The best known packings of unequal circles with integer radii in a circle (complete up to N = 200)

(a.k.a. Al Zimmermann's Programming Contests — Circle Packing)

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   101-112   113-124   125-136   137-148   149-160   161-172   173-184   185-196   197-199  

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 0.555469288332912726151680670796 9.0013977460502193186724442369 0.678801486618359574874925328840 6 2 3 [1]
6 0.542640687119285146405066172629 11.0570403997020210854826907276 0.744326702607593955856345537543 6 3 3 [2]
7 0.519977897050211400703405101999 13.4621106776083764315332377163 0.772505752630743560253951417467 10 2 5 [3]
8 0.493165141800723187457958263481 16.2217466765577238862694843261 0.775237794465856115965982966557 12 2 6 [3]
9 0.467941000491664114240991109597 19.2331939080861235778775740411 0.770445707200305981724184447115 12 3 6 [3]
10 0.454541466713966414248671660658 22.0001930127373705480969665696 0.795440588105562775239159646548 10 5 5 [4]
11 0.440693929195813169647285638219 24.9606342889111573930522301689 0.812155673143821999571809733927 14 4 7 [3]
12 0.422961308500480205032821823426 28.3713894364084035738803015551 0.807517878593645503862777325949 12 6 5 [4]
13 0.412098358011784183482521850271 31.5458670175731632372815314804 0.822998351583734223662462586187 16 5 8 [4]
14 0.398909868871450694437712843067 35.0956471435694689934204714083 0.824061325180018150758827180416 22 3 8 [5]
15 0.386219726423057712445676696713 38.8379955082079014895321489365 0.822068620342639618860726645080 14 8 6 [5]
16 0.376841964344851976863372323169 42.4581164356691300262522275884 0.829870154970956586622391362763 18 7 8 [5]
17 0.367239298376272037806176780909 46.2913421171550719193324546440 0.832987866973478766427625311970 16 9 8 [5]
18 0.359139769580141767404896552932 50.1197626234576999176979901660 0.839573203593855160813591562608 20 8 8 [6]
19 0.350293089176941293991678913509 54.2402935914121110891998555840 0.839562225382431103658456789154 20 9 9 [5]
20 0.342462425680645332532671642332 58.4005674790451137175957605817 0.841487680797038211616426398803 28 6 9 [7]
21 0.335683774638203635767197094004 62.5588770938767424283505524151 0.846021288423517960375027297964 26 8 8 [6]
22 0.329537232965746151587437705236 66.7602862414238840346547484610 0.851481859754512636177852613321 26 9 8 [5]
23 0.323036150563399487834699570817 71.1994616078920572697738619542 0.852967072144088217800948114383 30 8 12 [6]
24 0.316835269443004585524303028344 75.7491425818594156785683988320 0.853966112879868827578707216532 26 11 8 [11]
25 0.311387317984121974663661276624 80.2858644399730530763868433683 0.857144626323887082150972473282 34 8 12 [7]
26 0.305960855060902876640212799372 84.9781910657315417438460174133 0.858710488148435305960837487471 34 9 9 [27]
27 0.300832427788693775910674291185 89.7509626820049367574045052549 0.860310064186599190989539212196 32 11 10 [11]
28 0.296215183170731171241142692466 94.5258771015174002402895866014 0.863332723968201257707616656901 28 14 8 [11]
29 0.291506764758227393371207455091 99.4831115636451565653560844725 0.864413006911845332486123253826 38 10 11 [11]
30 0.286970495600954771173459073685 104.5403637651876706526971386013 0.865154197601446248693893845556 46 7 17 [28]
