804 200 2401 D2 [2]
2704 27.779462444871 0.902380043539 8113 204 2500 D1 [2]
2809 28.315628181871 0.902466082344 8428 208 2601 D2 [2]
2916 28.851930778431 0.902546554384 8749 212 2704 D1 [2]
3025 29.388098356776 0.902626417735 9076 216 2809 D2 [2]
3136 29.924401780096 0.902701236617 9409 220 2916 D1 [2]
3249 30.460572103527 0.902775572031 9748 224 3025 D2 [2]
3364 30.996875450684 0.902845308418 10093 228 3136 D1 [2]
3481 31.533047403339 0.902914664873 10444 232 3249 D2 [2]
3600 32.069351262839 0.902979823819 10801 236 3364 D1 [2]
3721 32.605525581976 0.903044689664 11164 240 3481 D2 [2]
3844 33.141829232378 0.903105703880 11533 244 3600 D1 [2]
3969 33.678005009071 0.903166497174 11908 248 3721 D2 [2]
4096 34.214308946266 0.903223753155 12289 252 3844 D1 [2]
4225 34.750486789519 0.903280849112 12676 256 3969 D2 [2]
4356 35.286790428296 0.903334680898 13069 260 4096 D1 [2]
4489 35.822969587360 0.903388403456 13468 264 4225 D2 [2]
4624 36.359273349608 0.903439111832 13873 268 4356 D1 [2]
4761 36.895454333047 0.903489754001 14284 272 4489 D2 [2]
4900 37.431757737679 0.903537600926 14701 276 4624 D1 [2]
5041 37.967939918159 0.903585417597 15124 280 4761 D2 [2]
5184 38.504243326714 0.903630640621 15553 284 4900 D1 [2]
5329 39.040427133252 0.903675863933 15988 288 5041 D2 [2]
5476 39.576730145858 0.903718671384 16429 292 5184 D1 [2]
5625 40.112915048219 0.903761504613 16876 296 5329 D2 [2]
5776 40.649217977288 0.903802086598 17329 300 5476 D1 [2]
5929 41.185404340592 0.903842716025 17788 304 5625 D2 [2]
6084 41.721706850114 0.903881240282 18253 308 5776 D1 [2]
6241 42.257894222053 0.903919830003 18724 312 5929 D2 [2]
6400 42.794196583905 0.903956450472 19201 316 6084 D1 [2]
6561 43.330385277980 0.903993151702 19684 320 6241 D2 [2]
6724 43.866687207154 0.904028004953 20173 324 6400 D1 [2]
6889 44.402876833941 0.904062951595 20668 328 6561 D2 [2]
7056 44.939178568456 0.904096163603 21169 332 6724 D1 [2]
7225 45.475369399047 0.904129479683 21676 336 6889 D2 [2]
7396 46.011670695164 0.904161162767 22189 340 7056 D1 [2]
7569 46.547862392224 0.904192958646 22708 344 7225 D2 [2]
7744 47.084163459560 0.904223216891 23233 348 7396 D1 [2]
7921 47.620356258529 0.904253595229 23764 352 7569 D2 [2]
8100 48.156656887070 0.904282521853 24301 356 7744 D1 [2]
8281 48.692850493910 0.904311574432 24844 360 7921 D2 [2]
8464 49.229150869190 0.904339256185 25393 364 8100 D1 [2]
8649 49.765345489899 0.904367068715 25948 368 8281 D2 [2]
8836 50.301645429788 0.904393583610 26509 372 8464 D1 [2]
9025 50.837840805753 0.904420233020 27076 376 8649 D2 [2]
9216 51.374140475396 0.904445653936 27649 380 8836 D1 [2]
9409 51.910336788398 0.904471212365 28228 384 9025 D2 [2]
9604 52.446636028618 0.904495605082 28813 388 9216 D1 [2]
9801 52.982833049976 0.904520137491 29404 392 9409 D2 [2]
10000 53.519132008453 0.904543563685 30001 396 9604 D1 [2]


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]–[15].
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.
24-Aug-2026: Some packings were rotated and/or mirrored to emphazise similarities between adjacent packings.
29-Aug-2026: For the following values of N=65, 95, 104, 110, 120, 125, 130, 135, 140, 145, 150, 155, 160, 170 considerable improvements could be achieved [2].
02-Sep-2026: Wes Sander, MoltFire, [14] has found ten new candidates for N=101–103, 105–109, 111, 114. The packings come from an LLM-evolved optimiser (penalty L-BFGS-B, basin hopping on the incumbent, contact-graph SLSQP polish).
03-Sep-2026: The next bundle of impressive improvements arrived today from Yue Huang, University of Notre Dame [15], for N=111, 113, 130, 140, 160, 181.