31 0.282771273549458347189500637659 109.6292406611024402846984210880 0.866658816020869491478987777092 38 12 12 [27]
32 0.278746094616351172986469308201 114.7998146630280658706319765668 0.868047819744773082420385262311 40 12 14 [28]
33 0.274849612282160404130602734329 120.0656596383415122093683784325 0.869118084589028194674566946998 42 12 12 [11]
34 0.271203877317730414753088529664 125.3669392055450240112052104274 0.870720473133783950653740933380 34 17 8 [11]
35 0.267483732648500407269287295755 130.8490787587198728562644440850 0.870835860590020359724510341041 46 12 12 [29]
36 0.264107928757250450076964537585 136.3079108203854190329810808949 0.872235405952007494910898227277 44 14 17 [27]
37 0.260960824665625021137105202555 141.7837334297549560117945821974 0.874263843373715273023256453093 40 17 15 [27]
38 0.257710780360268649583517595600 147.4521164651227438870763146764 0.874753436318585896667640957548 48 14 12 [11]
39 0.254569617067667141722997522171 153.1997433520647173486889022157 0.875153761501592507856961370018 42 18 12 [30]
40 0.251644994145 158.9540858378 0.876262497596 48 16 11 [33]
41 0.248777002524725865680996353728 164.8062304148271232722632734814 0.877026542643754002760280829193 60 11 15 [30]
42 0.246048849400964322773103897565 170.6978110332727782855530366578 0.878070764466559246567182029286 52 16 14 [30]
43 0.243305345567353410740085259165 176.7326562420160752816061674124 0.878325566834702973893786280211 38 24 10 [30]
44 0.240736277435061192530713372670 182.7726193525985544027347875604 0.879187844202763123397055915046 60 14 15 [30]
45 0.238139377033909560837998028693 188.9649690046526026113116134280 0.879220663239616505802911665142 56 17 16 [30]
46 0.235651027881733206951683866339 195.2039013514769789307222793032 0.879448477326981018315769207397 58 17 16 [30]
47 0.233266603646794826811708073584 201.4861933308120086082237696547 0.879874773754303487919544684269 58 18 16 [30]
48 0.230990400293647676448562117446 207.8008434072574668445413186071 0.880568588809814251644358466490 66 15 15 [30]
49 0.228778537726880973280327441828 214.1809301119709366011234455489 0.881228284544309554433452513741 50 24 11 [30]
50 0.226690133359610792105167039311 220.5654002624026944333044663742 0.882339112379809818448104632137 62 19 16 [11,31]
51 0.224065624360 227.6118889085 0.878758640293 58 22 27 [32]
52 0.221804895915532620814052495305 234.4402714167434577378062928804 0.877511528708134436196831868011 54 25 13 [32]
53 0.219796668907 241.1319528346 0.877794044584 74 16 27 [32]
54 0.218152316295 247.5334707292 0.880569896489 68 20 27 [32]
55 0.216119453948 254.4888902657 0.879801684294 70 20 28 [32]
56 0.214181755968 261.4601778140 0.879384833875 70 21 30 [32]
57 0.212402732101 268.3581300306 0.879872866750 60 27 29 [32]
58 0.210658921418 275.3265781940 0.880274947231 70 23 32 [32]
59 0.209017372107 282.2731881341 0.881170026992 74 22 28 [32]
60 0.207583470150 289.0403554614 0.883483087065 70 25 30 [32]
61 0.205797203570 296.4083036203 0.882459237959 72 25 32 [32]
62 0.204121443337 303.7407485778 0.882033101039 72 26 31 [32]
63 0.202622167430488774360968197387 310.9235322024312853301599978185 0.882807081861940534587063201642 73 26 14 [32]
64 0.201035272861 318.3520935864 0.882503364932 77 25 32 [32]
65 0.199651070037238199332104498903 325.5680021543407397697474529152 0.883677726373345410127411671569 76 27 15 [32]
66 0.198262542330 332.8919281689 0.884530065770 72 30 34 [32]
67 0.196788053142 340.4678227674 0.884329446643 70 32 34 [32]
68 0.195295102320946233231796918719 348.1910155035501357785275192200 0.883674247773672456787189383571 80 28 17 [32]
69 0.194034910001329766207838193104 355.6061123203403295238763915214 0.884855278964857253510405525763 80 29 17 [32]
70 0.192552171910498202942558970150 363.5378365533954593350583850241 0.883741020958291624377750846008 76 32 16 [32]
71 0.191191206408750053729835507028 371.3559913849187110024199816846 0.883476011720916721570337014134 82 30 17 [32]
72 0.190036942730 378.8737019527 0.884877567570 86 29 34 [32]
73 0.188784223792 386.6848539225 0.885128533646 88 29 35 [32]
74 0.187500297289706179390529919916 394.6660409058595206063958513400 0.884847611668159048595691435765 84 32 19 [32]
75 0.186508179241 402.1271362208 0.887102474227 99 25 31 [32]
76 0.185353205641 410.0279773266 0.887600455094 94 29 35 [32]
77 0.183978204338 418.5278374522 0.885762065430 84 35 38 [32]
78 0.182951705242 426.3420223209 0.887061670880 94 31 39 [32]
79 0.181753362232 434.6549578491 0.886489664904 94 32 39 [32]
80 0.180779365074 442.5283824146 0.887906777513 94 33 36 [32]
81 0.179639277530 450.9036170349 0.887498825603 90 36 39 [32]
82 0.178550944800 459.2526804696 0.887403707890 104 30 38 [32]
83 0.177471742758796632858723468704 467.6800864732928869824737129588 0.887206760713257905309547212587 92 37 21 [32]
84 0.176566354100 475.7418276451 0.888568662404 102 33 36 [32]
85 0.175474424410 484.4010760306 0.887875431147 106 32 35 [32]
86 0.174540925344543629086093274608 492.7211187303841642025994357569 0.888607966216220728038409117065 106 33 21 [32]
87 0.173551273236 501.2927786585 0.888599012265 100 37 40 [32]
88 0.172577066504650922452977829834 509.9171157694259163752229012521 0.888577916087923660972934217326 98 39 18 [32]
89 0.171695030498 518.3609551296 0.889343906646 112 33 39 [32]
90 0.170841904562 526.8028369906 0.890256218725 108 36 40 [32]
91 0.169908094410 535.5836654873 0.890172990023 110 36 41 [32]
92 0.168979051132 544.4461865769 0.889982226098 116 34 38 [32]
93 0.168122248183 553.1689053964 0.890400999805 125 30 36 [32]
94 0.167214958271 562.1506650614 0.890136388205 108 40 43 [32]
95 0.166390481042 570.9461226683 0.890608220070 112 39 45 [32]
96 0.165545312474 579.9016508838 0.890718219233 106 43 44 [32]
97 0.164663991211 589.0783970855 0.890297139736 110 42 44 [32]
98 0.163864419652 598.0553936483 0.890622008824 108 44 42 [32]
99 0.163051911247 607.1685958343 0.890673270455 118 40 40 [32]
100 0.162386105048 615.8162360652 0.892203476058 116 42 41 [32]
101 0.161514044660 625.3326155785 0.891341590438 108 47 43 [32]
102 0.160629554949310542914898684997 635.0014480970695501976183464657 0.890206120242899736820026473348 112 46 22 [32]
103 0.159933664428 644.0170077268 0.891035455717 114 46 44 [32]
104 0.159227313932 653.1542700292 0.891632999169 118 45 42 [32]
105 0.158434213124 662.7356423180 0.891139539526 114 48 44 [32]
106 0.157674108839 672.2727071692 0.890896054592 130 41 43 [32]
107 0.156951217634 681.7404898988 0.890956634149 120 47 48 [32]
108 0.156291213839 691.0177312435 0.891621134495 121 47 45 [32]
109 0.155653446430991443774946045125 700.2736045958665644815970033303 0.892434875909814449074203430384 126 46 19 [32]
110 0.154912565744 710.0779686362 0.891958422907 134 43 44 [32]