03-Sep-2026: For N=101–103, 105–109, 111, 114, Wes Sander [14] found new candidates.
04-Sep-2026: Henrik Tallbacka sent me an improvement for N=104 [16]. The construction was obtained by starting from the published csqv103 configuration and identifying an unused void where an additional circle could be inserted.
09-Sep-2026: Wilfred Heap [17] discovered several new candidates for N=68, 75, 84, 87, 98, 104, 110, 112, 115–120, 125, 135, 145, 150, 155, 170.
09-Sep-2026: David Cantrell [1] discovered in 2011/2012 configurations for which N is a square number. These nice and highly symmetric packings survived all attempts for any improvement. To continue his work, it was not hard to generalize the pattern: Take an odd square number N=(2p-1)2, i.e. N=81, and rotate the packing by 90 degrees clockwise, scale it down a little bit such that there is place for additional 4p+4 circles along the boundaries and let the solver do the rest. Similarily, generate the configuration for the next even square number N=(2p)2 by adding two columns of circles along two boundaries. In this manner, here are the new candidates for N=n2 for n=11–100 [2].
10-Sep-2026: Thanks to Anant Garg [18], who sent me candidates for N=48(!), 83, 98, 116, 122 124, 145.
10-Sep-2026: Thanks to Eddie Zhuang [19], who sent me a candidate for N=120.
12-Sep-2026: Many improvements arrived from Zeeshan Tariq [20] for N=102, 104, 110, 112, 114–120, 125, 135, 140, 144, 145, 150, 155, 169, 170, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 925.
12-Sep-2026: There are new candidates for N=53, 65, 68, 79 by Anant Garg [18].
13-Sep-2026: More new candidates for N=63, 77–79, 82, 121, 123, 221, 481 by Wilfred Heap [17].
13-Sep-2026: Some improvements for N=84, 121, 123, 124, 140, 170, 221, 265, 313, 365, 421 by Eckard Specht [2].
14-Sep-2026: One improvement for N=121 by Arnold Castro [21].
14-Sep-2026: Two more old candidates for N=120, 121, and three new candidates for N=141–143 by Arnold Castro [21].
16-Sep-2026: One improvement for N=119 by Arnold Castro [21].
19-Sep-2026: Old favorites in new guises for N=112, 117–120, 135, 140, 144, 150, 155, 169, 170, 265, 313, 365, 421, 545, 613, 685, 761, 925 by Zeeshan Tariq [20].
20-Sep-2026: What a incredible progress: 55 better candidates and 151 new ones by Jean-René Denoual [22] for N=54, 65–69, 73, 76, 77, 79, 82–89, 91–99, 101–104, 107, 111–115, 117, 119–129, 131–134, 136–143, 145–149, 151–154, 156–159, 161–168, 171–286, 288–291, 293–296, 298, 300. There are several new packings which show D1 symmetry, namely for N=69, 198, 227, 290. The candidate for N=69 is the very first which doesn't has a diagonal as a symmetry axes, but the midline between two opposite sides of the square. Are there more?
20-Sep-2026: Three improvements for N=97, 98, 118 by Arnold Castro [21].
20-Sep-2026: Some improvements for N=63, 89, 103, 116, 118–120, 123, 265 by Wilfred Heap [17].
20-Sep-2026: Further improvements for N=60–62, 65, 66, 68, 70, 77, 79, 82–84, 86, 87, 96–101, 107, 112, 121, 122, 124, 125, 141, 142, 421 by Wilfred Heap [17]. The second case with non-diagonal D1 symmetry was discovered for N=70.
20-Sep-2026: Yet another improvement for such a small value of N=46 by Wilfred Heap [17].
20-Sep-2026: Another update today for N=138, 179, 545, 613, 761, 925 by Wilfred Heap [17].
21-Sep-2026: Mostly all of the candidates from yesterday were blown away, namely for N=67, 68, 108, 127, 130–133, 136–139, 141, 142, 144, 146–149, 152–159, 162–170, 173–185, 189, 190, 196–224, 226–286, 288, 290, 291, 293–296, 298, 300 by Wilfred Heap [17].
22-Sep-2026: Here comes the next tranche: N=301–312, 314–360, 362–364, 366–400 by Jean-René Denoual [22]. He wrote that the computation took about 21 hours and the cost rises steeply with N.
22-Sep-2026: One improvement for N=84 by Arnold Castro [21].
23-Sep-2026: Several new candidates for N=120, 140, 147, 155, 164, 167, 183, 193, 197, 199, 200, 202–204, 206, 208, 210, 211, 215–217, 229, 234, 238, 242, 244, 246, 248, 250, 253–259, 263, 267, 269, 271, 275, 279, 282, 284, 286, 288, 290, 291, 298 by Eckard Specht [2].