111 0.154257165012 719.5775962276 0.892358464542 141 40 48 [32]
112 0.153651515581 728.9221949856 0.893234287215 118 53 49 [32]
113 0.152895257580 739.0680508263 0.892255111290 126 50 43 [32]
114 0.152220288685 748.9146222523 0.892118002823 127 50 44 [32]
115 0.151613984741 758.5052275793 0.892687395199 128 51 45 [32]
116 0.151033987613 768.0390475905 0.893474003532 145 43 43 [32]
117 0.150373761213 778.0612724983 0.893216799635 137 48 44 [32]
118 0.149778505335 787.8300009459 0.893636778852 130 53 48 [32]
119 0.149157137657 797.8163289347 0.893653203304 136 51 47 [32]
120 0.148472069232 808.2328253440 0.892810807956 142 49 47 [32]
121 0.147885988524 818.1978645031 0.893065969732 140 51 43 [32]
122 0.147313773519162609015717031409 828.1642448330210777980530337141 0.893401800568163231357071515980 126 59 19 [32]
123 0.146795224158 837.9019188477 0.894305969149 130 58 50 [32]
124 0.146180614363539976748841836691 848.2656919994980323017541472777 0.893955684124629628733954343108 149 48 23 [32]
125 0.145613812883 858.4350448974 0.894104233360 136 57 46 [32]
126 0.145036770954044133419738233510 868.7452097228773725958507425194 0.894043584426583542196896104288 142 55 23 [32]
127 0.144508299652 878.8422554683 0.894500856343 142 56 47 [32]
128 0.143930640646 889.3172393688 0.894268906172 146 54 50 [32]
129 0.143446996052 899.2868693695 0.895127854744 149 54 49 [32]
130 0.142866618581 909.9396436395 0.894702667014 150 55 52 [32]
131 0.142355293444 920.2327277791 0.895064588559 169 46 49 [32]
132 0.141795242773 930.9198067499 0.894737530073 162 51 50 [32]
133 0.141262651337 941.5085922656 0.894680300151 160 53 50 [32]
134 0.140864517471 951.2686544881 0.896258368204 156 56 52 [32]
135 0.140256069858 962.5251879420 0.895089599635 152 57 51 [32]
136 0.139713153416 973.4230219180 0.894679827998 146 63 49 [32]
137 0.139297170481 983.5088503751 0.895827834142 142 66 55 [32]
138 0.138765704801 994.4820314055 0.895423574489 153 60 52 [32]
139 0.138295732956 1005.0924712504 0.895743672642 154 62 52 [32]
140 0.137768626600 1016.1965278653 0.895255101556 152 64 49 [32]
141 0.137298606106 1026.9587142894 0.895440376228 166 57 52 [32]
142 0.136798788340 1038.0208898240 0.895170587521 156 64 53 [32]
143 0.136446611613 1048.0289565940 0.896773180626 167 58 50 [32]
144 0.135906804998 1059.5495935769 0.895848370850 156 66 52 [32]
145 0.135479617914 1070.2716927686 0.896343592873 180 54 52 [32]
146 0.134962079951 1081.7853433574 0.895579975156 166 63 54 [32]
147 0.134589461530 1092.2103285717 0.896679533951 164 64 51 [32]
148 0.134106175727 1103.6031651577 0.896246160128 160 68 55 [32]
149 0.133687668113 1114.5381029024 0.896618353865 172 63 56 [32]
150 0.133230745013 1125.8662554613 0.896416509245 170 65 54 [32]
151 0.132872130339 1136.4309401436 0.897482140316 190 56 54 [32]
152 0.132368534611 1148.3091540341 0.896532326962 185 59 54 [32]
153 0.131991698557 1159.1638085736 0.897242115142 179 62 55 [32]
154 0.131622973236294295159957515929 1170.0085191323984274421779364630 0.898010883406075423905042082167 162 73 24 [32]
155 0.131106561823 1182.2444112981 0.896707693241 187 60 53 [32]
156 0.130743269848 1193.1780517728 0.897442899668 174 69 59 [32]
157 0.130276549218 1205.1286355273 0.896704255383 186 63 56 [32]
158 0.129885832441 1216.4529189254 0.896956979137 175 69 55 [32]
159 0.129583620644 1227.0069257991 0.898385040132 178 70 57 [32]
160 0.129146306977 1238.9049578367 0.897891083661 178 71 56 [32]
161 0.128702024775 1250.9515703513 0.897245252868 185 68 58 [32]
162 0.128293129725 1262.7332449306 0.897039362097 173 75 59 [32]
163 0.127958072062 1273.8547664348 0.897817614869 188 69 55 [32]
164 0.127532083217 1285.9509220235 0.897271041329 186 71 59 [32]
165 0.127166905849772709644114868383 1297.5073891859963888258092197161 0.897530252628614894282309652667 194 68 27 [32]
166 0.126801425525 1309.1335472947 0.897738061109 208 61 54 [32]
167 0.126418558701 1321.0085743465 0.897652069517 192 70 58 [32]
168 0.126060506015 1332.6933653542 0.897871456624 189 73 59 [32]
169 0.125722844364 1344.2266666418 0.898336530648 220 59 58 [32]
170 0.125354172018 1356.1574957013 0.898313451514 199 70 60 [32]
171 0.124979724717 1368.2219286934 0.898161290551 180 80 62 [32]
172 0.124618980929 1380.2070817595 0.898160376928 212 65 57 [32]
173 0.124303789542 1391.7516162426 0.898773185935 207 69 57 [32]
174 0.123976924340 1403.4869870042 0.899175971964 199 74 58 [32]
175 0.123591123488 1415.9592943389 0.898679935585 196 77 61 [32]
176 0.123211542507 1428.4375994214 0.898228524941 202 75 65 [32]
177 0.122875311414 1440.4846503585 0.898365575124 199 76 59 [32]
178 0.122529147065 1452.7155722783 0.898315338339 192 82 60 [32]
179 0.122199906679333233044598532686 1464.8129025967001568196057468391 0.898471738338293727512431856593 186 86 24 [32]
180 0.121910880695 1476.4883903162 0.899180662686 214 73 55 [32]
181 0.121562065992 1488.9513313492 0.898968206988 192 84 61 [32]
182 0.121247207133 1501.0655033092 0.899217609380 217 72 58 [32]
183 0.120873406180 1513.9806660837 0.898551696131 212 76 59 [32]
184 0.120593106241 1525.7920268914 0.899236620765 210 79 61 [32]
185 0.120340363142 1537.3063132783 0.900298466570 226 70 59 [32]
186 0.119942882114 1550.7381240265 0.899156326561 208 82 60 [32]
187 0.119647753579 1562.9211113927 0.899508679172 216 78 59 [32]
188 0.119309762656 1575.7302320871 0.899178715727 214 81 60 [32]
189 0.118991641440 1588.3468596000 0.899109667420 203 87 57 [32]
190 0.118702879836 1600.6351342383 0.899447879963 224 78 68 [32]
191 0.118397846690 1613.2050146181 0.899503780289 206 88 62 [32]
192 0.118072914613 1626.1138350830 0.899220350896 228 78 62 [32]
193 0.117760761606 1638.9160308456 0.899094512616 211 87 67 [32]
194 0.117502311504 1651.0313500772 0.899754552282 228 79 59 [32]
195 0.117228452305 1663.4187022457 0.900146152694 218 85 64 [32]
196 0.116890578290 1676.7818490363 0.899519265331 229 78 62 [32]
197 0.116645310183 1688.8805875830 0.900283672753 232 81 65 [32]
198 0.116358654974 1701.6353449941 0.900377279810 233 80 67 [32]
199 0.116068459334 1714.5053974277 0.900382414348 230 84 66 [32]
200 0.115859388291 1726.2304155928 0.901616075438 224 87 63 [32]
300 0.094723759206 3167.1040350898 0.901750335862 357 121 79 [32]
400 0.082212380220 4865.4472590345 0.904565648787 471 161 86 [32]
600 0.067286559803 8917.0853994902 0.907761224219 698 248 115 [32]
800 0.058324477649 13716.3680199289 0.908833499176 925 329 138 [32]
1000 0.052189175983 19161.0612959174 0.909265672270 1189 396 152 [32]
2000 0.036948664246 54129.1557029731 0.910818575381 2416 757 224 [32]



Updates

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

23-Mar-2011: First complete presentation from N=1 to N=100.