23-Sep-2026: Many thanks to Jean-René Denoual [22], who closes the gap until 400 with N=287, 292, 297, 299.
24-Sep-2026: 23 improvements in the range 100–200 for N=117, 132, 134, 137, 141, 143, 145, 147, 148, 167–170, 174–179, 186, 187, 194, 195 by Eckard Specht [2].
24-Sep-2026: 33 improvements in the range 200–300 for N=203, 227–234, 236, 245, 246, 249–264, 270, 274, 294, 298, 300 by Jean-René Denoual [22].
24-Sep-2026: 33 only very small improvements in the range 300–400 for N=302, 306, 310, 312–314, 316, 328–330, 336, 337, 341, 343, 348, 350–352, 357, 358, 364, 366, 368, 370, 375, 378, 384, 387, 388, 391, 395, 397, 400 by Eckard Specht [2].
25-Sep-2026: Further improvements for N=99, 100, 118–120, 129, 133, 154, 156, 157, 164, 166, 178, 183, 185, 189, 196–198, 200–202, 205, 206, 209, 210, 214, 216–219, 227, 245, 247, 265, 272, 274, 279, 287, 290, 291, 293, 294, 296–298, 301, 307–309, 319–322, 355–359, 363, 364, 394, 396, 398, 399, 421 by Wilfred Heap [17].
28-Sep-2026: New candidates for N=79, 85, 89, 92, 94, 98–100, 121, 122, 131, 132, 135, 136, 139, 151, 153, 154, 157, 197–199, 202, 211, 212, 216, 238, 248, 255, 257, 275, 276, 278–280, 284–301, 309, 310, 314–316, 319–322, 325–334, 338, 340, 342, 344, 348–357, 359, 362, 364, 377, 379, 381–384, 386–391, 394–397, 421, 481 by Wilfred Heap [17].
29-Sep-2026: More improvements for N=178, 196, 200, 203, 204, 217, 219 by Wilfred Heap [17].
29-Sep-2026: Three new candidates for N=97, 104, 105 by Arnold Castro [21].
29-Sep-2026: The next bundle of candidates for N=80, 93, 191, 195, 253–256, 271, 282, 288, 289, 292, 304, 312, 314, 315, 317, 318, 323–329, 340, 346, 360–362, 366–368, 376–379, 395, 397, 398, 400, 441, 676, 729, 784, 841, 900, 961 by Wilfred Heap [17].
29-Sep-2026: Extension up to 500 and improvements for N=95, 96, 98, 101, 103, 118, 127–134, 138–140, 142, 144, 148–156, 162, 165–167, 170–181, 185, 186–190, 192–219, 221, 223, 224, 226–240, 242–252, 254, 257–263, 265–286, 295–333, 335–339, 341, 343, 345, 347, 350, 352–364, 366–382, 385–465, 469–488, 494–500 by Jean-René Denoual [22].
30-Sep-2026: Two candidates for N=84, 119 by Wilfred Heap [17].
30-Sep-2026: Improvements for N=97, 104, 158, 256, 290–294, 344, 346, 348, and a new candidate for N=466 by Jean-René Denoual [22].
30-Sep-2026: Improvements for N=100, 119, 147, 157, 334, 340, 342, 349, 351, 383, 384, and closure of gaps for N=467, 468, 489–492, such that the table is complete until 500 by Jean-René Denoual [22].
30-Sep-2026: Improvements for N=120, 250 by Elian Alfonso López Preciado [23]. — This was the last update until November 2026, happy searching ...
04-Oct-2026: Since the average (mean) radii rm follow a inverse square root law rm ∼ N-1/2, graphs with normed mean radius R(N) = sum_of_radii / N1/2 versus N can be plotted, which show a nearly constant function. But there are slight periodic oszillations, see Mean radii plots above. The packings at the maxima for N=58, 72, 90, 109, 128, 150, 174, 199, 226, 255, 288, 321, 354, 384, 426, 492 seems to be almost «regular».
04-Oct-2026: A similar behaviour show the standard deviations of radii versus N, see Standard deviation of radii plots above. The packings at the maxima show oppotunities for improvements.

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.
[14]   , discovery-loop and private communication, September 2026.
[15]   , University of Notre Dame, private communication, September 2026.
[16]   Henrik Tallbacka, private communication, September 2026.
[17]   , private communication, September 2026.
[18]   , private communication, September 2026.
[19]   Eddie Zhuang, private communication, September 2026.
[20]   , private communication, September 2026.
[21]   Arnold Castro, csqv-circle-packing and private communication, September 2026.
[22]   , private communication, September 2026.

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©  E. Specht    30-Sep-2026