31-Mar-2011: André Müller [26] sent me three excellent packings for N=60, 100 and 200. The last is the densest circle packing I have ever seen!
19-Jan-2012: What a surprise! Tao Ye [27] sent me twelve improved packings for N=26, 31, 32, 35, 36, 37, 39, 40, 43, 44, 47 and 48. After the long time since Al Zimmermann's contest has been finished it is unbelievable that such improvements are possible. Great result!
20-Jan-2012: Some of the new record packings, namely for N=35, 36, 37, 39 and 40, could be improved by applying simple exchange heuristics. Nevertheless, the credits should be given to Tao Ye and Wenqi Huang. For comparison, see here.
08-Mar-2012: Better packings were found for N=58, 63, 64, 67, 69, 71, 75, 78, 79, 80, 81, 82, 83, 84, 86, 88, 94 and 99 by Eckard Specht [31].
08-Mar-2012: Zhanghua Fu and Wenqi Huang [28] sent me two nice improvements for N=30 and 100.
08-Mar-2012: Unbelievable how some packings of this old famous contest can be improved nowadays, today for N=35, 36, 37, 39, 40, 42 and 46 by Tao Ye and Wenqi Huang [27].
09-Mar-2012: Some minor improvements for N=52, 54, 61, 63, 70, 78, 79, 84 and 85 by Eckard Specht [31].
09-Mar-2012: Further astonishing improvements for N=32, 36, 37, 41 and 60 by Zhanghua Fu and Wenqi Huang [28].
09-Mar-2012: A tiny improvement for N=60. The credits should be given to Zhanghua Fu and Wenqi Huang [28].
11-Mar-2012: Three new records for N=40, 43 and 45 by Tao Ye and Wenqi Huang [27].
13-Mar-2012: Nearly all of the packings in the range 51 ≤ N ≤ 100, namely for N=51, 52, 53, 55, 56, 57, 58, 59, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98 and 99, could be improved by Zhanghua Fu and Wenqi Huang [28]. That's great!
14-Mar-2012: But all these packings are still far from the putative optima, so improvements were found for N= 51, 52, 53, 55, 56, 57, 60, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 77, 79, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 97, 98 and 99 by Eckard Specht [31].
14-Mar-2012: Again, three new records for N=41, 46 and 48 by Tao Ye and Wenqi Huang [27]. Tao Ye was the first who sent me the coordinates in the new format right off, thanks!
22-Mar-2012: Yet another record for N=36 by Tao Ye and Wenqi Huang [27], but ...
23-Mar-2012: ... this was not the last word: a better packing by Tao Ye and Wenqi Huang [27].
18-Apr-2012: An essential improvement for N=35 by Tao Ye and Wenqi Huang [27].
24-Sep-2012: Four new record packings for N=41, 42, 45 and 46 by Tao Ye and Wenqi Huang [27].
27-Sep-2012: More new record packings for N=35, 37, 39 and 44 by Zhizhong Zeng, Wenqi Huang and Zhanghua Fu [29].
02-Oct-2012: Motivated by a question Tao Ye and Wenqi Huang [27] came up with 48 new record packings for N=51–99 (except N=58). Great!
17-Dec-2012: Two new record packings for N=41 and 43 went to Zhizhong Zeng, Wenqi Huang and Zhanghua Fu [29].
05-Feb-2013: Five new record packings for N=37, 39, 43, 44 and 49 by Tao Ye and Wenqi Huang [27].
06-Mar-2013: Four new record packings for N=45, 46, 47 and 48 by Zhizhong Zeng, Wenqi Huang and Zhanghua Fu [29].
21-Mar-2013: A new record packing once more for N=48 by Zhizhong Zeng, Wenqi Huang and Zhanghua Fu [29].
25-Mar-2013: Four new record packings for N=45, 46, 47 and 48 by Tao Ye and Wenqi Huang [27].
17-Jul-2014: After 16 months, all packings for N=39–49 are completely better than before. They are found by Zhizhong Zeng, Kun He, Xinguo Yu, Wenqi Huang, Zhanghua Fu (their work is partly inspired by Tao Ye et el.) [30].
16-Jun-2015: Some small improvements by rearrangements of circles for N=55, 56, 59, 68, 69, 72, 74, 75, 76, 78, 79, 81, 86, 87, 88, 90, 91, 93, 95, 96, 97, 98, 99 and 100 by Eckard Specht [31].
21-Oct-2015: What a sensation! André Müller [33] found lots of new records, namely for N=52, 53, 55–59, 61–99, 101–199, which are not only a little bit better than the previous, but significantly better. It seems that all other people will have no chance to beat these results for years (except for small improvements by rearrangements). So, congratulation André, for these milestones!
01-Jun-2024: Significant improvement for N= 40 by Jianrong Zhou, Jiyao He, and Kun He [33].
05-Jul-2026: André Müller has published his records on github in December 2025 [32]. I could improve most of them only by a small amount.

References

[1]   Klaus Nagel, Hugo Pfoertner, Al Zimmermann's Programming Contests — Circle Packing.
[2]   Fred Mellender, Al Zimmermann's Programming Contests — Circle Packing.
[3]   Gerrit de Blaauw, Al Zimmermann's Programming Contests — Circle Packing.
[4]   Steve Trevorrow, Al Zimmermann's Programming Contests — Circle Packing.
[5]   Tomas Rokicki, Al Zimmermann's Programming Contests — Circle Packing.
[6]   Boris von Loesch, Al Zimmermann's Programming Contests — Circle Packing.
[7]   Bernardetta Addis, Marco Locatelli, Fabio Schoen, Al Zimmermann's Programming Contests — Circle Packing.
[8]   Tobias Baumann, Al Zimmermann's Programming Contests — Circle Packing.
[9]   François Glineur, Al Zimmermann's Programming Contests — Circle Packing.
[10]   Paul Khuong, Joonas Pihlaja, Barkley Vowk, Al Zimmermann's Programming Contests — Circle Packing.
[11]   André Müller, Johannes Josef Schneider, Elmar Schömer, Packing a multidisperse system of hard discs in a circular environment, Phys. Rev. E 79, 021102 (2009).
[12]   Wenqi Huang, Ruchu Xu, Two personification strategies for solving circles packing problem, Science in China (Series E) 42 (1999), 595–602.
[13]   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.
[14]   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.
[15]   De-fu Zhang, Xin Li, A Personified Annealing Algorithm for Circles Packing Problem, Acta Automatica Sinica 31 (2005), 590–595.
[16]   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.
[17]   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.
[18]   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.
[19]   Zhipeng Lü, Wenqi Huang, PERM for solving circle packing problem, Computers & Operations Research 35 (2008), 1742–1755.
[20]   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.
[21]   Mhand Hifi, Rym M'Hallah, Adaptive and restarting techniques-based algorithms for circular packing problems, Comput. Optim. Appl. 39 (2008), 17–35.
[22]   Bernardetta Addis, Marco Locatelli, Fabio Schoen, Efficiently packing unequal disks in a circle, Operations Research Letters 36 (2008), 37–42.
[23]   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.
[24]   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.
[25]   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.
[26]   , private communication, March 2011.
[27]   , private communication, January 2012–February 2013.
[28]   , private communication, March 2012.
[29]   , private communication, September 2012–March 2013.
[30]   Zhizhong Zeng, Kun He, Xinguo Yu, Wenqi Huang, Zhanghua Fu, private communication, July 2014.
[31]   , program ccin, 2005–2024.
[32]   , github.com/muellan/packing, December 2025.
[33]   , Solution-Hashing Search Based on Layout-Graph Transformation for Unequal Circle Packing, submitted to European Journal of Operational Research, under review, May 2024.
[34]   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